Monoid generalizations of the Richard Thompson groups

TL;DR

This paper introduces monoid and inverse monoid generalizations of Richard Thompson groups, proves their simplicity and characterizes their Green relations, and analyzes their computational complexity and algebraic properties.

Contribution

It defines and studies the properties of monoids M_{k,1} and Inv_{k,1} as generalizations of Thompson groups, including simplicity, Green relations, and complexity results.

Findings

M_{k,1} and Inv_{k,1} are congruence-simple for all k.

They are J-0-simple with k-1 non-zero D-classes.

Their word problem is in P over finite sets and coNP-complete over certain infinite sets.

Abstract

The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_{k,1} have connections with circuit complexity (studied in another paper). Here we prove that M_{k,1} and Inv_{k,1} are congruence-simple for all k. Their Green relations J and D are characterized: M_{k,1} and Inv_{k,1} are J-0-simple, and they have k-1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra O_k. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets. Changes in this version: Section 4 has been thoroughly revised, and errors have been corrected; however, the main results of Section 4 do not change. Sections 1, 2, and 3 are unchanged, except for the proof of Theorem 2.3, which was incomplete; a complete proof was published in the Appendix of reference [6], and is also given here.

Full text

1 Thompson-Higman monoids

Since their introduction by Richard J. Thompson in the mid 1960s [29, 26, 30], the Thompson groups have had a great impact on infinite group theory. Graham Higman generalized the Thompson groups to an infinite family [20]. These groups and some of their subgroups have appeared in many contexts and have been widely studied; see for example [12, 8, 15, 10, 17, 18, 9, 11, 23].

The definition of the Thompson-Higman groups lends itself easily to generalizations to inverse monoids and to more general monoids. These monoids are also generalizations of the finite symmetric monoids (of all functions on a set), and this leads to connections with circuit complexity; more details on this appear in [2, 3, 5].

By definition the Thompson-Higman group $G_{k,1}$ consists of all maximally extended isomorphisms between finitely generated essential right ideals of $A^{*}$, where $A$ is an alphabet of cardinality $k$. The multiplication is defined to be composition followed by maximal extension: for any $\varphi,\psi\in G_{k,1}$, we have  $\varphi\cdot\psi=$ max$(\varphi\circ\psi)$. Every element $\varphi\in G_{k,1}$ can also be given by a bijection $\varphi:P\to Q$ where $P,Q\subset A^{*}$ are two finite maximal prefix codes over $A$; this bijection can be described concretely by a finite function table. For a detailed definition according to this approach, see [4] (which is also similar to [28], but with a different terminology); moreover, Subsection 1.1 gives all the needed definitions.

It is natural to generalize the maximally extended isomorphisms between finitely generated essential right ideals of $A^{*}$ to homomorphisms, and to drop the requirement that the right ideals be essential. It will turn out that this generalization leads to interesting monoids, or inverse monoids, which we call Thompson-Higman monoids. Our generalization of the Thompson-Higman groups to monoids will also generalize the embedding of these groups into the Cuntz algebras [4, 27], which provides an additional motivation for our definition. Moreover, since these homomorphisms are close to being arbitrary finite string transformations, there is a connection between these monoids and combinational boolean circuits; the study of the connection between Thompson-Higman groups and circuits was started in [5, 3] and will be developed more generally for monoids in [2]; the present paper lays some of the foundations for [2].

1.1 Definition of the Thompson-Higman groups and monoids

Before defining the Thompson-Higman monoids we need some basic definitions, that are similar to the introductory material that is needed for defining the Thompson-Higman groups $G_{k,1}$; we follow [4] (which is similar to [28]). We use an alphabet $A$ of cardinality $|A|=k$, and we list its elements as $A=\{a_{1},\ldots,a_{k}\}$. Let $A^{*}$ denote the set of all finite words over $A$ (i.e., all finite sequences of elements of $A$); this includes the empty word $\varepsilon$. The length of $w\in A^{*}$ is denoted by $|w|$; let $A^{n}$ denote the set of words of length $n$. For two words $u,v\in A^{*}$ we denote their concatenation by $uv$ or by $u\cdot v$; for sets $B,C\subseteq A^{*}$ the concatenation is $BC=\{uv:u\in B,v\in C\}$. A right ideal of $A^{*}$ is a subset $R\subseteq A^{*}$ such that $RA^{*}\subseteq R$. A generating set of a right ideal $R$ is a set $C$ such that $R$ is the intersection of all right ideals that contain $C$; equivalently, $R=CA^{*}$. A right ideal $R$ is called essential iff $R$ has a non-empty intersections with every right ideal of $A^{*}$. For words $u,v\in A^{*}$, we say that $u$ is a prefix of $v$ iff there exists $z\in A^{*}$ such that $uz=v$. A prefix code is a subset $C\subseteq A^{*}$ such that no element of $C$ is a prefix of another element of $C$. A prefix code is maximal iff it is not a strict subset of another prefix code. One can prove that a right ideal $R$ has a unique minimal (under inclusion) generating set, and that this minimal generating set is a prefix code; this prefix code is maximal iff $R$ is an essential right ideal.

For right ideals $R^{\prime}\subseteq R\subseteq A^{*}$ we say that $R^{\prime}$ is essential in $R$ iff $R^{\prime}$ intersects all right subideals of $R$ in a non-empty way.

Tree interpretation: The free monoid $A^{*}$ can be pictured by its right Cayley graph, which is the rooted infinite regular $k$-ary tree with vertex set $A^{*}$ and edge set $\{(v,va):v\in A^{*},a\in A\}$. We simply call this the tree of $A^{*}$. It is a directed tree, with all paths moving away from the root $\varepsilon$ (the empty word); by “path” we will always mean a directed path. A word $v$ is a prefix of a word $w$ iff $v$ is is an ancestor of $w$ in the tree. A set $P$ is a prefix code iff no two elements of $P$ are on the same path. A set $R$ is a right ideal iff any path that starts in $R$ has all its vertices in $R$. The prefix code that generates $R$ consists of the elements of $R$ that are maximal (within $R$) in the prefix order, i.e., closest to the root $\varepsilon$. A finitely generated right ideal $R$ is essential iff every infinite path of the tree eventually reaches $R$ (and then stays in it from there on). Similarly, a finite prefix code $P$ is maximal iff any infinite path starting at the root eventually intersects $P$. For two finitely generated right ideals $R^{\prime}\subset R$, $R^{\prime}$ is essential in $R$ iff any infinite path starting in $R$ eventually reaches $R^{\prime}$ (and then stays in $R^{\prime}$ from there on). In other words for finitely generated right ideals $R^{\prime}\subseteq R$, $R^{\prime}$ is essential in $R$ iff $R^{\prime}$ and $R$ have the same “ends”. For the prefix tree of $A^{*}$ we can consider also the “boundary” $A^{\omega}$ (i.e., all infinite words), a.k.a. the ends of the tree. In Thompson’s original definition [29, 30], $G_{2,1}$ was given by a total action on $\{0,1\}^{\omega}$. In [4] this total action was extended to a partial action on $A^{*}\cup A^{\omega}$; the partial action on $A^{*}\cup A^{\omega}$ is uniquely determined by the total action on $A^{\omega}$; it is also uniquely determined by the partial action on $A^{*}$. Here, as in [4], we only use the partial action on $A^{*}$.

Definition 1.1

A right ideal homomorphism of $A^{*}$ is a total function $\varphi:R_{1}\to A^{*}$ such that $R_{1}$ is a right ideal of $A^{*}$, and for all $x_{1}\in R_{1}$ and all $w\in A^{*}$:  $\varphi(x_{1}w)=\varphi(x_{1})\ w$.

For any partial function $f:A^{*}\to A^{*}$, let Dom$(f)$ denote the domain and let Im$(f)$ denote the image (range) of $f$. For a right ideal homomorphism $\varphi:R_{1}\to A^{*}$ it is easy to see that the image ${\rm Im}(\varphi)$ is also right ideal of $A^{*}$, which is finitely generated (as a right ideal) if the domain $R_{1}={\rm Dom}(\varphi)$ is finitely generated.

A right ideal homomorphism $\varphi:R_{1}\to R_{2}$, where $R_{1}={\rm Dom}(\varphi)$ and $R_{2}={\rm Im}(\varphi)$, can be described by a total surjective function $P_{1}\to S_{2}$, with $P_{1},S_{2}\subset A^{*}$; here $P_{1}$ is the prefix code (not necessarily maximal) that generates $R_{1}$ as a right ideal, and $S_{2}$ is a set (not necessarily a prefix code) that generates $R_{2}$ as a right ideal; so $R_{1}=P_{1}A^{*}$ and $R_{2}=S_{2}A^{*}$. The function $P_{1}\to S_{2}$ corresponding to $\varphi:R_{1}\to R_{2}$ is called the table of $\varphi$. The prefix code $P_{1}$ is called the domain code of $\varphi$ and we write $P_{1}={\rm domC}(\varphi)$. When $S_{2}$ is a prefix code we call $S_{2}$ the image code of $\varphi$ and we write $S_{2}={\rm imC}(\varphi)$. We denote the table size of $\varphi$ (i.e., the cardinality of ${\rm domC}(\varphi)$) by $\|\varphi\|$.

Definition 1.2

An injective right ideal homomorphism is called a right ideal isomorphism. A right ideal homomorphism $\varphi:R_{1}\to R_{2}$ is called total iff the domain right ideal $R_{1}$ is essential. And $\varphi$ is called surjective iff the image right ideal $R_{2}$ is essential.

The table $P_{1}\to P_{2}$ of a right ideal isomorphism $\varphi$ is a bijection between prefix codes (that are not necessarily maximal). The table $P_{1}\to S_{2}$ of a total right ideal homomorphism is a function from a maximal prefix code to a set, and the table $P_{1}\to S_{2}$ of a surjective right ideal homomorphism is a function from a prefix code to a set that generates an essential right ideal. The word “total” is justified by the fact that if a homomorphism $\varphi$ is total (and if ${\rm domC}(\varphi)$ is finite) then $\varphi(w)$ is defined for every word that is long enough (e.g., when $|w|$ is longer than the longest word in the domain code $P_{1}$); equivalently, $\varphi$ is defined from some point onward on every infinite path in the tree of $A^{*}$ starting at the root.

Definition 1.3

An essential restriction of a right ideal homomorphism $\varphi:R_{1}\to A^{*}$ is a right ideal homomorphism $\Phi:R^{\prime}_{1}\to A^{*}$ such that $R^{\prime}_{1}$ is essential in $R_{1}$, and such that for all $x^{\prime}_{1}\in R^{\prime}_{1}$:  $\varphi(x^{\prime}_{1})=\Phi(x^{\prime}_{1})$.

We say that $\varphi$ is an essential extension of $\Phi$ iff $\Phi$ is an essential restriction of $\varphi$.

Note that if $\Phi$ is an essential restriction of $\varphi$ then $R^{\prime}_{2}={\rm Im}(\Phi)$ will automatically be essential in $R_{2}={\rm Im}(\varphi)$. Indeed, if $I$ is any non-empty right subideal of $R_{1}$ then $I\cap R^{\prime}_{1}\neq\varnothing$, hence  $\varnothing\neq\Phi(I\cap R^{\prime}_{1})$ $\subseteq\Phi(I)\,\cap\,\Phi(R^{\prime}_{1})$ $=\Phi(I)\cap R^{\prime}_{2}$; moreover, any right subideal $J$ of $R_{2}$ is of the form $J=\Phi(I)$ where $I=\Phi^{-1}(J)$ is a right subideal of $R_{1}$; hence, for any right subideal $J$ of $R_{2}$, $\varnothing\neq J\cap R^{\prime}_{2}$.

Proposition 1.4

(1)* Let $\varphi,\Phi$ be homomorphisms between finitely generated right ideals of $A^{*}$, where $A=\{a_{1},\ldots,a_{k}\}$. Then $\Phi$ is an essential restriction of $\varphi$ iff $\Phi$ can be obtained from $\varphi$ by starting from the table of $\varphi$ and applying a finite number of restriction steps of the following form: Replace $(x,y)$ in a table by $\{(xa_{1},ya_{1}),\ldots,(xa_{k},ya_{k})\}$.

(2) Every homomorphism between finitely generated right ideals of $A^{*}$ has a unique maximal essential extension.*

Proof.  (1) Consider a homomorphism between finitely generated right ideals $\varphi:R_{1}\to R_{2}$, let $P_{1}$ be the finite prefix code that generates the right ideal $R_{1}$, and let $S_{2}=\varphi(P_{1})$, so $S_{2}$ generates the right ideal $R_{2}$.

If $x\in P_{1}$ and $y=\varphi(x)\in S_{2}$ then (since $\varphi$ is a right ideal homomorphism), $ya_{i}=\varphi(xa_{i})$ for $i=1,\ldots,k$. Then $R_{1}-\{x\}$ is a right ideal which is essential in $R_{1}$, and $R_{1}-\{x\}$ is generated by $(P_{1}-\{x\})\,\cup\,\{xa_{1},\ldots,xa_{k}\}$. Indeed, in the tree of $A^{*}$ every downward directed path starting at vertex $x$ goes through one of the vertices $xa_{i}$. Thus, removing $(x,y)$ from the graph of $\varphi$ is an essential restriction; for the table of $\varphi$, the effect is to replace the entry $(x,y)$ by the set of entries $\{(xa_{1},ya_{1}),\ldots,(xa_{k},ya_{k})\}$. If finitely many restriction steps of the above type are carried out, the result is again an essential restriction of $\varphi$.

Conversely, let us show that if $\Phi$ is an essential restriction of $\varphi$ then $\Phi$ can be obtained by a finite number of replacement steps of the form “replace $(x,y)$ by $\{(xa_{1},ya_{1}),\ldots,(xa_{k},ya_{k})\}$ in the table”.

Using the tree of $A^{*}$ we have: If $R$ and $R^{\prime}$ are right ideals of $A^{*}$ generated by the finite prefix codes $P$, respectively $P^{\prime}$, and if $R^{\prime}$ is essential in $R$ then every infinite path from $P$ intersects $P^{\prime}$. It follows from this characterization of essentiality and from the finiteness of $P_{1}$ and $P^{\prime}_{1}$ that $R_{1}-R^{\prime}_{1}$ is finite. Hence $\varphi$ and $\Phi$ differ only in finitely many places, i.e., one can transform $\varphi$ into $\Phi$ in a finite number of restriction steps.

So, the restriction $\Phi$ of $\varphi$ is obtained by removing a finite number of pairs $(x,y)$ from $\varphi$; however, not every such removal leads to a right ideal homomorphism or an essential restriction of $\varphi$. If $(x_{0},y_{0})$ is removed from $\varphi$ then $x_{0}$ is removed from $R_{1}$ (since $\varphi$ is a function). Also, since $R^{\prime}_{1}$ is a right ideal, when $x_{0}$ is removed then all prefixes of $x_{0}$ (equivalently, all ancestor vertices of $x_{0}$ in the tree of $A^{*}$) have to be removed. So we have the following removal rule (still assuming that domain and image right ideals are finitely generated):

*If $\Phi$ is an essential restriction of $\varphi$ then $\varphi$ can be transformed into $\Phi$ by removing a finite set of strings from $R_{1}$, with the following restriction: If a string $x_{0}$ is removed then all prefixes of $x_{0}$ are also removed from $R_{1}$; moreover, $x_{0}$ is removed from $R_{1}$ iff $(x_{0},\varphi(x_{0}))$ is removed from $\varphi$. *

As a converse of this rule, we claim that if the transformation from $\varphi$ to $\Phi$ is done according to this rule, then $\Phi$ is an essential restriction of $\varphi$. Indeed, $\Phi$ will be a right ideal homomorphism: if $\Phi(x_{1})$ is defined then $\Phi(x_{1}z)$ will also be defined (if it were not, the prefix $x_{1}$ of $x_{1}z$ would have been removed), and $\Phi(x_{1}z)=\varphi(x_{1}z)=$ $\varphi(x_{1})\ z=\Phi(x_{1})\ z$. Moreover, ${\rm Dom}(\Phi)=R^{\prime}_{1}$ will be essential in $R_{1}$: every directed path starting at $R_{1}$ eventually meets $R^{\prime}_{1}$ because only finitely many words were removed from $R_{1}$ to form $R^{\prime}_{1}$. Hence by the tree characterization of essentiality, $R^{\prime}_{1}$ is essential in $R_{1}$.

In summary, if $\Phi$ is an essential restriction of $\varphi$ then $\Phi$ is obtained from $\varphi$ by a finite sequence of steps, each of which removes one pair $(x,\varphi(x))$. In ${\rm Dom}(\varphi)$ the string $x$ is removed. The domain code becomes $(P_{1}-\{x\})\,\cup\,\{xa_{1},\ldots,xa_{k}\}$, since $\{xa_{1},\ldots,xa_{k}\}$ is the set of children of $x$ in the tree of $A^{*}$. This means that in the table of $\varphi$, the pair $(x,\varphi(x))$ is replaced by $\{(xa_{1},\varphi(x)\,a_{1}),\ldots,(xa_{k},\varphi(x)\,a_{k})\}$.

(2) Uniqueness of the maximal essential extension:  By (1) above, essential extensions are obtained by the set of rewrite rules of the form  $\{(xa_{1},ya_{1}),\ldots,(xa_{k},ya_{k})\}\to(x,y)$, applied to tables. This rewriting system is locally confluent (because different rules have non-overlapping left sides) and terminating (because they decrease the length); hence maximal essential extensions exist and are unique.    $\Box$

Proposition 1.4 yields another tree interpretation of essential restriction: Assume first that a total order $a_{1}<a_{2}<\ldots<a_{k}$ has been chosen for the alphabet $A$; this means that the tree of $A^{*}$ is now an oriented rooted tree, i.e., the children of each vertex $v$ have a total order (namely, $va_{1}<va_{2}<\ldots<va_{k}$). The rule “replace $(x,y)$ in the table by $\{(xa_{1},ya_{1}),\ldots,(xa_{k},ya_{k})\}$” has the following tree interpretation: Replace $x$ and $y=\varphi(x)$ by the children of $x$, respectively of $y$, matched according to the order of the children.

Important remark:

As we saw, every right ideal homomorphism can be described by a table $P\to S$ where $P$ is a prefix code and $S$ is a set. But we also have: Every right ideal homomorphism $\varphi$ has an essential restriction $\varphi^{\prime}$ whose table $P^{\prime}\to Q^{\prime}$ is such that both $P^{\prime}$ and $Q^{\prime}$ are prefix codes; moreover, $Q^{\prime}$ can be chosen to be a subset of $A^{n}$ for some $n\leq{\rm max}\{|s|:s\in S\}$. Example (with alphabet $A=\{a,b\}$):

\left(\begin{array}[]{r|r}a&b\\ a&aa\end{array}\right)  has an essential restriction  \left(\begin{array}[]{r|r|r}aa&ab&b\\ aa&ab&aa\end{array}\right). Theorem 4.5B gives a tighter result with polynomial bounds.

Definition 1.5

The Thompson-Higman partial function monoid $M_{k,1}$ consists of all maximal essential extensions of homomorphisms between finitely generated right ideals of $A^{*}$. The multiplication is composition followed by maximal essential extension.

In order to prove associativity of the multiplication of $M_{k,1}$ we define the following and we prove a few Lemmas.

Definition 1.6

By RIk we denote the monoid of all right ideal homomorphisms between finitely generated right ideals of $A^{*}$, with function composition as multiplication. We consider the equivalence relation $\equiv$ defined for $\varphi_{1},\varphi_{2}\in{\it RI}_{k}$ by:  $\varphi_{1}\equiv\varphi_{2}$  iff  ${\rm max}(\varphi_{1})={\rm max}(\varphi_{2})$.

It is easy to prove that ${\it RI}_{k}$ is closed under composition. Moreover, by existence and uniqueness of the maximal essential extension (Prop. 1.4(2)) each $\equiv$-equivalence class contains exactly one element of $M_{k,1}$. We want to prove:

Proposition 1.7

The equivalence relation $\equiv$ is a monoid congruence on ${\it RI}_{k}$, and $M_{k,1}$ is isomorphic (as a monoid) to ${\it RI}_{k}/\!\equiv$.  Hence, $M_{k,1}$ is associative.

First some Lemmas.

Lemma 1.8

If $R^{\prime}_{i}\subseteq R_{i}$ ($i=1,2$) are finitely generated right ideals with $R^{\prime}_{i}$ essential in $R_{i}$, then $R^{\prime}_{1}\cap R^{\prime}_{2}$ is essential in $R_{1}\cap R_{2}$.

Proof. We use the tree characterization of essentiality. Any infinite path $p$ in $R_{1}\cap R_{2}$ is also in $R_{i}$ ($i=1,2$), hence $p$ eventually enters into $R^{\prime}_{i}$. Thus $p$ eventually meets $R^{\prime}_{1}$ and $R^{\prime}_{2}$, i.e., $p$ meets $R^{\prime}_{1}\cap R^{\prime}_{2}$.     $\Box$

Lemma 1.9

*All $\varphi_{1},\varphi_{2}\in{\it RI}_{k}$ have restrictions $\Phi_{1},\Phi_{2}\in{\it RI}_{k}$ (not necessarily essential restrictions) such that:

$\bullet$   $\Phi_{2}\circ\Phi_{1}=\varphi_{2}\circ\varphi_{1}$,  and

$\bullet$   ${\rm Dom}(\Phi_{2})={\rm Im}(\Phi_{1})=$ ${\rm Dom}(\varphi_{2})\,\cap\,{\rm Im}(\varphi_{1})$.*

Proof. Let $R={\rm Dom}(\varphi_{2})\cap{\rm Im}(\varphi_{1})$. This is a right ideal which is finitely generated since ${\rm Dom}(\varphi_{2})$ and ${\rm Im}(\varphi_{1})$ are finitely generated (see Lemma 3.3 of [4]). Now we restrict $\varphi_{1}$ to $\Phi_{1}$ in such a way that ${\rm Im}(\Phi_{1})=R$ and ${\rm Dom}(\Phi_{1})=\varphi_{1}^{-1}(R)$, and we restrict $\varphi_{2}$ to $\Phi_{2}$ in such a way that ${\rm Dom}(\Phi_{2})=R$ and ${\rm Im}(\Phi_{2})=\varphi_{2}(R)$. Then $\Phi_{2}\circ\Phi_{1}(.)$ and $\varphi_{2}\circ\varphi_{1}(.)$ agree on $\varphi_{1}^{-1}(R)$; moreover, ${\rm Dom}(\Phi_{2}\circ\Phi_{1})=\varphi_{1}^{-1}(R)$. Since $\varphi_{2}\circ\varphi_{1}(x)$ is only defined when $\varphi_{1}(x)\in R$, we have $\Phi_{2}\circ\Phi_{1}=\varphi_{2}\circ\varphi_{1}$. Also, by the definition of $R$ we have ${\rm Dom}(\Phi_{2})={\rm Im}(\Phi_{1})$.     $\Box$

Lemma 1.10

For all $\varphi_{1},\varphi_{2}\in{\it RI}_{k}$ we have:

${\rm max}(\varphi_{2}\circ\varphi_{1})=$* ${\rm max}({\rm max}(\varphi_{2})\circ\varphi_{1})=$ ${\rm max}(\varphi_{2}\circ{\rm max}(\varphi_{1}))$.*

Proof. We only prove the first equality; the proof of the second one is similar. By Lemma 1.9 we can restrict $\varphi_{1}$ and $\varphi_{2}$ to $\varphi^{\prime}_{1}$, respectively $\varphi^{\prime}_{2}$, so that  $\varphi^{\prime}_{2}\circ\varphi^{\prime}_{1}=\varphi_{2}\circ\varphi_{1}$, and  ${\rm Dom}(\varphi^{\prime}_{2})={\rm Im}(\varphi^{\prime}_{1})=$ ${\rm Dom}(\varphi_{2})\cap{\rm Im}(\varphi_{1})$; let $R^{\prime}={\rm Dom}(\varphi_{2})\cap{\rm Im}(\varphi_{1})$.

Similarly we can restrict $\varphi_{1}$ and ${\rm max}(\varphi_{2})$ to $\varphi^{\prime\prime}_{1}$, respectively $\varphi^{\prime\prime}_{2}$, so that  $\varphi^{\prime\prime}_{2}\circ\varphi^{\prime\prime}_{1}={\rm max}(\varphi_{2})\circ\varphi_{1}$, and  ${\rm Dom}(\varphi^{\prime\prime}_{2})={\rm Im}(\varphi^{\prime\prime}_{1})=$ ${\rm Dom}({\rm max}(\varphi_{2}))\cap{\rm Im}(\varphi_{1})$; let $R^{\prime\prime}={\rm Dom}({\rm max}(\varphi_{2}))\cap{\rm Im}(\varphi_{1})$.

Obviously, $R^{\prime}\subseteq R^{\prime\prime}$ (since $\varphi_{2}$ is a restriction of ${\rm max}(\varphi_{2})$). Moreover, $R^{\prime}$ is essential in $R^{\prime\prime}$, by Lemma 1.8; indeed, ${\rm Dom}(\varphi_{2})$ is essential in ${\rm Dom}({\rm max}(\varphi_{2}))$ since ${\rm max}(\varphi_{2})$ is an essential extension of $\varphi_{2}$. Since $R^{\prime}$ is essential in $R^{\prime\prime}$, $\varphi_{2}\circ\varphi_{1}$ is an essential restriction of ${\rm max}(\varphi_{2})\circ\varphi_{1}$. Hence by uniqueness of the maximal essential extension, ${\rm max}({\rm max}(\varphi_{2})\circ\varphi_{1})=$ ${\rm max}(\varphi_{2}\circ{\rm max}(\varphi_{1}))$.    $\Box$

Proof of Prop. 1.7:  If $\varphi_{2}\equiv\psi_{2}$ then, by definition, ${\rm max}(\varphi_{2})={\rm max}(\psi_{2})$, hence by Lemma 1.10:

${\rm max}(\varphi_{2}\circ\varphi)=$ ${\rm max}({\rm max}(\varphi_{2})\circ\varphi)=$ ${\rm max}({\rm max}(\psi_{2})\circ\varphi)=$ ${\rm max}(\psi_{2}\circ\varphi)$,

for all $\varphi\in{\it RI}_{k}$. Thus (by the definition of $\equiv$), $\varphi_{2}\circ\varphi\equiv\psi_{2}\circ\varphi$, so $\equiv$ is a right congruence. Similarly one proves that $\equiv$ is a left congruence. Thus, ${\it RI}_{k}/\!\!\equiv\,$ is a monoid.

Since every $\equiv$-equivalence class contains exactly one element of $M_{k,1}$ there is a one-to-one correspondence between ${\it RI}_{k}/\!\!\equiv\,$ and $M_{k,1}$. Moreover, the map  $\varphi\in{\it RI}_{k}\longmapsto{\rm max}(\varphi)\in M_{k,1}$  is a homomorphism, by Lemma 1.10 and by the definition of multiplication in $M_{k,1}$. Hence ${\it RI}_{k}/\!\!\equiv\,$ is isomorphic to $M_{k,1}$.    $\Box$

1.2 Other Thompson-Higman monoids

We now introduce a few more families of Thompson-Higman monoids, whose definition comes about naturally in analogy with $M_{k,1}$.

Definition 1.11

The Thompson-Higman total function monoid ${\it tot}M_{k,1}$ and the Thompson-Higman surjective function monoid ${\it sur}M_{k,1}$ consist of maximal essential extensions of homomorphisms between finitely generated right ideals of $A^{*}$ where the domain, respectively, the image ideal, is an essential right ideal.

The Thompson-Higman inverse monoid ${\it Inv}_{k,1}$ consists of all maximal essential extensions of isomorphisms between finitely generated (not necessarily essential) right ideals of $A^{*}$.

Every element $\varphi\in{\it tot}M_{k,1}$ can be described by a function $P\to Q$, called the table of $\varphi$, where $P,Q\subset A^{*}$ with $P$ a finite maximal prefix code over $A$. A similar description applies to ${\it sur}M_{k,1}$ but now with $Q$ a finite maximal prefix code. Every $\varphi\in{\it Inv}_{k,1}$ can be described by a bijection $P\to Q$ where $P,Q\subset A^{*}$ are two finite prefix codes (not necessarily maximal).

It is easy to prove that essential extension and restriction of right ideal homomorphisms, as well as composition of such homomorphisms, preserve injectiveness, totality, and surjectiveness. Thus ${\it tot}M_{k,1}$, ${\it sur}M_{k,1}$, and ${\it Inv}_{k,1}$ are submonoids of $M_{k,1}$.

We also consider the intersection ${\it tot}M_{k,1}\cap{\it sur}M_{k,1}$, i.e., the monoid of all maximal essential extensions of homomorphisms between finitely generated essential right ideals of $A^{*}$; we denote this monoid by ${\it totsur}M_{k,1}$. The monoids $M_{k,1}$, ${\it tot}M_{k,1}$, ${\it sur}M_{k,1}$, and ${\it totsur}M_{k,1}$ are regular monoids. (A monoid $M$ is regular iff for every $m\in M$ there exists $x\in M$ such that $mxm=m$.) The monoid ${\it Inv}_{k,1}$ is an inverse monoid. (A monoid $M$ is inverse iff for every $m\in M$ there exists one and only one $x\in M$ such that $mxm=m$ and $x=xmx$.)

We consider the submonoids ${\it totInv}_{k,1}$ and ${\it surInv}_{k,1}$ of ${\it Inv}_{k,1}$, described by bijections $P\to Q$ where $P,Q\subset A^{*}$ are two finite prefix codes with $P$, respectively $Q$ maximal. The (unique) inverses of elements in ${\it totInv}_{k,1}$ are in ${\it surInv}_{k,1}$, and vice versa, so these submonoids of ${\it Inv}_{k,1}$ are not regular monoids. We have  ${\it totInv}_{k,1}\cap{\it surInv}_{k,1}\ =\ G_{k,1}$  (the Thompson-Higman group).

It is easy to see that for all $n>0$, $M_{k,1}$ contains the symmetric monoids ${\it PF}_{k^{n}}$ of all partial functions on $k^{n}$ elements, represented by all elements of $M_{k,1}$ with a table $P\to Q$ where $P,Q\subseteq A^{n}$. Hence $M_{k,1}$ contains all finite monoids. Similarly, ${\it tot}M_{k,1}$ contains the symmetric monoids $F_{k^{n}}$ of all total functions on $k^{n}$ elements. And ${\it Inv}_{k,1}$ contains ${\mathcal{I}}_{k^{n}}$ (the finite symmetric inverse monoid of all injective partial functions on $A^{n}$).

1.3 Cuntz algebras and Thompson-Higman monoids

All the monoids, inverse monoids, and groups, defined above, are submonoids of the multiplicative part of the Cuntz algebra ${\cal O}_{k}$.

The Cuntz algebra ${\cal O}_{k}$, introduced by Dixmier [16] (for $k=2$) and Cuntz [14], is a $k$-generated star-algebra (over the field of complex numbers) with identity element 1 and zero 0, given by the following finite presentation. The generating set is $A=\{a_{1},\ldots,a_{k}\}$. Since this is defined as a star-algebra, we automatically have the star-inverses $\{\overline{a}_{1},\ldots,\overline{a}_{k}\}$; for clarity we use overlines rather than stars.

Relations of the presentation:

$\overline{a}_{i}a_{i}={\bf 1}$,   for $i=1,\ldots,k$;

$\overline{a}_{i}a_{j}={\bf 0}$,   when $i\neq j$, $1\leq i,j\leq k$;

$a_{1}\overline{a}_{1}+\ldots+a_{k}\overline{a}_{k}={\bf 1}$.

It is easy to verify that this defines a star-algebra. The Cuntz algebras are actually C∗-algebras with many remarkable properties (proved in [14]), but here we only need them as star-algebras, without their norm and Cauchy completion.

In [4] and independently in [27] it was proved that the Thompson-Higman group $G_{k,1}$ is the subgroup of ${\cal O}_{k}$ consisting of the elements that have an expression of the form  $\sum_{x\in P}f(x)\ \overline{x}$  where we require the following: $P$ and $Q$ range over all finite maximal prefix codes over the alphabet $\{a_{1},\ldots,a_{k}\}$, and $f$ is any bijection $P\to Q$. Another proof is given in [22]. More generally we also have:

Theorem 1.12

The Thompson-Higman monoid $M_{k,1}$ is a submonoid of the multiplicative part of the Cuntz algebra ${\cal O}_{k}$.

Proof outline. The Thompson-Higman partial function monoid $M_{k,1}$ is the set of all elements of ${\cal O}_{k}$ that have an expression of the form  $\sum_{x\in P}f(x)\ \overline{x}$  where $P\subset A^{*}$ ranges over all finite prefix codes, and $f$ ranges over functions $P\to A^{*}$.

The details of the proof are very similar to the proofs in [4, 27]; the definition of essential restriction (and extension) and Proposition 1.4 insure that the same proof goes through.    $\Box$

The embeddability into the Cuntz algebra is a further justification of the definitional choices that we made for the Thompson-Higman monoid $M_{k,1}$.

2 Structure and simplicity of the Thompson-Higman monoids

We give some structural properties of the Thompson-Higman monoids; in particular, we show that $M_{k,1}$ and ${\it Inv}_{k,1}$ are simple for all $k$.

2.1 Group of units, $J$-relation, simplicity

By definition, the group of units of a monoid $M$ is the set of invertible elements (i.e., the elements $u\in M$ for which there exists $x\in M$ such that $xu=ux={\bf 1}$, where 1 is the identity element of $M$).

Proposition 2.1

The Thompson-Higman group $G_{k,1}$ is the group of units of the monoids $M_{k,1}$, ${\it tot}M_{k,1}$, ${\it sur}M_{k,1}$, ${\it totsur}M_{k,1}$, and ${\it Inv}_{k,1}$.

Proof. It is obvious that the groups of units of the above monoids contain $G_{k,1}$. Conversely, we want to show that that if $\varphi\in M_{k,1}$ (and in particular, if $\varphi$ is in one of the other monoids) and if $\varphi$ has a left inverse and a right inverse, then $\varphi\in G_{k,1}$.

First, it follows that $\varphi$ is injective, i.e., $\varphi\in$ ${\it Inv}_{k,1}$. Indeed, existence of a left inverse implies that for some $\alpha\in M_{k,1}$ we have $\alpha\ \varphi={\bf 1}$; hence, if $\varphi(x_{1})=\varphi(x_{2})$ then $x_{1}=\alpha\ \varphi(x_{1})=$ $\alpha\ \varphi(x_{2})=x_{2}$.

Next, we show that ${\rm domC}(\varphi)$ is a maximal prefix code, hence $\varphi\in{\it totInv}_{k,1}$. Indeed, we can again consider $\alpha\in M_{k,1}$ such that $\alpha\ \varphi={\bf 1}$. For any essential restriction of 1 the domain code is a maximal prefix code, hence ${\rm domC}(\alpha\circ\varphi)$ is maximal (where $\circ$ denotes functional composition). Moreover, ${\rm domC}(\alpha\circ\varphi)$ is also contained in the domain code of some restriction of $\varphi$, since $\varphi(x)$ must be defined when $\alpha\circ\varphi(x)$ is defined. Hence domC($\varphi^{\prime}$), for some restriction $\varphi^{\prime}$ of $\varphi$, is a maximal prefix code; it follows that domC($\varphi$) is a maximal prefix code.

If we apply the reasoning of the previous paragraph to $\varphi^{-1}$ (which exists since we saw that $\varphi$ is injective), we conclude that ${\rm domC}(\varphi^{-1})={\rm imC}(\varphi)$ is a maximal prefix code. Thus, $\varphi\in{\it surInv}_{k,1}$.

We proved that if $\varphi$ has a left inverse and a right inverse then $\varphi\in{\it totInv}_{k,1}\cap{\it surInv}_{k,1}$. Since ${\it totInv}_{k,1}\cap{\it surInv}_{k,1}=G_{k,1}$ we conclude that $\varphi\in G_{k,1}$.    $\Box$

We now characterize some of the Green relations of $M_{k,1}$ and of ${\it Inv}_{k,1}$, and we prove simplicity.

By definition, two elements $x,y$ of a monoid $M$ are $J$-related (denoted $x\equiv_{J}y$) iff $x$ and $y$ belong to exactly the same ideals of $M$. More generally, the $J$-preorder of $M$ is defined as follows: $x\leq_{J}y$ iff $x$ belongs to every ideal that $y$ belongs to. It is easy to see that $x\equiv_{J}y$ iff $x\leq_{J}y$ and $y\leq_{J}x$; moreover, $x\leq_{J}y$ iff there exist $\alpha,\beta\in M$ such that $x=\alpha y\beta$. A monoid $M$ is called $J$-simple iff $M$ has only one $J$-class (or equivalently, $M$ has only one ideal, namely $M$ itself). A monoid $M$ is called [math]-$J$-simple iff $M$ has exactly two $J$-classes, one of which consist of just a zero element (equivalently, $M$ has only two ideals, one of which is a zero element, and the other is $M$ itself). See [13, 19] for more information on the $J$-relation. Cuntz [14] proved that the multiplicative part of the $C^{*}$-algebra ${\cal O}_{k}$ is a [math]-$J$-simple monoid, and that as an algebra ${\cal O}_{k}$ is simple. We will now prove similar results for the Thompson-Higman monoids.

Proposition 2.2

The inverse monoid ${\it Inv}_{k,1}$ and the monoid $M_{k,1}$ are [math]-$J$-simple. The monoid ${\it tot}M_{k,1}$ is $J$-simple.

Proof. Let $\varphi\in M_{k,1}$ (or $\in{\it Inv}_{k,1}$). When $\varphi$ is not the empty map there are $x_{0},y_{0}\in A^{*}$ such that $y_{0}=\varphi(x_{0})$. Let us define $\alpha,\beta\in{\it Inv}_{k,1}$ by the tables $\alpha=\{(\varepsilon\mapsto x_{0})\}$ and $\beta=\{(y_{0}\mapsto\varepsilon)\}$. Recall that $\varepsilon$ denotes the empty word. Then  $\beta\,\varphi\,\alpha(.)=$ $\{(\varepsilon\mapsto\varepsilon)\}={\bf 1}$. So, every non-zero element of $M_{k,1}$ (and of ${\it Inv}_{k,1}$) is in the same $J$-class as the identity element.

In the case of ${\it tot}M_{k,1}$ we can take $\alpha=\{(\varepsilon\mapsto x_{0})\}$ as before (since the domain code of $\alpha$ is $\{\varepsilon\}$, which is a maximal prefix code), and we take $\beta^{\prime}:Q\mapsto\{\varepsilon\}$ (i.e., the map that sends every element of $Q$ to $\varepsilon$), where $Q$ is any finite maximal prefix code containing $y_{0}$. Then again,  $\beta^{\prime}\,\varphi\,\alpha(.)=$ $\{(\varepsilon\mapsto\varepsilon)\}={\bf 1}$.    $\Box$

Thompson proved that $V$ ($=G_{2,1}$) is a simple group; Higman proved more generally that when $k$ is even then $G_{k,1}$ is simple, and when $k$ is odd then $G_{k,1}$ contains a simple normal subgroup of index 2. We will show next that in the monoid case we have simplicity for all $k$ (not only when $k$ is even). For a monoid $M$, “simple”, or more precisely, “congruence-simple” is defined to mean that the only congruences on $M$ are the trivial congruences (i.e., the equality relation, and the congruence that lumps all elements of $M$ into one congruence class).

Theorem 2.3

The Thompson-Higman monoids ${\it Inv}_{k,1}$ and $M_{k,1}$ are congruence-simple for all $k$.

Proof.  Let $\equiv$ be any congruence on $M_{k,1}$ that is not the equality relation. We will show that then the whole monoid is congruent to the empty map 0. We will make use of [math]-$\cal J$-simplicity.

Case 0:  Assume that $\Phi\equiv$ 0 for some element $\Phi\neq$ 0 of $M_{k,1}$. Then for all $\alpha,\beta\in M_{k,1}$ we have obviously $\alpha\,\Phi\,\beta\equiv$ 0. Moreover, by [math]-$\cal J$-simplicity of $M_{k,1}$ we have  $M_{k,1}$ $=\{\alpha\,\Phi\,\beta:\alpha,\beta\in M_{k,1}\}$  since $\Phi\neq 0$. Hence in this case all elements of $M_{k,1}$ are congruent to 0.

For the remainder we suppose that $\varphi\equiv\psi$ and $\varphi\neq\psi$, for some elements $\varphi,\psi$ of $M_{k,1}-\{{\bf 0}\}$.

For a right ideal $R\subseteq A^{*}$ generated by a prefix code $P$ we call $PA^{\omega}$ the set of ends of $R$. We call two right ideals $R_{1},R_{2}$ essentially equal iff $R_{1}$ and $R_{2}$ have the same ends, and we denote this by $R_{1}=_{\sf ess}R_{2}$. This is equivalent to the following property: Every right ideal that intersects $R_{1}$ also intersects $R_{2}$, and vice versa (see [6] and [7]).

Case 1:  ${\rm Dom}(\varphi)\neq_{\sf ess}{\rm Dom}(\psi)$.

Then there exists $x_{0}\in A^{*}$ such that $x_{0}A^{*}\subseteq{\rm Dom}(\varphi)$, but  ${\rm Dom}(\psi)\cap x_{0}A^{*}=\varnothing$; or, vice versa, there exists $x_{0}\in A^{*}$ such that $x_{0}A^{*}\subseteq{\rm Dom}(\psi)$, but  ${\rm Dom}(\varphi)\cap x_{0}A^{*}=\varnothing$. Let us assume the former. Letting $\beta=(x_{0}\mapsto x_{0})$, we have $\varphi\,\beta(.)=(x_{0}\mapsto\varphi(x_{0}))$. We also have $\psi\,\beta(.)=$ 0, since $x_{0}A^{*}\cap{\rm Dom}(\psi)=\varnothing$. So, $\varphi\,\beta\equiv\psi\,\beta={\bf 0}$, but $\varphi\,\beta\neq$ 0. Hence case 0, applied to $\Phi=\varphi\,\beta$, implies that the entire monoid $M_{k,1}$ is congruent to 0.

Case 2.1:  ${\rm Im}(\varphi)\neq_{\sf ess}{\rm Im}(\psi)$  and  ${\rm Dom}(\varphi)=_{\sf ess}{\rm Dom}(\psi)$.

Then there exists $y_{0}\in A^{*}$ such that $y_{0}A^{*}\subseteq{\rm Im}(\varphi)$, but  ${\rm Im}(\psi)\cap y_{0}A^{*}=\varnothing$; or, vice versa, $y_{0}A^{*}\subseteq{\rm Im}(\psi)$, but  ${\rm Im}(\varphi)\cap y_{0}A^{*}=\varnothing$. Let us assume the former. Let $x_{0}\in A^{*}$ be such that $y_{0}=\varphi(x_{0})$. Then  $(y_{0}\mapsto y_{0})\circ\varphi\circ(x_{0}\mapsto x_{0})$ $\ =\ (x_{0}\mapsto y_{0})$.

On the other hand,  $(y_{0}\mapsto y_{0})\circ\psi\circ(x_{0}\mapsto x_{0})\ =\ {\bf 0}$. Indeed, if $x_{0}A^{*}\cap{\rm Dom}(\psi)=\varnothing$ then for all $w\in A^{*}:$  $\psi\circ(x_{0}\mapsto x_{0})(x_{0}w)\ =\ \psi(x_{0}w)$ \ =\ $\varnothing$. And if $x_{0}A^{*}\cap{\rm Dom}(\psi)\neq\varnothing$ then for those $w\in A^{*}$ such that $x_{0}w\in{\rm Dom}(\psi)$ we have  (y_{0}\mapsto y_{0})\circ\psi\circ(x_{0}\mapsto x_{0})(x_{0}w)\ =\ $(y_{0}\mapsto y_{0})(\psi(x_{0}w))\ =\ \varnothing$, since ${\rm Im}(\psi)\cap y_{0}A^{*}=\varnothing$. Now case 0 applies to ${\bf 0}\neq\Phi=$ $(y_{0}\mapsto y_{0})\circ\varphi\circ(x_{0}\mapsto x_{0})$ $\equiv{\bf 0}$; hence all elements of $M_{k,1}$ are congruent to 0.

Case 2.2:  ${\rm Im}(\varphi)=_{\sf ess}{\rm Im}(\psi)$  and  ${\rm Dom}(\varphi)=_{\sf ess}{\rm Dom}(\psi)$.

Then after restricting $\varphi$ and $\psi$ to ${\rm Dom}(\varphi)\cap{\rm Dom}(\psi)$ ($=_{\sf ess}$ ${\rm Dom}(\varphi)=_{\sf ess}{\rm Dom}(\psi)$), we have: $\,{\rm domC}(\varphi)={\rm domC}(\psi)$, and there exist $x_{0}\in{\rm domC}(\varphi)={\rm domC}(\psi)$ and $y_{0}\in{\rm Im}(\varphi)$, $y_{1}\in{\rm Im}(\psi)$ such that $\varphi(x_{0})=y_{0}\neq y_{1}=\psi(x_{0})$. We have two sub-cases.

Case 2.2.1:  $y_{0}$ and $y_{1}$ are not prefix-comparable.

Then  $(y_{0}\mapsto y_{0})\circ\varphi\circ(x_{0}\mapsto x_{0})$ $\ =\ (x_{0}\mapsto y_{0})$.

On the other hand,  $(y_{0}\mapsto y_{0})\circ\psi\circ(x_{0}\mapsto x_{0})(x_{0}w)\ =$  $(y_{0}\mapsto y_{0})(y_{1}w)\ =\ \varnothing$  for all $w\in A^{*}$  (since $y_{0}$ and $y_{1}$ are not prefix-comparable). So  $(y_{0}\mapsto y_{0})\circ\psi\circ(x_{0}\mapsto x_{0})\ =\ \ {\bf 0}$. Hence case 0 applies to  {\bf 0}\neq\Phi\ =\ $(y_{0}\mapsto y_{0})\circ\varphi\circ(x_{0}\mapsto x_{0})$ $\equiv{\bf 0}$.

Case 2.2.2:  $y_{0}$ is a prefix of $y_{1}$, and $y_{0}\neq y_{1}$. (The case where $y_{0}$ is a prefix of $y_{1}$ is similar.)

Then $y_{1}=y_{0}au_{1}$ for some $a\in A$, $u_{1}\in A^{*}$. Letting $b\in A-\{a\}$, and $y_{2}=y_{0}b$, we obtain a string $y_{2}$ that is not prefix-comparable with $y_{1}$. Now, $\,(y_{2}\mapsto y_{2})\circ\varphi\circ(x_{0}\mapsto x_{0})(x_{0}v_{2})$ \ =\ $(y_{2}\mapsto y_{2})(y_{0}v_{2})\ =\ y_{2}$. But for all $w\in A^{*}$, \,(y_{2}\mapsto y_{2})\circ\psi\circ(x_{0}\mapsto x_{0})(x_{0}w)\ =\ $(y_{2}\mapsto y_{2})(y_{1}w)\ =\ \varnothing$, since $y_{2}$ and $y_{1}$ are not prefix-comparable. Thus, case 0 applies to  ${\bf 0}\neq\Phi=$ $(y_{2}\mapsto y_{2})\circ\varphi\circ(x_{0}\mapsto x_{0})$ $\equiv{\bf 0}$.

The same proof works for ${\it Inv}_{k,1}$ since all the multipliers used in the proof (of the form $(u\mapsto v)$ for some $u,v\in A^{*}$) belong to ${\it Inv}_{k,1}$.    $\Box$

2.2 $D$-relation

Besides the $J$-relation and the $J$-preorder, based on ideals, there are the $R$- and $L-$relations and $R$- and $L-$preorders, based on right (or left) ideals. Two elements $x,y\in M$ are $R$-related (denoted $x\equiv_{R}y$) iff $x$ and $y$ belong to exactly the same right ideals of $M$. The $R$-preorder is defined as follows: $x\leq_{R}y$ iff $x$ belongs to every right ideal that $y$ belongs to. It is easy to see that $x\equiv_{R}$ iff $x\leq_{R}y$ and $y\leq_{R}x$; also, $x\leq_{R}y$ iff there exists $\alpha\in M$ such that $x=y\alpha$. In a similar way one defines $\equiv_{L}$ and $\leq_{L}$. Finally, there is the $D$-relation of $M$, which is defined as follows: $x\equiv_{D}y$ iff there exists $s\in M$ such that $x\equiv_{R}s\equiv_{L}y$; this is easily seen to be equivalent to saying that there exists $t\in M$ such that $x\equiv_{L}t\equiv_{R}y$. For more information on these definitions see for example [13, 19].

The $D$-relation of $M_{k,1}$ and ${\it Inv}_{k,1}$ has an interesting characterization, as we shall prove next. We will represent all elements of $M_{k,1}$ by tables of the from $\varphi:P\to Q$, where both $P$ and $Q$ are finite prefix codes over $A$ (with $|A|=k$). For such a table we also write $P={\rm domC}(\varphi)$ (the domain code of $\varphi$) and $Q={\rm imC}(\varphi)$ (the image code of $\varphi$). In general, tables of elements of $M_{k,1}$ have the form $P\to S$, where $P$ is a finite prefix code and $S$ is a finite set; but by using essential restrictions, if necessary, every element of $M_{k,1}$ can be given a table $P\to Q$, where both $P$ and $Q$ are finite prefix codes.

Note the following invariants with respect to essential restrictions:

Proposition 2.4

Let $\varphi_{1}:P_{1}\to Q_{1}$ be a table for an element of $M_{k,1}$, where $P_{1},Q_{1}\subset A^{*}$ are finite prefix codes. Let $\varphi_{2}:P_{2}\to Q_{2}$ be another finite table for the same element of $M_{k,1}$, obtained from the table $\varphi_{1}$ by an essential restriction. Then $P_{2},Q_{2}\subset A^{*}$ are finite prefix codes and we have

$|P_{1}|\equiv|P_{2}|$*  mod $(k-1)$    and*

$|Q_{1}|\equiv|Q_{2}|$*  mod $(k-1)$.*

These modular congruences also hold for essential extensions, provided that we only extend to tables in which the image is a prefix code.

Proof. An essential restriction consists of a finite sequence of essential restriction steps; an essential restriction step consists of replacing a table entry $(x,y)$ of $\varphi_{1}$ by $\{(xa_{1},ya_{1}),\ldots,(xa_{k},ya_{k})\}$ (according to Proposition 1.4). For a finite prefix code $Q\subset A^{*}$, and $q\in Q$, the finite set $(Q-\{q\})\cup\{qa_{1},\ldots,qa_{k}\}$ is also a prefix code, as is easy to prove. In this process, the cardinalities change as follows:  $|P_{1}|$ becomes $|P_{1}|-1+k$ and $|Q_{1}|$ becomes $|Q_{1}|-1+k$. Indeed (looking at $Q_{1}$ for example), first an element $y$ is removed from $Q_{1}$, then the $k$ elements $\{ya_{1},\ldots,ya_{k}\}$ are added. The elements $ya_{i}$ that are added are all different from the elements that are already present in $Q_{1}-\{y\}$; in fact, more strongly, $ya_{i}$ and the elements of $Q_{1}-\{y\}$ are not prefixes of each other.    $\Box$

As a consequence of Prop. 2.4 it makes sense, for any $\varphi\in M_{k,1}$, to talk about $|{\rm domC}(\varphi)|$ and $|{\rm imC}(\varphi)|$ as elements of ${\mathbb{Z}}_{k-1}$, independently of the representation of $\varphi$ by a right-ideal homomorphism.

Theorem 2.5

For any non-zero elements $\varphi,\psi$ of $M_{k,1}$ (or of ${\it Inv}_{k,1}$) the $D$-relation is characterized as follows:

$\varphi\equiv_{D}\psi$*    iff    $|{\rm imC}(\varphi)|\equiv|{\rm imC}(\psi)|$  mod $(k-1)$.*

Hence, $M_{k,1}$ and ${\it Inv}_{k,1}$ have $k-1$ non-zero $D$-classes. In particular, $M_{2,1}$ and ${\it Inv}_{2,1}$ are [math]-$D$-simple (also called [math]-bisimple).

The proof of Theorem 2.5 uses several Lemmas.

Lemma 2.6

([5] Lemma 6.1; Arxiv version of [5] Lemma 9.9).*  For every finite alphabet $A$ and every integer $i\geq 0$ there exists a maximal prefix code of cardinality  $1+(|A|-1)\,i$. And every finite maximal prefix code over $A$ has cardinality  $1+(|A|-1)\,i$, for some integer $i\geq 0$.*

It follows that when $|A|=2$, there are finite prefix codes over $A$ of every finite cardinality.      $\Box$

As a consequence of this Lemma we have for all $\varphi\in G_{k,1}$:  $\|\varphi\|\equiv 1$ mod $(k-1)$. Thus, except for the Thompson group $V$ (when $k=2$), there is a constraint on the table size of the elements of the group.

In the following ${\rm id}_{Q}$ denotes the element of ${\it Inv}_{k,1}$ given by the table $\{(x\mapsto x):x\in Q\}$ where $Q\subset A^{*}$ is any finite prefix code.

Lemma 2.7

(1)* For any $\varphi\in M_{k,1}$ (or $\in{\it Inv}_{k,1}$) with table $P\to Q$ (where $P,Q$ are finite prefix codes) we have:  $\varphi\equiv_{R}{\rm id}_{Q}$.

(2) If $S,T$ are finite prefix codes with $|S|=|T|$ then  ${\rm id}_{S}\equiv_{D}{\rm id}_{T}$.

(3) If $\varphi_{1}:P_{1}\to Q_{1}$ and $\varphi_{2}:P_{2}\to Q_{2}$ are such that $|Q_{1}|=|Q_{2}|$ then  $\varphi_{1}\equiv_{D}\varphi_{2}$.*

Proof. (1) Let $P^{\prime}\subseteq P$ be a set of representatives modulo $\varphi$ (i.e., we form $P^{\prime}$ by choosing one element in every set $\varphi^{-1}\varphi(x)$ as $x$ ranges over $P$). So, $|P^{\prime}|=|Q|$. Let $\alpha\in{\it Inv}_{k,1}$ be given by a table $Q\to P^{\prime}$; the exact map does not matter, as long as $\alpha$ is bijective. Then $\varphi\circ\alpha(.)$ is a permutation of $Q$, and $\varphi\circ\alpha\equiv_{R}$ $\varphi\circ\alpha\circ(\varphi\circ\alpha)^{-1}={\rm id}_{Q}$.

Now, $\varphi\geq_{R}\varphi\circ\alpha\geq_{R}$ $\varphi\circ\alpha\circ(\varphi\circ\alpha)^{-1}\circ\varphi=$ ${\rm id}_{Q}\circ\varphi=\varphi$, hence $\varphi\equiv_{R}\varphi\circ\alpha$  ($\equiv_{R}{\rm id}_{Q}$).

(2) Let $\alpha:S\to T$ be a bijection (which exists since $|S|=|T|$); so $\alpha$ represents an element of ${\it Inv}_{k,1}$. Then $\alpha=\alpha\circ{\rm id}_{S}(.)$ and ${\rm id}_{S}=\alpha^{-1}\circ\alpha(.)$; hence, $\alpha\equiv_{L}{\rm id}_{S}$.

Also, $\alpha={\rm id}_{T}\circ\alpha(.)$ and ${\rm id}_{T}=\alpha\circ\alpha^{-1}(.)$; hence, $\alpha\equiv_{R}{\rm id}_{T}$. Thus, ${\rm id}_{S}\equiv_{L}\alpha\equiv_{R}{\rm id}_{T}$.

(3) If $|Q_{1}|=|Q_{2}|$ then ${\rm id}_{Q_{1}}\equiv_{D}{\rm id}_{Q_{2}}$ by (2). Moreover, $\varphi_{1}\equiv_{D}{\rm id}_{Q_{1}}$ and $\varphi_{2}\equiv_{D}{\rm id}_{Q_{2}}$ by (1). The result follows by transitivity of $\equiv_{D}$.    $\Box$

Lemma 2.8

(1)* For any $m\geq k$ let $i$ be the residue of $m$ modulo $k-1$ in the range $2\leq i\leq k$, and let us write $m=i+(k-1)j$, for some $j\geq 0$. Then there exists a prefix code $Q_{i,j}$ of cardinality $|Q_{i,j}|=m$, such that ${\rm id}_{Q_{i,j}}$ is an essential restriction of ${\rm id}_{\{a_{1},\ldots,a_{i}\}}$. Hence,  ${\rm id}_{Q_{i,j}}={\rm id}_{\{a_{1},\ldots,a_{i}\}}$  as elements of ${\it Inv}_{k,1}$.

(2) In $M_{k,1}$ and in ${\it Inv}_{k,1}$ we have  ${\rm id}_{\{a_{1}\}}\equiv_{D}{\rm id}_{\{a_{1},\ldots,a_{k}\}}={\bf 1}$.*

Proof. (1) For any $m\geq k$ there exist $i,j\geq 0$ such that $1\leq i\leq k$ and $m=i+(k-1)j$. We consider the prefix code

Q_{i,j}\ =\ \{a_{2},\ldots,a_{i}\}\ \ \cup\ \ $\bigcup_{r=1}^{j-1}a_{1}^{r}(A-\{a_{1}\})\ \ \cup\ \ a_{1}^{j}A$.

It is easy to see that $Q_{i,j}$ is a prefix code, which is maximal iff $i=k$; see Fig. 1 below. Clearly, $|Q_{i,j}|=i+(k-1)j$. Since $Q_{i,j}$ contains $a_{1}^{j}A$, we can perform an essential extension of ${\rm id}_{Q_{i,j}}$ by replacing the table entries  $\{(a_{1}^{j}a_{1},a_{1}^{j}a_{1}),(a_{1}^{j}a_{2},a_{1}^{j}a_{2}),\ldots,$ $(a_{1}^{j}a_{k},a_{1}^{j}a_{k})\}$  by $(a_{1}^{j},a_{1}^{j})$. This replaces $Q_{i,j}$ by $Q_{i,j-1}$. So, ${\rm id}_{Q_{i,j}}$ can be essentially extended to ${\rm id}_{Q_{i,j-1}}$. By repeating this we find that ${\rm id}_{Q_{i,j}}$ is the same element (in $M_{k,1}$ and in ${\it Inv}_{k,1}$) as ${\rm id}_{Q_{i,0}}={\rm id}_{\{a_{1},\ldots,a_{i}\}}$.

(2) By essential restriction,  ${\rm id}_{\{a_{1}\}}={\rm id}_{\{a_{1}a_{1},a_{1}a_{2},\ldots,a_{1}a_{k}\}}$, in $M_{k,1}$ and in ${\it Inv}_{k,1}$. And by Lemma 2.7(2),  ${\rm id}_{\{a_{1}a_{1},a_{1}a_{2},\ldots,a_{1}a_{k}\}}\equiv_{D}$ ${\rm id}_{\{a_{1},\ldots,a_{k}\}}$; the latter, by essential extension, is 1.    $\Box$

\begin{picture}(150.0,100.0)\par\put(120.0,90.0){\makebox(0.0,0.0)[cc]{$\varepsilon$}} \put(123.0,87.0){\line(1,-2){2.0}} \put(136.0,79.0){\makebox(0.0,0.0)[cc]{$\{a_{2},\ldots,a_{i}\}$}} \put(124.0,87.0){\line(2,-1){8.0}} \put(128.0,83.0){\makebox(0.0,0.0)[cc]{$\ldots$}} \par\put(117.0,87.0){\line(-1,-2){2.1}} \put(114.0,80.0){\makebox(0.0,0.0)[cc]{$a_{1}$}} \par\put(117.0,77.0){\line(1,-2){2.2}} \put(133.0,70.0){\makebox(0.0,0.0)[cc]{$a_{1}\,(A-\{a_{1}\})$}} \put(118.0,77.0){\line(2,-1){8.0}} \put(122.0,73.0){\makebox(0.0,0.0)[cc]{$\ldots$}} \par\put(111.0,77.0){\line(-1,-2){2.0}} \put(107.0,70.0){\makebox(0.0,0.0)[cc]{.}} \put(106.0,68.0){\makebox(0.0,0.0)[cc]{.}} \put(105.0,66.0){\makebox(0.0,0.0)[cc]{.}} \put(103.0,62.0){\line(-1,-2){2.0}} \par\put(99.0,54.0){\makebox(0.0,0.0)[cc]{$a_{1}^{r}$}} \put(102.0,50.0){\line(1,-2){2.0}} \put(119.0,43.0){\makebox(0.0,0.0)[cc]{$a_{1}^{r}\,(A-\{a_{1}\})$}} \put(103.0,50.0){\line(2,-1){8.0}} \put(107.0,46.0){\makebox(0.0,0.0)[cc]{$\ldots$}} \par\put(96.0,50.0){\line(-1,-2){2.0}} \put(92.0,43.0){\makebox(0.0,0.0)[cc]{.}} \put(91.0,41.0){\makebox(0.0,0.0)[cc]{.}} \put(90.0,39.0){\makebox(0.0,0.0)[cc]{.}} \put(88.0,35.0){\line(-1,-2){2.0}} \par\put(84.0,26.0){\makebox(0.0,0.0)[cc]{$a_{1}^{j-1}$}} \put(85.0,22.0){\line(1,-2){2.7}} \put(104.0,13.0){\makebox(0.0,0.0)[cc]{$a_{1}^{j-1}\,(A-\{a_{1}\})$}} \put(86.0,22.0){\line(2,-1){10.0}} \put(91.0,17.0){\makebox(0.0,0.0)[cc]{$\ldots$}} \par\par\par\put(79.0,22.0){\line(-1,-2){2.2}} \put(73.0,13.0){\makebox(0.0,0.0)[cc]{$a_{1}^{j}$}} \par\put(71.0,9.0){\line(-1,-2){2.2}} \put(75.0,9.0){\line(1,-2){2.2}} \put(73.0,5.0){\makebox(0.0,0.0)[cc]{$\ldots$}} \par\put(73.0,1.0){\makebox(0.0,0.0)[cc]{$A$}} \par\par\par\par\put(20.0,-6.0){\makebox(0.0,0.0)[cc]{{\sf Fig.\ 1: \ The prefix tree of $Q_{i,j}$.}}} \par\end{picture}

\varepsilon$$\{a_{2},\ldots,a_{i}\}$$\ldots

$a_{1}$

a_{1}\,(A-\{a_{1}\})$$\ldots

...

a_{1}^{r}$$a_{1}^{r}\,(A-\{a_{1}\})$$\ldots

...

a_{1}^{j-1}$$a_{1}^{j-1}\,(A-\{a_{1}\})$$\ldots

$a_{1}^{j}$

$\ldots$

$A$

Fig. 1:  The prefix tree of $Q_{i,j}$.

Lemma 2.9

For all $\varphi,\psi\in{\it Inv}_{k,1}$: If  $\varphi\geq_{L(M_{k,1})}\psi$, where $\geq_{L(M_{k,1})}$ is the $L$-preorder of $M_{k,1}$, then  $\varphi\geq_{L(I_{k,1})}\psi$, where $\geq_{L(I_{k,1})}$ is the $L$-preorder of ${\it Inv}_{k,1}$.

The same holds with $\geq_{L}$ replaced by $\equiv_{L}$, $\geq_{R}$, $\equiv_{R}$, $\equiv_{D}$, $\geq_{J}$ and $\equiv_{J}$.

Proof. If $\psi=\alpha\,\varphi$ for some $\alpha\in M_{k,1}$ then let us define $\alpha^{\prime}$ by  $\alpha^{\prime}=\alpha\ {\rm id}_{{\rm Im}(\varphi)}$. Then we have:  $\psi\ \varphi^{-1}=\alpha\ \varphi\ \varphi^{-1}=$ $\alpha\ {\rm id}_{{\rm Im}(\varphi)}=\alpha^{\prime}$, hence $\alpha^{\prime}\in{\it Inv}_{k,1}$ (since $\varphi,\psi\in{\it Inv}_{k,1}$). Moreover, $\alpha^{\prime}\,\varphi=$ $\alpha\ {\rm id}_{{\rm Im}(\varphi)}\ \varphi=\alpha\,\varphi=$ $\psi$.    $\Box$

So far our Lemmas imply that in $M_{k,1}$ and in ${\it Inv}_{k,1}$, every non-zero element is $\equiv_{D}$ to one of the $k-1$ elements ${\rm id}_{\{a_{1},\ldots,a_{i}\}}$, for $i=1,\ldots,k-1$. Moreover the Lemmas show that if two elements of $M_{k,1}$ (or of ${\it Inv}_{k,1}$) are given by tables $\varphi_{1}:P_{1}\to Q_{1}$ and $\varphi_{2}:P_{2}\to Q_{2}$, where $P_{1}$, $Q_{1}$, $P_{2}$ and $Q_{2}$ are finite prefix codes, then we have:  If  $|Q_{1}|\equiv|Q_{2}|$ mod $(k-1)$  then  $\varphi_{1}\equiv_{D}\varphi_{2}$.

We still need to prove the converse of this. It is sufficient to prove the converse for ${\it Inv}_{k,1}$, by Lemma 2.9 and because every element of $M_{k,1}$ is $\equiv_{D}$ to an element of ${\it Inv}_{k,1}$ (namely ${\rm id}_{\{a_{1},\ldots,a_{i}\}}$).

Lemma 2.10

Let $\varphi,\psi\in{\it Inv}_{k,1}$. If $\varphi\equiv_{D}\psi$ in ${\it Inv}_{k,1}$, then  $\|\varphi\|\equiv\|\psi\|$ mod $(k-1)$.

Proof. (1) We first prove that if $\varphi\equiv_{L}\psi$ then $|{\rm domC}(\varphi)|\equiv|{\rm domC}(\psi)|$ mod $(k-1)$.

By definition, $\varphi\equiv_{L}\psi$ iff $\varphi=\beta\,\psi$ and $\psi=\alpha\,\varphi$ for some $\alpha,\beta\in{\it Inv}_{k,1}$. By Lemma 1.9  there are restrictions $\beta^{\prime}$ and $\psi^{\prime}$ of $\beta$, respectively $\psi$, and an essential restriction $\Phi$ of $\varphi$ such that:

$\Phi=\beta^{\prime}\circ\psi^{\prime}$,  and  ${\rm Dom}(\beta^{\prime})={\rm Im}(\psi^{\prime})$.

It follows that ${\rm Dom}(\Phi)\subseteq{\rm Dom}(\psi^{\prime})$, since if $\psi^{\prime}(x)$ is not defined then $\Phi(x)=\beta^{\prime}\circ\psi^{\prime}(x)$ is not defined either. Similarly, there is an essential restriction $\Psi$ of $\psi$ and a restriction $\varphi^{\prime}$ of $\varphi$ and such that ${\rm Dom}(\Psi)\subseteq{\rm Dom}(\varphi^{\prime})$.

Thus, the restriction of both $\varphi$ and $\psi$ to the intersection ${\rm Dom}(\Phi)\cap{\rm Dom}(\Psi)$ yields restrictions $\varphi^{\prime\prime}$, respectively $\psi^{\prime\prime}$ such that ${\rm Dom}(\varphi^{\prime\prime})={\rm Dom}(\psi^{\prime\prime})$.

Claim:  $\varphi^{\prime\prime}$ and $\psi^{\prime\prime}$ are essential restrictions of $\varphi$, respectively $\psi$.

Indeed, every right ideal $R$ of $A^{*}$ that intersects ${\rm Dom}(\psi)$ also intersects ${\rm Dom}(\Psi)$ (since $\Psi$ is an essential restriction of $\psi$). Since ${\rm Dom}(\Psi)\subseteq{\rm Dom}(\varphi^{\prime})\subseteq$ ${\rm Dom}(\varphi)$, it follows that $R$ also intersects ${\rm Dom}(\varphi)$. Moreover, since $\Phi$ is an essential restriction of $\varphi$, $R$ also intersects ${\rm Dom}(\Phi)$. Thus, ${\rm Dom}(\Phi)$ is essential in ${\rm Dom}(\psi)$. Since ${\rm Dom}(\Psi)$ is also essential in ${\rm Dom}(\psi)$, it follows that ${\rm Dom}(\Phi)\cap{\rm Dom}(\Psi)$ is essential in ${\rm Dom}(\psi)$; indeed, in general, the intersection of two right ideals $R_{1},R_{2}$ that are essential in a right ideal $R_{3}$, is essential in $R_{3}$ (this is a special case of Lemma 1.8). This means that $\psi^{\prime\prime}$ is an essential restriction of $\psi$. Similarly, one proves that $\varphi^{\prime\prime}$ is an essential restriction of $\varphi$.    [This proves the Claim.]

So, $\varphi^{\prime\prime}$ and $\psi^{\prime\prime}$ are essential restrictions such that ${\rm Dom}(\varphi^{\prime\prime})={\rm Dom}(\psi^{\prime\prime})$. Hence, ${\rm domC}(\varphi^{\prime\prime})={\rm domC}(\psi^{\prime\prime})$; Proposition 2.4 then implies that  $|{\rm domC}(\varphi)|\equiv|{\rm domC}(\varphi^{\prime\prime})|=$ $|{\rm domC}(\psi^{\prime\prime})|\equiv|{\rm domC}(\psi)|$  mod  $(k-1)$.

(2) Next, let us prove that if $\varphi\equiv_{R}\psi$ then $|{\rm imC}(\varphi)|\equiv|{\rm imC}(\psi)|$ mod $(k-1)$. In ${\it Inv}_{k,1}$ we have $\varphi\equiv_{R}\psi$ iff $\varphi^{-1}\equiv_{L}\psi^{-1}$. Also, ${\rm imC}(\varphi)={\rm domC}(\varphi^{-1})$. Hence, (2) follows from (1).

The Lemma now follows from (1) and (2), since for elements of ${\it Inv}_{k,1}$, $|{\rm imC}(\varphi)|=|{\rm domC}(\varphi)|=$ $\|\varphi\|$, and since the $D$-relation is the composite of the $L$-relation and the $R$-relation.    $\Box$

Proof of Theorem 2.5. We saw already (in the observations before Lemma 2.10 and in the preceding Lemmas) that for $\varphi_{1}:P_{1}\to Q_{1}$ and $\varphi_{2}:P_{2}\to Q_{2}$ (where $P_{1}$, $Q_{1}$, $P_{2}$ and $Q_{2}$ are non-empty finite prefix codes) we have:  If  $|Q_{1}|\equiv|Q_{2}|$ mod $(k-1)$  then  $\varphi_{1}\equiv_{D}\varphi_{2}$. In particular, when $|Q_{1}|\equiv i$ mod $(k-1)$ then $\varphi_{1}\equiv_{D}{\rm id}_{\{a_{1},\ldots,a_{i}\}}$.

It follows from Lemma 2.10 that the elements ${\rm id}_{\{a_{1},\ldots,a_{i}\}}$ (for $i=1,\ldots,k-1$) are all in different $D$-classes.    $\Box$

So far we have characterized the $D$- and $J$-relations of $M_{k,1}$ and ${\it Inv}_{k,1}$. We leave the general study of the Green relations of $M_{k,1}$, ${\it Inv}_{k,1}$, and the other Thompson-Higman monoids for future work. The main result of this paper, to be proved next, is that the Thompson-Higman monoids $M_{k,1}$ and ${\it Inv}_{k,1}$ are finitely generated and that their word problem over any finite generating set is in P.

3 Finite generating sets

We will show that ${\it Inv}_{k,1}$ and $M_{k,1}$ are finitely generated. An application of the latter fact is that a finite generating set of $M_{k,1}$ can be used to build combinational circuits for finite boolean functions that do not have fixed-length inputs or outputs. In engineering, non-fixed length inputs or outputs make sense, for example, if the inputs or outputs are handled sequentially, and if the possible input strings form a prefix code.

First we need some more definitions about prefix codes. The prefix tree of a prefix code $P\subset A^{*}$ is, by definition, a tree whose vertex set is the set of all the prefixes of the elements of $P$, and whose edge set is $\{(x,xa):a\in A,\ xa$ is a prefix of some element of $P\}$. The tree is rooted, with root $\varepsilon$ (the empty word). Thus, the prefix tree of $P$ is a subtree of the tree of $A^{*}$. The set of leaves of the prefix tree of $P$ is $P$ itself. The vertices that are not leaves are called internal vertices. We will say more briefly an “internal vertex of $P$” instead of internal vertex of the prefix tree of $P$. An internal vertex has between 1 and $k$ children; an internal vertex is called saturated iff it has $k$ children.

One can prove easily that a prefix code $P$ is maximal iff every internal vertex of the prefix tree of $P$ is saturated. Hence, every prefix code $P$ can be embedded in a maximal prefix code (which is finite when $P$ is finite), obtained by saturating the prefix tree of $P$. Moreover we have:

Lemma 3.1

For any two finite non-maximal prefix codes $P_{1},P_{2}\subset A^{*}$ there are finite maximal prefix codes $P^{\prime}_{1},P^{\prime}_{2}\subset A^{*}$ such that $P_{1}\subset P^{\prime}_{1}$, $P_{2}\subset P^{\prime}_{2}$, and $|P^{\prime}_{1}|=|P^{\prime}_{2}|$.

Proof. First we saturate $P_{1}$ and $P_{2}$ to obtain two maximal prefix codes $P^{\prime\prime}_{1}$ and $P^{\prime\prime}_{2}$ such that $P_{1}\subset P^{\prime\prime}_{1}$, and $P_{2}\subset P^{\prime\prime}_{2}$. If $|P^{\prime\prime}_{1}|\neq|P^{\prime\prime}_{2}|$ (e.g., if $|P^{\prime\prime}_{1}|<|P^{\prime\prime}_{2}|$) then $|P^{\prime\prime}_{1}|$ and $|P^{\prime\prime}_{2}|$ differ by a multiple of $k-1$ (by Prop. 2.4). So, in order to make $|P^{\prime\prime}_{1}|$ equal to $|P^{\prime\prime}_{2}|$ we repeat the following (until $|P^{\prime\prime}_{1}|=|P^{\prime\prime}_{2}|$): consider a leaf of the prefix tree of $P^{\prime\prime}_{1}$ that does not belong to $P_{1}$, and attach $k$ children at that leaf; now this leaf is no longer a leaf, and the net increase in the number of leaves is $k-1$.    $\Box$

Lemma 3.2

Let $P$ and $Q$ be finite prefix codes of $A^{*}$ with $|P|=|Q|$. If $P$ and $Q$ are both maximal prefix codes, or if both are non-maximal, then there is an element of $G_{k,1}$ that maps $P$ onto $Q$. On the other hand, if one of $P$ and $Q$ is maximal and the other one is not maximal, then there is no element of $G_{k,1}$ that maps $P$ onto $Q$.

Proof. When $P$ and $Q$ are both maximal then any one-to-one correspondence between $P$ and $Q$ is an element of $G_{k,1}$.

When $P$ and $Q$ are both non-maximal, we use Lemma 3.1 above to find two maximal prefix codes $P^{\prime}$ and $Q^{\prime}$ such that $P\subset P^{\prime}$, $Q\subset Q^{\prime}$, and $|P^{\prime}|=|Q^{\prime}|$. Consider now any bijection from $P^{\prime}$ onto $Q^{\prime}$ that is also a bijection from $P$ onto $Q$. This is an element of $G_{k,1}$.

When $P$ is maximal and $Q$ is non-maximal, then every element $\varphi\in M_{k,1}$ that maps $P$ onto $Q$ will satisfy ${\rm domC}(\varphi)=P$; since $\varphi$ is onto $Q$, we have ${\rm imC}(\varphi)=Q$. Hence, $\varphi\not\in G_{k,1}$ since ${\rm imC}(\varphi)$ is a non-maximal prefix code. A similar reasoning shows that no element of $G_{k,1}$ maps $P$ onto $Q$ if $P$ is non-maximal and $Q$ is maximal.    $\Box$

Notation: For $u,v\in A^{*}$, the element of ${\it Inv}_{k,1}$ with one-element domain code $\{u\}$ and one-element image code $\{v\}$ is denoted by $(u\mapsto v)$. When $(u\mapsto v)$ is composed with itself $j$ times the resulting element of ${\it Inv}_{k,1}$ is denoted by $(u\mapsto v)^{j}$.

Lemma 3.3

(1)*   For all $j>0$:   $(a_{1}\mapsto a_{1}a_{1})^{j}\ =\ (a_{1}\mapsto a_{1}^{j+1})$.

(2)  Let  $S={\{a_{1}^{j}a_{1},a_{1}^{j}a_{2},\ldots,a_{1}^{j}a_{i}\}}$, for some $1\leq i\leq k-1$, $0\leq j$. Then ${\rm id}_{S}$ is generated by the $k+1$ elements  \{(a_{1}\mapsto a_{1}a_{1}),\ (a_{1}a_{1}\mapsto a_{1})\}\ \cup\ $\{{\rm id}_{\{a_{1}a_{1},\ a_{1}a_{2},\ \ldots,\ a_{1}a_{i}\}}:1\leq i\leq k-1\}$.

(3)  For all $j\geq 2$:   $(\varepsilon\mapsto a_{1}^{j})(.)\ =\ (a_{1}\mapsto a_{1}a_{1})^{j-1}\,\cdot$ $(\varepsilon\mapsto a_{1})(.)$.*

Proof. (1) We prove by induction that  $(a_{1}\mapsto a_{1}a_{1})^{j}=(a_{1}\mapsto a_{1}a_{1}^{j})$  for all $j\geq 1$.

Indeed,  $(a_{1}\mapsto a_{1}a_{1})^{j+1}(.)=(a_{1}\mapsto a_{1}a_{1})\,\cdot$ $(a_{1}\mapsto a_{1}a_{1}^{j})(.)$,  and by essential restriction this is

\left(\begin{array}[]{l|l}a_{1}a_{1}^{j}&\ a_{1}w\ \ \ \ (w\in A^{j}-\{a_{1}^{j}\})\\ a_{1}a_{1}a_{1}^{j}&\ a_{1}a_{1}w\end{array}\right)\cdot(a_{1}\mapsto a_{1}a_{1}^{j})(.) $\ =\ (a_{1}\mapsto a_{1}a_{1}a_{1}^{j})(.)$.

(2) For $S={\{a_{1}^{j}a_{1},a_{1}^{j}a_{2},\ldots,a_{1}^{j}a_{i}\}}$ we have

{\rm id}_{S}\ =\ \left(\begin{array}[]{l|l|l|l}a_{1}a_{1}&a_{1}a_{2}&\ \ldots&a_{1}a_{i}\\ a_{1}^{j}a_{1}&a_{1}^{j}a_{2}&\ \ldots&a_{1}^{j}a_{i}\end{array}\right)\cdot \left(\begin{array}[]{l|l|l|l}a_{1}^{j}a_{1}&a_{1}^{j}a_{2}&\ \ldots&a_{1}^{j}a_{i}\\ a_{1}a_{1}&a_{1}a_{2}&\ \ldots&a_{1}a_{i}\end{array}\right)(.)

and

\left(\begin{array}[]{l|l|l|l}a_{1}a_{1}&a_{1}a_{2}&\ \ldots&a_{1}a_{i}\\ a_{1}^{j}a_{1}&a_{1}^{j}a_{2}&\ \ldots&a_{1}^{j}a_{i}\end{array}\right)

\ =\ \left(\begin{array}[]{l|l|l|l|l|l|l}a_{1}a_{1}&a_{1}a_{2}&\ \ldots&a_{1}a_{i}&a_{1}a_{i+1}&\ \ldots&a_{1}a_{k}\\ a_{1}^{j}a_{1}&a_{1}^{j}a_{2}&\ldots&a_{1}^{j}a_{i}&a_{1}^{j}a_{i+1}&\ \ldots&a_{1}^{j}a_{k}\end{array}\right)\cdot ${\rm id}_{\{a_{1}a_{1},\ a_{1}a_{2},\ \ldots,\ a_{1}a_{i}\}}(.)$

$\ =\ (a_{1}\mapsto a_{1}^{j})\cdot$ ${\rm id}_{\{a_{1}a_{1},\ a_{1}a_{2},\ \ldots,\ a_{1}a_{i}\}}$ \ =\ $(a_{1}\mapsto a_{1}a_{1})^{j-1}\cdot$ ${\rm id}_{\{a_{1}a_{1},\ a_{1}a_{2},\ \ldots,\ a_{1}a_{i}\}}$.

The map id${}_{\{a_{1}a_{1}\}}$ is redundant as a generator since $(a_{1}a_{1}\mapsto a_{1}a_{1})=(a_{1}a_{1}\mapsto a_{1})\,$ $(a_{1}\mapsto a_{1}a_{1})(.)$.

(3) By (1) we have  $(\varepsilon\mapsto a_{1}^{j})=(a_{1}\mapsto a_{1}^{j})\,\cdot$ $(\varepsilon\mapsto a_{1})(.)$, and  $(a_{1}\mapsto a_{1}^{j})=$ $(a_{1}\mapsto a_{1}a_{1})^{j-1}$.    $\Box$

Theorem 3.4

The inverse monoid ${\it Inv}_{k,1}$ is finitely generated.

Proof. Our strategy for finding a finite generating set for ${\it Inv}_{k,1}$ is as follows: We will use the fact that the Thompson-Higman group $G_{k,1}$ is finitely generated. Hence, if $\varphi\in{\it Inv}_{k,1}$,   $g_{1},g_{2}\in G_{k,1}$, and if $g_{2}\varphi g_{1}$ can be expressed as a product $p$ over a fixed finite set of elements of ${\it Inv}_{k,1}$, then it follows that $\varphi=g_{2}^{-1}p\,g_{1}^{-1}$ can also be expressed as a product over a fixed finite set of elements of ${\it Inv}_{k,1}$. We assume that a finite generating set for $G_{k,1}$ has been chosen.

For any element $\varphi\in{\it Inv}_{k,1}$ with domain code domC$(\varphi)=P$ and image code imC$(\varphi)=Q$, we distinguish four cases, depending on the maximality or non-maximality of $P$ and $Q$.

(1) If $P$ and $Q$ are both maximal prefix codes then $\varphi\in G_{k,1}$, and we can express $\varphi$ over a finite fixed generating set of $G_{k,1}$.

(2) Assume $P$ and $Q$ are both non-maximal prefix codes. By Lemma 3.1 there are finite maximal prefix codes $P^{\prime},Q^{\prime}$ such that $P\subset P^{\prime}$, $Q\subset Q^{\prime}$, and $|P^{\prime}|=|Q^{\prime}|$; and by Lemma 2.6, $|P^{\prime}|=|Q^{\prime}|=1+(k-1)N$ for some $N\geq 0$. Consider the following maximal prefix code $C$, of cardinality $|P^{\prime}|=|Q^{\prime}|=1+(k-1)N$:

$C\ =\ \bigcup_{r=0}^{N-2}a_{1}^{r}(A-\{a_{1}\})\ \ \cup\ a_{1}^{N-1}A$.

The maximal prefix code $C$ is none other than the code $Q_{i,j}$ when $i=k$ and $j=N-1$   (introduced in the proof of Lemma 2.8, Fig. 1). The elements $g_{1}:C\to P^{\prime}$ and $g_{2}:Q^{\prime}\to C$ of $G_{k,1}$ can be chosen so that $\psi=g_{2}\varphi g_{1}(.)$ is a partial identity with ${\rm domC}(\psi)={\rm imC}(\psi)\subset C$ consisting of the $|P|$ first elements of $C$ in the dictionary order. So, $\psi$ is the identity map restricted to these $|P|$ first elements of $C$, and $\psi$ is undefined on the rest of $C$. To describe ${\rm domC}(\psi)={\rm imC}(\psi)$ in more detail, let us write $|P|=i+(k-1)\,\ell$, for some $i,\ell$ with $1\leq i<k$ and $0\leq\ell\leq N-1$. Then

{\rm domC}(\psi)={\rm imC}(\psi)\ =\ a_{1}^{N-1}A\ \cup\ \bigcup_{r=j+1}^{N-2}a_{1}^{r}(A-\{a_{1}\})\ \ \cup\ $a_{1}^{j}\ \{a_{2},\ldots,a_{i}\}$.

where $j=N-1-\ell$. Since  $\psi={\rm id}_{{\rm domC}(\psi)}$, we claim:

By essential maximal extension

$\psi\ =\ {\rm id}_{S}$  (as elements of ${\it Inv}_{k,1}$),   where  $S={\{a_{1}^{j}a_{1},a_{1}^{j}a_{2},\ldots,a_{1}^{j}a_{i}\}}$,

with $i,j$ as in the description of ${\rm domC}(\psi)={\rm imC}(\psi)$ above, i.e., $1<i<k$,  $N-1\geq j=N-1-\ell\geq 0$, and $|P|=i+(k-1)\,\ell$.

Indeed, if $|P|<k$ then $S$ is just ${\rm domC}(\psi)$, with $i=|P|$, and $\ell=0$ (hence $j=N-1$). If $|P|\geq k$ then the maximum essential extension of $\psi$ will replace the $1+(k-1)\,\ell$ elements  $a_{1}^{N-1}A\ \cup\ \ \bigcup_{r=N-j+1}^{N-2}a_{1}^{r}(A-\{a_{1}\})$  by the single element $a_{1}^{N-\ell+1}=a_{1}^{j+1}$. What remains is the set

$S\ =\ \{a_{1}^{j+1}\}\ \cup\ a_{1}^{j}\,\{a_{2},\ldots,a_{i}\}$.

Finally, by Lemma 3.3, ${\rm id}_{S}$ (where $S={\{a_{1}^{j}a_{1},a_{1}^{j}a_{2},\ldots,a_{1}^{j}a_{i}\}}$) can be generated by the $k+1$ elements   \{(a_{1}\mapsto a_{1}a_{1}),\ (a_{1}a_{1}\mapsto a_{1})\}\ \cup\ $\{{\rm id}_{\{a_{1}a_{1},\ a_{1}a_{2},\ \ldots,\ a_{1}a_{i}\}}:$ $1\leq i\leq k-1\}$.

(3) Assume $P$ is a maximal prefix code and $Q$ is non-maximal. Let $Q^{\prime}$ be the finite maximal prefix code obtained by saturating the prefix tree of $Q$. Then $Q\subset Q^{\prime}$,  $|Q^{\prime}|=1+(k-1)N^{\prime}$, and $|P|=1+(k-1)N$ for some $N^{\prime}>N\geq 0$. We consider the maximal prefix codes $C$ and $C^{\prime}$ as defined in the proof of (2), using $N^{\prime}$ for defining $C^{\prime}$. We can choose $g_{1}:C\to P$ and $g_{2}:Q^{\prime}\to C^{\prime}$ in $G_{k,1}$ so that $\psi=g_{2}\varphi g_{1}(.)$ is the dictionary-order preserving map that maps $C$ to the first $|C|$ elements of $C^{\prime}$. So we have

${\rm domC}(\psi)=C$,  and

${\rm imC}(\psi)=S_{0}$ , where $S_{0}\subset C^{\prime}$ consist of the $|C|$ first elements of $C^{\prime}$, in dictionary order.

Since $|C|=1+(k-1)\,N$, we can describe $S_{0}$ in more detail by

$S_{0}\ =\ \bigcup_{r=N^{\prime}-N}^{N^{\prime}-2}a_{1}^{r}(A-\{a_{1}\})\ \ \cup\ a_{1}^{N^{\prime}-1}A$.

Next, by essential maximal extension we now obtain  $\psi\ =\ (\varepsilon\mapsto a_{1}^{N^{\prime}-N})$.

Indeed, we saw that $|P|=1+(k-1)\,N$. If $|P|=1$ then $P=\{\varepsilon\}$, and $\psi\ =\ (\varepsilon\mapsto a_{1}^{N^{\prime}})$. If $|P|\geq k$ then maximum essential extension of $\psi$ will replace all the elements of $C$ by the single element $\varepsilon$, and it will replace all the elements of $S_{0}$ by the single element $a_{1}^{N^{\prime}-N}$.

Finally, by Lemma 3.3, $(\varepsilon\mapsto a_{1}^{N^{\prime}-N})$ is generated by the two elements $(\varepsilon\mapsto a_{1})$ and $(a_{1}\mapsto a_{1}a_{1})$.

(4) The case where $P$ is a non-maximal maximal prefix code and $Q$ is maximal can be derived from case (3) by taking the inverses of the elements from case (3).    $\Box$

Theorem 3.5

The monoid $M_{k,1}$ is finitely generated.

Proof. Let $\varphi:P\to Q$ be the table of any element of $M_{k,1}$, mapping $P$ onto $Q$, where $P,Q\subset A^{*}$ are finite prefix codes. The map described by the table is total and surjective, so if $|P|=|Q|$ (and in particular, if $\varphi$ is the empty map) then $\varphi\in{\it Inv}_{k,1}$, hence $\varphi$ can be expressed over the finite generating set of ${\it Inv}_{k,1}$. In the rest of the proof we assume $|P|>|Q|$. The main observation is the following.

Claim.  $\varphi$ can be written as the composition of finitely many elements $\varphi_{i}\in M_{k,1}$ with tables $P_{i}\to Q_{i}$ such that $0\leq|P_{i}|-|Q_{i}|\leq 1$.

Proof of the Claim: We use induction on $|P|-|Q|$. There is nothing to prove when $|P|-|Q|\leq 1$, so we assume now that $|P|-|Q|\geq 2$.

If $\varphi(x_{1})=\varphi(x_{2})=\varphi(x_{3})=y_{1}$ for some $x_{1},x_{2},x_{3}\in P$ (all three being different) and $y_{1}\in Q$, then we can write $\varphi$ as a composition $\varphi(.)=\psi_{2}\circ\psi_{1}(.)$, as follows. The map $\psi_{1}:P\longrightarrow P-\{x_{1}\}$ is defined by $\psi_{1}(x_{1})=\psi_{1}(x_{2})=x_{2}$, and acts as the identity everywhere else on $P$. The map $\psi_{2}:P-\{x_{1}\}\longrightarrow Q$ is defined by $\psi_{2}(x_{2})=\psi_{2}(x_{3})=y_{1}$, and acts in the same way as $\varphi$ everywhere else on $P-\{x_{1}\}$. Then for $\psi_{1}$ we have   $|P|-|P-\{x_{1}\}|\ <\ |P|-|Q|$,  and for $\psi_{2}$ we have   $|P-\{x_{1}\}|-|Q|\ <\ |P|-|Q|$.

If $\varphi(x_{1})=\varphi(x_{2})=y_{1}$ and $\varphi(x_{3})=\varphi(x_{4})=y_{2}$ for some $x_{1},x_{2},x_{3},x_{4}\in P$ (all four being different) and $y_{1},y_{2}\in Q$ ($y_{1}\neq y_{2}$), then we can write $\varphi$ as a composition $\varphi(.)=\psi_{2}\circ\psi_{1}(.)$, as follows. First the map $\psi_{1}:P\longrightarrow P-\{x_{1}\}$ is defined by $\psi_{1}(x_{1})=\psi_{1}(x_{2})=x_{2}$, and acts as the identity everywhere else on $P$. Second, the map $\psi_{2}:P-\{x_{1}\}\longrightarrow Q$ is defined by $\psi_{2}(x_{2})=y_{1}$ and $\psi_{2}(x_{3})=\psi_{2}(x_{4})=y_{2}$, and acts like $\varphi$ everywhere else on $P-\{x_{1}\}$. Again, for $\psi_{1}$ we have    $|P|-|P-\{x_{1}\}|\ <\ |P|-|Q|$  and for $\psi_{2}$ we have    $|P-\{x_{1}\}|-|Q|\ <\ |P|-|Q|$.      [End, proof of the Claim.]

Because of the Claim we now only need to consider elements $\varphi\in M_{k,1}$ with tables $P\to Q$ such that the prefix codes $P,Q$ satisfy $|P|=|Q|+1$. We denote $P=\{p_{1},\ldots,p_{n}\}$ and $Q=\{q_{1},\ldots,q_{n-1}\}$, with $\varphi(p_{j})=q_{j}$ for $1\leq j\leq n-1$, and $\varphi(p_{n-1})=\varphi(p_{n})=q_{n-1}$. We define the following prefix code $C$ with $|C|=|P|$:

$\bullet$  if $|P|=i\leq k$ then   $C\ =\ \{a_{1},\ldots,a_{i}\}$;  note that $i\geq 2$, since $|P|>|Q|>0$;

$\bullet$  if $|P|>k$ then   C\ =\ \{a_{2},\ldots,a_{i}\}\ \cup\ $\bigcup_{r=1}^{j-1}a_{1}^{r}(A-\{a_{1}\})\ \ \cup\ \ a_{1}^{j}A$,

where $i,j$ are such that $|P|=i+(k-1)j$,  $2\leq i\leq k$, and $1\leq j$ (see Fig. 1). Let us write $C$ in increasing dictionary order as $C=\{c_{1},\ldots,c_{n}\}$. The last element of $C$ in the dictionary order is thus $c_{n}=a_{i}$.

We now write $\varphi(.)=\psi_{3}\,\psi_{2}\,\psi_{1}(.)$ where $\psi_{1}$, $\psi_{2}$, $\psi_{3}$ are as follows:

$\bullet$  $\psi_{1}:P\longrightarrow C$  is bijective and is defined by $p_{j}\mapsto c_{j}$ for $1\leq j\leq n$;

$\bullet$  $\psi_{2}:C\longrightarrow C-\{a_{i}\}$  is the identity map on $\{c_{1},\ldots,c_{n-1}\}$, and $\psi_{2}(c_{n})=c_{n-1}$.

$\bullet$  $\psi_{3}:C-\{a_{i}\}\longrightarrow Q$  is bijective and is defined by $c_{j}\mapsto q_{j}$ for $1\leq j\leq n-1$.

It follows that $\psi_{1}$ and $\psi_{3}$ can be expressed over the finite generating set of ${\it Inv}_{k,1}$. On the other hand, $\psi_{2}$ has a maximum essential extension, as follows.

$\bullet$  If  $2\leq|P|=i\leq k$  then

\psi_{2}\ =\ \left(\begin{array}[]{l|l|l|l|l}a_{1}&\ \ldots&a_{i-2}&a_{i-1}&a_{i}\\ a_{1}&\ \ldots&a_{i-2}&a_{i-1}&a_{i-1}\end{array}\right)\ =\ \left(\begin{array}[]{l|l}{\rm id}_{\{a_{1},\ \ldots,\ a_{i-1}\}}&a_{i}\\ &a_{i-1}\end{array}\right).

$\bullet$  If  $|P|=i+(k-1)j>k$  and if $i>2$ then, after maximal essential extension, $\psi_{2}$ also becomes

\max(\psi_{2})\ =\ \left(\begin{array}[]{l|l}{\rm id}_{\{a_{1},\ \ldots,\ a_{i-1}\}}&a_{i}\\ &a_{i-1}\end{array}\right).

$\bullet$  If  $|P|=i+(k-1)j>k$  and if $i=2$ then, after essential extensions,

\max(\psi_{2})\ =\ \left(\begin{array}[]{l|l|l|l|l|l}a_{1}a_{1}&\ \ldots&a_{1}a_{k-2}&a_{1}a_{k-1}&a_{1}a_{k}&a_{2}\\ a_{1}a_{1}&\ \ldots&a_{1}a_{k-2}&a_{1}a_{k-1}&a_{1}a_{k}&a_{1}a_{k}\end{array}\right)\ =\ \left(\begin{array}[]{l|l}{\rm id}_{a_{1}A}&a_{2}\\ &a_{1}a_{k}\end{array}\right)\ =\ \left(\begin{array}[]{l|l}a_{1}&a_{2}\\ a_{1}&a_{1}a_{k}\end{array}\right).

In summary, we have factored $\varphi$ over a finite set of generators of ${\it Inv}_{k,1}$ and $k$ additional generators in $M_{k,1}$.    $\Box$

Factorization algorithm: The proofs of Theorems 3.4 and 3.5 are constructive; they provide algorithms that, given $\varphi\in{\it Inv}_{k,1}$ or $\in M_{k,1}$, output a factorization of $\varphi$ over the finite generating set of ${\it Inv}_{k,1}$, respectively $M_{k,1}$.

In [20] (p. 49) Higman introduces a four-element generating set for $G_{2,1}$; a special property of these generators is that their domain codes and their image codes only contain words of length $\leq 2$, and that \big{|}\,|\gamma(x)|-|x|\,\big{|}\leq 1 for every generator $\gamma$ and every $x\in{\rm domC}(\gamma)$. The generators in the finite generating set of $M_{k,1}$ that we introduced above also have those properties. Thus we obtain:

Corollary 3.6

The monoid $M_{2,1}$ has a finite generating set such that all the generators have the following property: The domain codes and the image codes only contain words of length $\leq 2$, and  \big{|}\,|\gamma(x)|-|x|\,\big{|}\leq 1 for every generator $\gamma$ and every $x\in{\rm domC}(\gamma)$.      $\Box$

For reference we list an explicit finite generating set for $M_{2,1}$. It consists, first, of the Higman generators of $G_{2,1}$ ([20] p. 49):

Not $=$ \left(\begin{array}[]{l|l}0&1\\ 1&0\end{array}\right),   $(01\leftrightarrow 1)=$ \left(\begin{array}[]{r|r|r}00&01&1\\ 00&1&01\end{array}\right),   $(0\leftrightarrow 10)=$ \left(\begin{array}[]{r|r|r}0&10&11\\ 10&0&11\end{array}\right),   and

$\tau_{1,2}=$ \left(\begin{array}[]{r|r|r|r}00&01&10&11\\ 00&10&01&11\end{array}\right);

the additional generators for ${\it Inv}_{2,1}$:

$(\varepsilon\to 0)$,   $(0\to\varepsilon)$,   $(0\to 00)$,   $(00\to 0)$;

the additional generators for $M_{2,1}$:

\left(\begin{array}[]{l|l}0&1\\ 0&0\end{array}\right),  and   \left(\begin{array}[]{r|r}0&1\\ 0&01\end{array}\right) =\left(\begin{array}[]{r|r|r}00&01&1\\ 00&01&01\end{array}\right) .

Observe that Higman’s generators of $G_{k,1}$ (in [20] p. 27) have domain and image codes with at most 3 internal vertices. We observe that the additional generators that we introduced for ${\it Inv}_{k,1}$ and $M_{k,1}$ have domain and image codes have at most 2 internal vertices.

The following problem remains open: Are ${\it Inv}_{k,1}$ and $M_{k,1}$ finitely presented?

4 The word problem of the Thompson-Higman monoids

We saw that the Thompson-Higman monoid $M_{k,1}$ is finitely generated. We want to show now that the word problem of $M_{k,1}$ over any finite generating set can be decided in deterministic polynomial time, i.e., it belongs to the complexity class P. 222  This section has been revised in depth, to correct errors.

In [4] it was shown that the word problem of the Thompson-Higman group $G_{k,1}$ over any finite generating set is in P. In fact, it is in the parallel complexity class AC1 [4], and it is co-context-free [25]. In [5] it was shown that the word problem of the Thompson-Higman group $G_{k,1}$ over the infinite generating set  $\Gamma_{k,1}\cup\{\tau_{i,i+1}:i>0\}$  is coNP-complete, where $\Gamma_{k,1}$ is any finite generating set of $G_{k,1}$; the position transposition $\tau_{i,i+1}\in G_{k,1}$ has  ${\rm domC}(\tau_{i,i+1})={\rm imC}(\tau_{i,i+1})=A^{i+1}$, and is defined by  $u\alpha\beta\mapsto u\beta\alpha$  for all letters $\alpha,\beta\in A$ and all words $u\in A^{i-1}$. We will see below that the word problem of $M_{k,1}$ over $\Gamma_{k,1}\cup\{\tau_{i,i+1}:i>0\}$ is also coNP-complete, where $\Gamma_{k,1}$ is any finite generating set of $M_{k,1}$.

4.1 The image code formula

Our proof in [4] that the word problem of the Thompson-Higman group $G_{k,1}$ (over any finite generating set) is in P, was based on the following fact (the table size formula):

$\forall\varphi,\psi\in G_{k,1}$:   $\|\psi\circ\varphi\|\leq\|\psi\|+\|\varphi\|$.

Here $\|\varphi\|$ denotes the table size of $\varphi$, i.e., the cardinality of ${\rm domC}(\varphi)$. See Proposition 3.5, Theorem 3.8, and Proposition 4.2 in [4]. In $M_{k,1}$ the above formula does not hold in general, as the following example shows. We give some definitions and notation first.

Definition 4.1

For any finite set $S\subseteq A^{*}$ we denote the length of the longest word in $S$ by $\ell(S)$. The cardinality of $S$ is denoted by $|S|$.

The table of a right-ideal morphism $\varphi$ is the set $\{(x,\varphi(x)):x\in{\rm domC}(\varphi)\}$.

Proposition 4.2

For every $n>0$ there exists $\Phi_{n}=\varphi_{2}^{n-1}\varphi_{1}\in M_{2,1}$ (for some $\varphi_{1},\varphi_{2}\in M_{2,1}$) with the following properties:

The table sizes are $\|\Phi_{n}\|=2^{n}$, and  $\|\varphi_{2}\|=\|\varphi_{1}\|=2$. So, $\|\Phi_{n}\|$ is exponentially larger than   $(n-1)\cdot\|\varphi_{2}\|+\|\varphi_{1}\|$. Hence the table size formula does not hold in $M_{2,1}$.

The word lengths of $\varphi_{1},\varphi_{2}$, and $\Phi_{n}$ (over the finite generating set $\Gamma$ of $M_{2,1}$ from Section 3 in [1]) satisfy  $|\varphi_{1}|_{{}_{\Gamma}}=1$,  $|\varphi_{2}|_{{}_{\Gamma}}\leq 2$, and $|\Phi_{n}|_{{}_{\Gamma}}<2n$. So the table size of $\Phi_{n}$ is exponentially larger than its word length:  $\|\Phi_{n}\|\,>\,\sqrt{2}^{\,|\Phi_{n}|_{{}_{\Gamma}}}$.

Proof. Consider $\varphi_{1},\varphi_{2}\in M_{2,1}$ given by the tables  $\varphi_{1}=\{(0\mapsto 0),\ (1\mapsto 0)\}$, and  $\varphi_{2}=\{(00\mapsto 0),\ (01\mapsto 0)\}$. One verifies that  $\Phi_{n}=\varphi_{2}^{n-1}\circ\varphi_{1}(.)$  sends every bitstring of length $n$ to the word [math]; its domain code is $\{0,1\}^{n}$, its image code is $\{0\}$, and it is its maximum essential extension. Thus, $\|\varphi_{2}^{n-1}\circ\varphi_{1}\|\ =\ 2^{n}$, whereas  $(n-1)\cdot\|\varphi_{2}\|+\|\varphi_{1}\|=2n$. Also,  $\varphi_{2}(.)\,=\,$ $(0\mapsto 0,1\mapsto 0)\cdot(0\mapsto\varepsilon)$, so $|\varphi_{1}|_{{}_{\Gamma}}=1$,  $|\varphi_{2}|_{{}_{\Gamma}}\leq 2$, and $|\Phi_{n}|_{{}_{\Gamma}}\leq 2n-1$; hence $\|\Phi_{n}\|>2^{|\Phi_{n}|_{{}_{\Gamma}}/2}$.    $\Box$

We will use the following facts that are easy to prove: If $R\subset A^{*}$ is a right ideal and $\varphi$ is a right-ideal morphism then $\varphi(R)$ and $\varphi^{-1}(R)$ are right ideals. The intersection and the union of right ideals are right ideals. We also need the following result.

(Lemma 3.3 of [4])  *If $P,Q,S\subseteq A^{*}$ are such that $PA^{*}\cap QA^{*}=SA^{*}$, and if $S$ is a prefix code then $S\subseteq P\cup Q$. *    $\Box$

Lemma 4.3

Let $\theta$ be a right-ideal morphism, and assume $SA^{*}\subseteq{\rm Dom}(\theta)$, where $S\subset A^{*}$ is a finite prefix code. Then there is a finite prefix code $R\subset A^{*}$ such that $\,\theta(SA^{*})=RA^{*}\,$ and $\,R\subseteq\theta(S)$.

Proof. Since $\theta$ is a right-ideal morphism we have  $\theta(SA^{*})=\theta(S)\ A^{*}$. Since $\theta(S)$ might not be a prefix code we take  $R=\{r\in\theta(S):r$ is minimal (shortest) in the prefix order within $\theta(S)\}$. Then $R$ is a prefix code that has the required properties.    $\Box$

Lemma 4.4

333  This Lemma was incorrect in the earlier versions of this paper and in [1].

For any right-ideal morphism $\theta$ and any prefix code $Z\subset A^{*}$, $\theta^{-1}(Z)$ is a prefix code.

In particular, $\theta^{-1}({\rm imC}(\theta))$ is a prefix code, and $\,\theta^{-1}({\rm imC}(\theta))\subseteq{\rm domC}(\theta)$. There exist right-ideal morphisms $\theta$ with finite domain code, such that $\,\theta^{-1}({\rm imC}(\theta))\neq{\rm domC}(\theta)$.

Proof. First, $\theta^{-1}(Z)$ is a prefix code. Indeed, if we had $x_{1}=x_{2}u$ for some $x_{1},x_{2}\in\theta^{-1}(Z)$ with $u$ non-empty, then $\theta(x_{1})=\theta(x_{2})\ u$, with $\theta(x_{1}),\theta(x_{2})\in Z$. This would contradict the assumption that $Z$ is a prefix code.

Second, let $Q={\rm imC}(\theta)$; then $\theta^{-1}(Q)\,A^{*}\subseteq\theta^{-1}(QA^{*})$. Indeed, if $x\in\theta^{-1}(Q)$, then $x=pw$ for some $p\in{\rm domC}(\theta)$ and $w\in A^{*}$. Hence, $\theta(x)=\theta(p)\,w$, and $\theta(x)\in Q$. Since $\theta(p)\,w\in Q$ and $\theta(p)\in QA^{*}$, we have $\theta(p)\,w=\theta(p)$ (since $Q$ is a prefix code). So $w$ is empty, hence $x=pw=p\in{\rm domC}(\theta)$.

Example: Let $A=\{0,1\}$, and let $\theta$ be the right-ideal morphism defined by ${\rm domC}(\theta)=\{01,1\}$, ${\rm imC}(\theta)=\{\varepsilon\}$, and $\,\theta(01)=0$, $\,\theta(1)=\varepsilon$. Then, $\theta^{-1}({\rm imC}(\theta))=\{1\}\neq{\rm domC}(\theta)$.    $\Box$

The following generalizes the “table size formula” of $G_{k,1}$ to the monoid $M_{k,1}$.

Theorem 4.5

(Generalized image code formulas).* 444  This Theorem was incorrect in the previous versions and in [1]; this is a corrected (and expanded) version.

Let $\varphi_{i}$ be right-ideal morphism with finite domain codes, for $i=1,2,\ldots,n$. Then*

(1)*     \big{|}{\rm imC}(\varphi_{n}\circ\ \ldots\ \circ\varphi_{1})\big{|} \ \leq\ |{\rm imC}(\varphi_{1})|\ +\ $\sum_{i=2}^{n}|\varphi_{i}({\rm domC}(\varphi_{i}))|$,*

(2)*     \ell\big{(}{\rm domC}(\varphi_{n}\circ\ \ldots\ \circ\varphi_{1})\big{)} \ \leq\ $\sum_{i=1}^{n}\ell({\rm domC}(\varphi_{i}))$,*

(3)*     \ell\big{(}\varphi_{n}\ldots\varphi_{1}({\rm domC}(\varphi_{n}\circ\ \ldots\ \circ\varphi_{1}))\big{)} \ \leq\ $\sum_{i=1}^{n}\ell(\varphi_{i}({\rm domC}(\varphi_{i})))$,*

(4)*     \ell\big{(}{\rm imC}(\varphi_{n}\circ\ \ldots\ \circ\varphi_{1})\big{)} \ \leq\ \ell({\rm imC}(\varphi_{1}))\ +\ $\sum_{i=2}^{n}\ell(\varphi_{i}({\rm domC}(\varphi_{i})))$,*

(5)*     \big{|}\varphi_{n}\ldots\varphi_{1}({\rm domC}(\varphi_{n}\circ\ \ldots\ \circ\varphi_{1}))\big{|} \ \leq\ $\sum_{i=1}^{n}|(\varphi_{i}({\rm domC}(\varphi_{i}))|$,  and*

$\varphi_{n}\ldots\varphi_{1}({\rm domC}(\varphi_{n}\circ\ \ldots\,$* $\circ$ $\varphi_{1}))$ \ \subseteq\ \ $\bigcup_{i=1}^{n}$ $\varphi_{n}\ldots\varphi_{i}({\rm domC}(\varphi_{i}))$.*

Proof.  Let $P_{i}={\rm domC}(\varphi_{i})$ and $Q_{i}={\rm imC}(\varphi_{i})$.

(1)   The proof is similar to the proof of Proposition 3.5 in [4]. We have   ${\rm Dom}(\varphi_{2}\circ\varphi_{1})=$ $\varphi_{1}^{-1}(Q_{1}A^{*}\cap P_{2}A^{*})$  and  ${\rm Im}(\varphi_{2}\circ\varphi_{1})=\varphi_{2}(Q_{1}A^{*}\cap P_{2}A^{*})$. So the following maps are total and onto on the indicated sets:

$\varphi_{1}^{-1}(Q_{1}A^{*}\cap P_{2}A^{*})\ \stackrel{{\scriptstyle\varphi_{1}}}{{\longrightarrow}}$ \ Q_{1}A^{*}\cap P_{2}A^{*}\ \stackrel{{\scriptstyle\varphi_{2}}}{{\longrightarrow}}\ $\varphi_{2}(Q_{1}A^{*}\cap P_{2}A^{*})$.

By Lemma 3.3 of [4] (quoted above) we have $Q_{1}A^{*}\cap P_{2}A^{*}=SA^{*}$ for some finite prefix code $S$ with $S\subseteq Q_{1}\cup P_{2}$. Moreover, by Lemma 4.3 we have $\varphi_{2}(SA^{*})=R_{2}A^{*}$ for some finite prefix code $R_{2}$ such that $R_{2}\subseteq\varphi_{2}(S)$. Now, since $S\subseteq Q_{1}\cup P_{2}$ we have  $R_{2}\subseteq\varphi_{2}(S)\subseteq$ $\varphi_{2}(Q_{1})\,\cup\,\varphi_{2}(P_{2})$. Thus,  $|{\rm imC}(\varphi_{2}\circ\varphi_{1})|=$ $|R_{2}|\leq|\varphi_{2}(P_{2})|+|\varphi_{2}(Q_{1})|$. Since $|\varphi_{2}(Q_{1})|\leq|Q_{1}|$, we have $|R_{2}|\leq|\varphi_{2}(P_{2})|+|Q_{1}|$.

By induction for $n>2$, $|{\rm imC}(\varphi_{n}$ $\circ$ $\varphi_{n-1}\circ\ \ldots\ \circ\varphi_{1})|$ $\leq$ $|\varphi_{n}({\rm domC}(\varphi_{n}))|$ $+$ $|{\rm imC}(\varphi_{n-1}\circ\ \ldots\ \circ\varphi_{1})|$ $\leq$ $|\varphi_{n}({\rm domC}(\varphi_{n}))|$ $+$ $\sum_{i=2}^{n-1}|\varphi_{i}({\rm domC}(\varphi_{i}))|\ +\ |{\rm imC}(\varphi_{1})|$ .

(2)  We prove the formula when $n=2$; the general formula then follows immediately by induction. Let $x\in{\rm domC}(\varphi_{2}\circ\varphi_{1})$; then $\varphi_{1}(x)$ is defined, hence $x=p_{1}u$ for some $p_{1}\in P_{1}$, $u\in A^{*}$. And $\varphi_{2}$ is defined on $\varphi_{1}(x)=\varphi_{1}(p_{1})\,u$, so $\varphi_{1}(x)\in P_{2}A^{*}={\rm Dom}(\varphi_{2})$. Hence there exist $p_{2}\in P_{2}$ and $v\in A^{*}$ such that

$(\star)$             $\varphi_{1}(p_{1})\,u=p_{2}v$  $\in\varphi_{1}(P_{1})\,A^{*}\cap P_{2}A^{*}$.

It follows that $u$ and $v$ are suffix-comparable.

Claim.  *The words $u$ and $v$ in $(\star)$ satisfy: $\,u=\varepsilon$, or $v=\varepsilon$. *

Proof of the Claim: Since $u$ and $v$ are suffix-comparable, let us first consider the case where $v$ is a suffix of $u$, i.e., $u=tv$ for some $t\in A^{*}$. Then $\varphi_{1}(x)=\varphi_{1}(p_{1})\,tv=p_{2}v$, hence $\varphi_{1}(p_{1})\,t=p_{2}$, hence $\varphi_{2}$ is defined on $\varphi_{1}(p_{1})\,t=p_{2}$. So, $\varphi_{2}\circ\varphi_{1}$ is defined on $p_{1}t$, i.e., $p_{1}t\in{\rm domC}(\varphi_{2}\circ\varphi_{1})$. But we also have $x=p_{1}tv\in{\rm domC}(\varphi_{2}\circ\varphi_{1})$. Since ${\rm domC}(\varphi_{2}\circ\varphi_{1})$ is a prefix code, it follows that $v=\varepsilon$.

Let us next consider the other case, namely where $u$ is a suffix of $v$, i.e., $v=su$ for some $s\in A^{*}$. Then $\varphi_{1}(x)=\varphi_{1}(p_{1})\,u=p_{2}su$, hence $\varphi_{1}(p_{1})=p_{2}s$, hence $\varphi_{2}$ is defined on $\varphi_{1}(p_{1})=p_{2}s$, hence $p_{1}\in{\rm domC}(\varphi_{2}\circ\varphi_{1})$. But we also have $x=p_{1}u\in{\rm domC}(\varphi_{2}\circ\varphi_{1})$. Since ${\rm domC}(\varphi_{2}\circ\varphi_{1})$ is a prefix code, it follows that $u=\varepsilon$.    [This proves the Claim.]

Now for $x\in{\rm domC}(\varphi_{2}\circ\varphi_{1})$ we have $x=p_{1}u$, and $\varphi_{1}(p_{1})\,u=p_{2}v$, hence $|x|=|p_{1}|+|u|$ and $|\varphi_{1}(p_{1})|+|u|=|p_{2}|+|v|$. By the Claim, either $|u|=0$ or $|v|=0$.

If $\,|u|=0\,$ then $\,|x|=|p_{1}|\leq\ell({\rm domC}(\varphi_{1}))$.

If $\,|v|=0\,$ then $\,|x|=|p_{1}|+|u|$ $=$ $|p_{1}|+|p_{2}|+|v|-|\varphi_{1}(p_{1})|$ $=$ $|p_{1}|+|p_{2}|-|\varphi_{1}(p_{1})|\leq|p_{1}|+|p_{2}|$ $\leq$ $\ell({\rm domC}(\varphi_{1}))+\ell({\rm domC}(\varphi_{2}))$.

(3)  As in the proof of (2) we only need to consider $n=2$. Let $x\in$ ${\rm domC}(\varphi_{2}\varphi_{1})$, hence $\varphi_{2}\varphi_{1}(x)\in$ $\varphi_{2}\varphi_{1}({\rm domC}(\varphi_{2}\varphi_{1}))$. By $(\star)$ (and with the notation of the proof of (2)) we have $\varphi_{2}\varphi_{1}(x)=\varphi_{2}(\varphi_{1}(p_{1})\,u)$ $=$ $\varphi_{2}(p_{2})\,v\in\varphi_{2}(\varphi_{1}(P_{1})\,A^{*}\cap P_{2}A^{*})=$ ${\rm Im}(\varphi_{2}\varphi_{1})$. By the reasoning of the proof of (2), we have two cases:

If $|u|=0$ then $|v|=|\varphi_{1}(p_{1})|+|u|-|p_{2}|=|\varphi_{1}(p_{1})|-|p_{2}|$ $\leq$ $|\varphi_{1}(p_{1})|$. Hence, $|\varphi_{2}\varphi_{1}(x)|=|\varphi_{2}(p_{2})|+|v|\leq$ |\varphi_{2}(p_{2})|+|\varphi_{1}(p_{1})|\ \leq\ $\ell(\varphi_{2}({\rm domC}(\varphi_{2}))$ $+$ $\ell(\varphi_{1}({\rm domC}(\varphi_{1}))$.

If $|v|=0$ then $\varphi_{2}\varphi_{1}(x)=\varphi_{2}(p_{2})$, hence  $|\varphi_{2}\varphi_{1}(x)|=|\varphi_{2}(p_{2})|\leq$ $\ell(\varphi_{2}({\rm domC}(\varphi_{2}))$.

(4)  We first consider the case $n=2$. As we saw in the proof of (1), ${\rm imC}(\varphi_{2}\varphi_{1})=R_{2}$ where $R_{2}\subseteq\varphi_{2}(S)$, and where $S$ is a prefix code such that $S\subseteq Q_{1}\cap P_{2}$. Hence $R_{2}\subseteq$ $\varphi_{2}(Q_{1})\cup\varphi_{2}(P_{2})$.

Hence for any $z\in R_{2}$, either $z\in\varphi_{2}(P_{2})$ or $z\in\varphi_{2}(Q_{1})$. If $z\in\varphi_{2}(P_{2})$ then $|z|\leq\ell(\varphi_{2}(P_{2}))$. If $z\in\varphi_{2}(Q_{1})$, then $z=\varphi_{2}(q_{1})$ for some $q_{1}\in Q_{1}\cap P_{2}A^{*}$, so $q_{1}=p_{2}u$ for some $p_{2}\in P_{2}$ and $u\in A^{*}$. We have $q_{1}\in P_{2}A^{*}$ ($={\rm Im}(\varphi_{2})$), so $q_{1}\in{\rm Im}(\varphi_{2})$. Now $|z|=|\varphi_{2}(p_{2})|+|u|$, and $|u|=|q_{1}|-|p_{2}|\leq|q_{1}|\leq\ell({\rm imC}(\varphi_{1}))$. Thus, $|z|\leq|\varphi_{2}(p_{2})|+\ell({\rm imC}(\varphi_{1}))$ $\leq$ $\ell(\varphi_{2}({\rm domC}(\varphi_{2})))$ $+$ $\ell({\rm imC}(\varphi_{1}))$.

The formula for $n>2$ now follows by induction in the same way as in the proof of (1).

(5)  We first prove the formula for $n=2$. As we saw in the proof of (2), if $x\in{\rm domC}(\varphi_{2}\varphi_{1})$ then there exist $u,v\in A^{*}$, $p_{1}\in P_{1}$, $p_{2}\in P_{2}$, such that $x=p_{1}u$ and $\varphi_{1}(x)=\varphi_{1}(p_{1})\,u=p_{2}v$. Moreover, by the Claim in (2) we have $u=\varepsilon$ or $v=\varepsilon$. Also, $\varphi_{2}\varphi_{1}(x)=\varphi_{2}(\varphi_{1}(p_{1})\,u)=$ $\varphi_{2}(p_{2})\,v$.

If $v=\varepsilon$ then $\varphi_{2}\varphi_{1}(x)=\varphi_{2}(p_{2})$ $\in\varphi_{2}({\rm domC}(\varphi_{2}))$. If $u=\varepsilon$ then $\varphi_{2}\varphi_{1}(x)=$ $\varphi_{2}\varphi_{1}(p_{1})\in\varphi_{2}\varphi_{1}({\rm domC}(\varphi_{1}))$. Thus we proved the following fact:

$\varphi_{2}\varphi_{1}({\rm domC}(\varphi_{2}\varphi_{1}))$ \,\subseteq\ $\varphi_{2}({\rm domC}(\varphi_{2}))$ \ \cup\ $\varphi_{2}\varphi_{1}({\rm domC}(\varphi_{1}))$.

Now, since $|\varphi_{2}\varphi_{1}({\rm domC}(\varphi_{1}))|\leq$ $|\varphi_{1}({\rm domC}(\varphi_{1}))|$, the fact implies that $\,|\varphi_{2}\varphi_{1}({\rm domC}(\varphi_{2}\varphi_{1}))|$ \ \leq\ $|\varphi_{2}({\rm domC}(\varphi_{2}))|$ $+$ $|\varphi_{1}({\rm domC}(\varphi_{1}))|$. By induction we immediately obtain

\big{|}\varphi_{n}\ldots\varphi_{1}({\rm domC}(\varphi_{n}\circ\ \ldots\ \circ\varphi_{1}))\big{|} \ \leq\ $\sum_{i=1}^{n}|(\varphi_{i}({\rm domC}(\varphi_{i}))|$,  and

$\varphi_{n}\ldots\varphi_{1}({\rm domC}(\varphi_{n}\circ\ \ldots\,$ $\circ$ $\varphi_{1}))$

\ \subseteq\ \ $\varphi_{n}({\rm domC}(\varphi_{n}))$ \ \cup\ $\varphi_{n}\varphi_{n-1}({\rm domC}(\varphi_{n-1}))$ $\ \cup\ \ \ldots\ \ \ldots\ \ \ldots$

\cup\ $\varphi_{n}\ldots\varphi_{i}({\rm domC}(\varphi_{i}))$ \ \cup\ \ \ldots\ \ \ldots\ \ \ldots\ \ \cup\ $\varphi_{n}\ldots\varphi_{i}\ldots\varphi_{1}({\rm domC}(\varphi_{1}))$.      $\Box$

Remarks. Obviously, ${\rm Dom}(\varphi_{2}\varphi_{1})$ $\subseteq$ ${\rm Dom}(\varphi_{1})$; however, in infinitely many cases (in “most” cases), ${\rm domC}(\varphi_{2}\varphi_{1})$ $\not\subseteq$ ${\rm domC}(\varphi_{1})$. Instead, we have the more complicated formula of Theorem 4.5(5).

By Prop. 4.2, we cannot have a formula for $|{\rm domC}(\varphi_{n}\ldots\varphi_{1})|$ of a similar nature as the formulas in Theorem 4.5.

The following class of right-ideal morphisms plays an important role here (as well as in Section 5 of [7], where it was introduced). 555  Def. 4.5A, Theorem 4.5B, and Cor. 4.5C are new in this version.

Definition 4.5A (Normal). *A right-ideal morphism $\varphi$ is called normal iff $\,\varphi({\rm domC}(\varphi))={\rm imC}(\varphi)$. *

By Lemma 5.7 of [7] we also have: $\,\varphi$ is normal iff $\varphi^{-1}({\rm imC}(\varphi))={\rm domC}(\varphi)$. In other words, $\varphi$ is normal iff $\varphi$ is entirely determined by the way it maps ${\rm domC}(\varphi)$ onto ${\rm imC}(\varphi)$.

For example, every injective right-ideal morphism is normal (by Lemma 5.1 in [7]). The finite generating set $\Gamma$ of $M_{k,1}$, constructed in Section 3, consist entirely of normal right-ideal morphisms.

On the other hand, the composition of two normal right-ideal morphisms does not always result in a normal morphism, as is shown by the following example: $\,{\rm domC}(f)=\{0,1\}\,$ and $\,f(0)=0$, $f(1)=10$; $\,{\rm domC}(g)=\{0,1\}\,$ and $\,g(0)=g(1)=0$; so $f$ and $g$ are normal. But ${\rm domC}(gf)=\{0,1\}$ and $gf(0)=0$, $\,gf(1)=00$; so $gf$ is not normal (for more details, see Prop. 5.8 in [7]).

The next result (Theorem 4.5B) shows that every element of $M_{k,1}$ can be represented by a normal right-ideal morphism. So one can say informally that “from the point of view of $M_{k,1}$, all right-ideal morphisms are normal”. For proving this we need some definitions. We always assume $|A|\geq 2$.

Definitions and notation.  If $x_{1},x_{2}\in A^{*}$ are such that $x_{1}$ is a prefix of $x_{2}$, i.e., $x_{2}\in x_{1}A^{*}$, we denote this by $x_{1}\leq_{\rm pref}x_{2}$.

For $Z\subseteq A^{*}$, the set of prefixes of $Z$ is $\,{\sf pref}(Z)=\{v\in A^{*}:v\leq_{\rm pref}z$ for some $z\in Z\}$.

For a set $X\subseteq A^{*}$ and a word $v\in A^{*}$, $v^{-1}X$ denotes the set $\{s\in A^{*}:vs\in X\}$.

The tree of $A^{*}$ has root $\varepsilon$, vertex set $A^{*}$, and edge set $\,\{(w,wa):w\in A^{*},\,a\in A\}$.

*A subtree of the tree of $A^{*}$ has as root any string $r\in A^{*}$, and as vertex set any subset $V\subseteq rA^{*}$, such that the following holds for all $v\in V$ and $u\in A^{*}$:  $r\leq_{\rm pref}u\leq_{\rm pref}v$ implies $u\in V$. *

The following is a slight generalization of the classical notion of a prefix tree.

Definition (Prefix tree).  *Let $Z\subseteq A^{*}$, and let $q\in A^{*}$. The prefix tree $T(q,Z)$ is the subtree of the tree of $A^{*}$ with root $q$ and vertex set   $V_{q,Z}=\{v\in A^{*}:q\leq_{\rm pref}v$, and $v\leq_{\rm pref}z$ for some $z\in Z\}$. *

Remark. Let $L$ be the set of leaves of $T(q,Z)$; then $L$ and $q^{-1}L$ are prefix codes.

Definition (Saturated tree).  *A subtree $T$ of the tree of $A^{*}$ is saturated iff for every vertex $v$ of $T$ we have: $v$ has no child in $T$ (i.e., $v$ is a leaf), or $v$ has $|A|$ children in $T$. *

Definition (Tree saturation).  *Let $T$ be a subtree of the tree of $A^{*}$, with root $q$, set of vertices $V$, and set of leaves $L$. The saturation of $T$ is the smallest (under inclusion) saturated subtree of the tree of $A^{*}$ with root $q$, that contains $T$. In other words, if $T$ is just $\{q\}$, it is its own saturation; otherwise the saturation has root $q$ and has vertex set $\,V\cup(V-L)\cdot A$. We denote the saturation of $T$ by ${\rm s}T$. *

Remark. (1) The prefix tree $T(q,Z)$ and its saturation have the same depth (i.e., length of a longest path from the root). Every leaf of $T(q,Z)$ is also a leaf of ${\rm s}T(q,Z)$, but unless $T(q,Z)$ is already saturated, ${\rm s}T(q,Z)$ has more leaves than $T(q,Z)$. The non-leaf vertices of $T(q,Z)$ and ${\rm s}T(q,Z)$ are the same.

(2) The number of leaves in the saturated tree ${\rm s}T(q,Z)$ is $\,<|V_{q,Z}|\cdot|A|$.

(3) Let $L$ be the leaf set of the saturated tree ${\rm s}T(q,Z)$; if $Z$ is finite then $q^{-1}L$ is a maximal prefix code.

Theorem 4.5B (Equivalent normal morphism). For every right-ideal morphism $\varphi$ with finite domain code there exists a normal right-ideal morphism $\varphi_{0}$ with finite domain code, such that $\varphi=\varphi_{0}$ in $M_{k,1}$. Moreover,

$|{\rm imC}(\varphi_{0})|\ =\ |\varphi_{0}({\rm domC}(\varphi_{0}))|$* \ \leq\ $|A|\cdot(\ell(\varphi(P))+1)\cdot|\varphi(P)|$,*

$|{\rm domC}(\varphi_{0})|$* \ \leq\ $|P|\cdot|A|\cdot(\ell(\varphi(P))+1)\cdot|\varphi(P)|$,*

$\ell({\rm imC}(\varphi_{0}))\,=\,\ell(\varphi_{0}({\rm domC}(\varphi_{0})))$* $\,=\,\ell(\varphi({\rm domC}(\varphi)))$,*

\ell({\rm domC}(\varphi_{0}))\ \leq\* $\ell({\rm domC}(\varphi))\,+\,\ell(\varphi({\rm domC}(\varphi)))$. *

Proof. Let $P={\rm domC}(\varphi)$, $Q={\rm imC}(\varphi)$, $P_{0}={\rm domC}(\varphi_{0})$, $Q_{0}={\rm imC}(\varphi_{0})$. For each $p\in P$, let $\varphi(p)\,W_{\varphi(p)}$ be the the set of leaves of the saturated tree \,{\rm s}T\big{(}\varphi(p),\,\varphi(P)\cap\varphi(p)\,A^{*}\big{)}. By Remark (3) above, $W_{\varphi(p)}$ is a finite maximal prefix code. Now we define $\varphi_{0}$ as follows:

$\varphi_{0}$ is the restriction of $\varphi$ to $\ \bigcup_{p\in P}\,p\,W_{\varphi(p)}\,A^{*}$.

Let us verify that $\varphi_{0}$ has the required properties. Since $P$ and $W_{\varphi(p)}$ are finite prefix codes, $\bigcup_{p\in P}p\,W_{\varphi(p)}$ is a finite prefix code. So,

${\rm domC}(\varphi_{0})\ =\ \bigcup_{p\in P}\,p\,W_{\varphi(p)}$.

Since each $W_{\varphi(p)}$ is a maximal prefix code, the right ideal $\,\bigcup_{p\in P}p\,W_{\varphi(p)}\,A^{*}$ is essential in the right ideal $PA^{*}$; hence $\varphi$ and $\varphi_{0}$ are equal as elements of $M_{k,1}$. Finally, let us show that $\varphi_{0}({\rm domC}(\varphi_{0}))$ is a prefix code. We have

$\varphi_{0}({\rm domC}(\varphi_{0}))$ \ =\ $\bigcup_{p\in P}\,\varphi(p)\ W_{\varphi(p)}$,

which is the set of leaves of the union of the saturated prefix trees \,{\rm s}T\big{(}\varphi(p),\,\varphi(P)\cap\varphi(p)\,A^{*}\big{)}, for $p$ ranging over $P$. For each $p\in P$, the leaves of \,{\rm s}T\big{(}\varphi(p),\,\varphi(P)\cap\varphi(p)\,A^{*}\big{)} form the prefix code $\varphi(p)\,W_{\varphi(p)}$. For $p_{1}\neq p_{2}$ in $P$, if $\varphi(p_{1})$ is a prefix of $\varphi(p_{2})$ then the leaves of \,{\rm s}T\big{(}\varphi(p_{2}),\,\varphi(P)\cap\varphi(p_{2})\,A^{*}\big{)} are a subset of the leaves of \,{\rm s}T\big{(}\varphi(p_{1}),\,\varphi(P)\cap\varphi(p_{1})\,A^{*}\big{)}, so the union of these two leaf sets is just the leaf set of \,{\rm s}T\big{(}\varphi(p_{1}),\,\varphi(P)\cap\varphi(p_{1})\,A^{*}\big{)}; a similar thing happens if $\varphi(p_{2})$ is a prefix of $\varphi(p_{1})$. So in $\,\bigcup_{p\in P}\,\varphi(p)\ W_{\varphi(p)}\,$ we can ignore elements $p$ of $P$ for which $\varphi(p)$ is a strict prefix of another element of $\varphi(P)$. If $\varphi(p_{1})$ and $\varphi(p_{2})$ are not prefix-comparable, then the leaves of \,{\rm s}T\big{(}\varphi(p_{i}),\,\varphi(P)\cap\varphi(p_{i})\,A^{*}\big{)} have $\varphi(p_{i})$ as a prefix, so these two trees have leaf sets that are two-by-two prefix-incomparable (namely the sets $\varphi(p_{1})\,W_{\varphi(p_{1})}$ and $\varphi(p_{2})\,W_{\varphi(p_{2})}$). The union of prefix codes that are two-by-two prefix-incomparable forms a prefix code; hence, $\,\bigcup_{p\in P}\,\varphi(p)\ W_{\varphi(p)}\,$ is a prefix code.

Now, since $\varphi_{0}({\rm domC}(\varphi_{0}))$ is a prefix code it follows that $\,{\rm imC}(\varphi_{0})=\varphi_{0}({\rm domC}(\varphi_{0}))$, so $\varphi_{0}$ is normal. This proves the first part of the theorem.

Let us prove the formulas. We saw that $\,{\rm imC}(\varphi_{0})=\varphi_{0}({\rm domC}(\varphi_{0}))$ $=$ $\,\bigcup_{p\in P}\,\varphi(p)\ W_{\varphi(p)}$, and $\varphi(p)\ W_{\varphi(p)}$ is the leaf set of the saturated tree \,{\rm s}T\big{(}\varphi(p),\varphi(P)\cap\varphi(p)A^{*}\big{)}. By the definition of prefix trees, the vertices of all the (non-saturated) trees \,T\big{(}\varphi(p),\varphi(P)\cap\varphi(p)A^{*}\big{)}\, are subsets of ${\sf pref}(\varphi(P))$. By Remark (2) above, the number of leaves in a saturated tree \,{\rm s}T\big{(}\varphi(p),\varphi(P)\cap\varphi(p)A^{*}\big{)}\, is at most $|A|$ times the number of vertices of the non-saturated tree. Hence, $|{\rm imC}(\varphi_{0})|\leq|A|\cdot|{\sf pref}(\varphi(P))|$. Moreover, for any finite $Z\subset A^{*}$, $\,|{\sf pref}(Z)|\leq(1+\ell(Z))\cdot|Z|$, hence, $|{\rm imC}(\varphi_{0})|\leq$ $|A|\cdot(\ell(\varphi(P))+1)\cdot|\varphi(P)|$.

We have $\,{\rm domC}(\varphi_{0})$ $=$ $\bigcup_{p\in P}p\,W_{\varphi(p)}$, and $\varphi(p)\,W_{\varphi(p)}$ is the leaf set of \,{\rm s}T\big{(}\varphi(p),\varphi(P)\cap\varphi(p)A^{*}\big{)}. Hence by the same reasoning as for $|{\rm imC}(\varphi_{0})|$: $\,|W_{\varphi(p)}|=|\varphi(p)\,W_{\varphi(p)}|$ $\,\leq\,$ $|A|\cdot(\ell(\varphi(P))+1)\cdot|\varphi(P)|$. Hence, $|{\rm domC}(\varphi_{0})|$ $\,\leq\,$ $\sum_{p\in P}|W_{\varphi(p)}|$ $\,\leq\,$ $\sum_{p\in P}\,|A|\cdot(\ell(\varphi(P))+1)\cdot|\varphi(P)|$ $\,\leq\,$ $|P|\cdot|A|\cdot(\ell(\varphi(P))+1)\cdot|\varphi(P)|$.

We have $\,{\rm imC}(\varphi_{0})=\bigcup_{p\in P}\varphi(p)\,W_{\varphi(p)}$, and $\varphi(p)\,W_{\varphi(p)}$ is the leaf set of \,{\rm s}T\big{(}\varphi(p),\,\varphi(P)\cap\varphi(p)\,A^{*}\big{)}. Hence, $\ell({\rm imC}(\varphi_{0}))\leq\ell(\varphi(P))$; indeed, tree saturation does not increase the depth of a tree, and the depth of \,T\big{(}\varphi(p),\,\varphi(P)\cap\varphi(p)\,A^{*}\big{)}\, is $\,\leq\ell(\varphi(P))$.

We have $\,{\rm domC}(\varphi_{0})=\bigcup_{p\in P}p\,W_{\varphi(p)}$. And $\ell(W_{\varphi(p)})\leq\ell(\varphi(p)\,W_{\varphi(p)})$ $\leq$ $\ell(\varphi(P))$, since $\varphi(p)\,W_{\varphi(p)}$ is the leaf set of \,{\rm s}T\big{(}\varphi(p),\,\varphi(P)\cap\varphi(p)\,A^{*}\big{)}. Hence, for every $x\in{\rm domC}(\varphi_{0})$ we have $x\in pW_{\varphi(p)}$ for some $p\in P$, so $\,|x|\leq|p|+\ell(W_{\varphi(p)})$. Therefore, $\ell({\rm domC}(\varphi_{0}))\,\leq\,\ell(P)+\ell(\varphi(P))$.    $\Box$

Theorem 4.5B tells us that as far as $M_{k,1}$ is concerned, all right-ideal morphisms are normal. 666  The concept of normal morphism and Theorem 4.5 enable us to rehabilitate the image code formula (which was incorrect as stated in Theorem 4.5 of [1], but which is correct when one adds the hypothesis that the morphisms $\varphi_{i}$ are normal).

Corollary 4.5C (Image code formula).

Let $\varphi_{i}$ be a right-ideal morphism (for $i=1,\ldots,n$), and let $\,\Phi=\varphi_{n}\circ\ \ldots\ \circ\varphi_{1}$.

(1)* If $\varphi_{i}$ is normal for $2\leq i\leq n$, then*

$|{\rm imC}(\Phi)|$* \ \leq\ $\sum_{i=1}^{n}|{\rm imC}(\varphi_{i})|$ .*

(2)* If all $\varphi_{i}$ are normal (for $1\leq i\leq n$), then*

\ell({\rm domC}(\Phi)\cup{\rm imC}(\Phi))\ \leq\* $\sum_{i=1}^{n}\ell({\rm domC}(\varphi_{i})\cup{\rm imC}(\varphi_{i}))$ . *

Proof. (1) follows immediately from Theorem 4.5(1), and (2) follows from 4.5(2) and 4.5(4).    $\Box$

Counter-examples:

(1) The following shows that the image code formula of Corollary 4.5C(1) is wrong in some examples when $\varphi_{2}$ is not normal (but $\varphi_{1}$ is normal). Let $A=\{0,1\}$, $n\geq 2$, and

$\varphi_{1}\ =\ \{(01,00),\,(00,01),\,(10,1011),\,(11,1100)\}$,  and

$\varphi_{2}\ =\ \{(00u0,000u1):u\in\{0,1\}^{n-1}\}$ \ \cup\ $\{(01v0,001v1):v\in\{0,1\}^{n-1}\}$ \ \cup\ $\{(10,000),\,(11,001)\}$.

So, ${\rm imC}(\varphi_{1})=\{00,01,1011,1100\}$, and ${\rm imC}(\varphi_{2})=\{000,001\}$, hence $|{\rm imC}(\varphi_{1})|+|{\rm imC}(\varphi_{2})|=6$. Note that the right-ideal morphisms $\varphi_{1}$ and $\varphi_{2}$ are in maximally extended form.

Now, $\varphi_{2}\circ\varphi_{1}:\ 01u0\mapsto 00u0\mapsto 000u1$ and $\varphi_{2}\circ\varphi_{1}:\ 00v0\mapsto 01v0\mapsto 001v1$, for all $u,v\in\{0,1\}^{n-1}$; and $\varphi_{2}\circ\varphi_{1}:\ 10\mapsto 1011\mapsto 00011$, $\varphi_{2}\circ\varphi_{1}:\ 11\mapsto 1100\mapsto 00100$. Note that $\varphi_{2}\circ\varphi_{1}$ is in maximally extended form.

Then $\,{\rm imC}(\varphi_{2}\circ\varphi_{1})=\{00011,00100\}$ $\cup$ $000\,\{00,01,11\}\,\{0,1\}^{n-2}$ $\cup$ $001\,\{00,01,11\}\,\{0,1\}^{n-2}$. Thus when $n\geq 2$:  $2+6\cdot 2^{n-2}=|{\rm imC}(\varphi_{2}\circ\varphi_{1})|$ \ \not\leq\ $|{\rm imC}(\varphi_{1})|+|{\rm imC}(\varphi_{2})|=6$.    $\Box$

(2) The following shows that the formula of Corollary 4.5C(2) is wrong in some examples when $\varphi_{2}$ is not normal (but $\varphi_{1}$ is normal). We abbreviate $\ell({\rm domC}(\varphi)\cup{\rm imC}(\varphi))$ by $\ell(\varphi)$. Let $A=\{0,1\}$, $n\geq 2$, and

$\varphi_{1}\ =\ \{(0,0^{n})\}$,  and

$\varphi_{2}\ =\ \{(0,0^{n+1}),\,(1,0)\}$.

So, $\ell(\varphi_{1})=n$, and $\ell(\varphi_{2})=1$ since ${\rm imC}(\varphi_{2})=\{0\}$. Now, $\varphi_{2}\circ\varphi_{1}=\{(0,0^{2n})\}$. Thus when $n\geq 2$:  2n=\ell(\varphi_{2}\circ\varphi_{1})\ \not\leq\ $\ell(\varphi_{2})+\ell(\varphi_{1})=n+1$.    $\Box$

For elements of ${\it Inv}_{k,1}$ the image code has the same size as the domain code, which is also the table size. Moreover, injective right-ideal morphisms are normal, thus Corollary 4.5C implies:

Corollary 4.6

For all injective right-ideal morphisms $\varphi,\psi$:   $\|\psi\circ\varphi\|\leq\|\psi\|+\|\varphi\|$.      $\Box$

In other words, the table size formula holds for ${\it Inv}_{k,1}$. Another immediate consequence of Theorem 4.5 is the following.

Corollary 4.7

Let $\varphi_{i}$ be normal right-ideal morphisms for $i=1,\ldots,n$, and let $c_{1},c_{2}$ be positive constants.

(1)*     If $\,|{\rm imC}(\varphi_{i})|\leq c_{1}$ for all $i$ then  $|{\rm imC}(\varphi_{n}\circ\ldots\circ\varphi_{1})|\ \leq\ c_{1}\,n$.*

(2)*     If $\,\ell({\rm imC}(\varphi_{i}))\leq c_{2}$ for all $i$ then  $\ell({\rm imC}(\varphi_{n}\circ\ldots\circ\varphi_{1}))\ \leq\ c_{2}\,n$.          $\Box$*

The position transposition $\tau_{i,j}$ (with $0<i<j$) is, by definition, the partial permutation of $A^{*}$ which transposes the letters at positions $i$ and $j$; $\tau_{i,j}$ is undefined on words of length $<j$. More precisely, we have  ${\rm domC}(\tau_{i,j})={\rm imC}(\tau_{i,j})=A^{j}$, and  $u\alpha v\beta\mapsto u\beta v\alpha$  for all letters $\alpha,\beta\in A$ and all words $u\in A^{i-1}$ and $v\in A^{j-i-1}$. In this form, $\tau_{i,j}$ is equal to its maximum essential extension.

Corollary 4.8

The word-length of $\tau_{i,j}$ over any finite generating set of $M_{k,1}$ is exponential.

Proof. We have $|{\rm imC}(\tau_{i,j})|=k^{j}$. The Corollary follows then from Corollary 4.7(1).    $\Box$

4.2 Some algorithmic problems about right-ideal morphisms

We consider several problems about right-ideal morphisms of $A^{*}$ and show that they have deterministic polynomial-time algorithms. We also show that the word problem of $M_{k,1}$ over $\Gamma_{k,1}\cup$ $\{\tau_{i,i+1}:0<i\}$ is coNP-complete, where $\Gamma_{k,1}$ is any finite generating set of $M_{k,1}$. We saw that $\Gamma_{k,1}$ can be chosen so as to consist of normal right-ideal morphisms.

Lemma 4.9

*There are deterministic polynomial time algorithms for the following problems.

Input: Two finite prefix codes $P_{1},P_{2}\subset A^{*}$, given explicitly by lists of words.

Output 1: The finite prefix code $\Pi\subset A^{*}$ such that $\Pi A^{*}=P_{1}A^{*}\cap P_{2}A^{*}$, where $\Pi$ is produced explicitly as a list of words.

Question 2: Is $P_{1}A^{*}\cap P_{2}A^{*}$ essential in $P_{1}A^{*}$ (or in $P_{2}A^{*}$, or in both)?*

Proof. We saw already that $\Pi$ exists and $\Pi\subseteq P_{1}\cup P_{2}$; see Lemma 3.3 of [4] (quoted before Lemma 4.3 above).

Algorithm for Output 1:  Since $\Pi\subseteq P_{1}\cup P_{2}$, we just need to search for the elements of $\Pi$ within $P_{1}\cup P_{2}$. For each $x\in P_{1}$ we check whether $x$ also belongs to $P_{2}A^{*}$ (by checking whether any element of $P_{2}$ is a prefix of $x$). Since $P_{1}$ and $P_{2}$ are explicitly given as lists, this takes polynomial time. Similarly, for each $x\in P_{2}$ we check whether $x$ also belongs to $P_{1}A^{*}$. Thus, we have computed the set   $\Pi_{1}=(P_{1}\cap P_{2}A^{*})\,\cup\,(P_{2}\cap P_{1}A^{*})$. Now, $\Pi$ is obtained from $\Pi_{1}$ by eliminating every word that has another word of $\Pi_{1}$ as a prefix. Since $\Pi_{1}$ is explicitly listed, this takes just polynomial time.

Algorithm for Question 2:  We first compute $\Pi$ by the previous algorithm. Next, we check whether every $p_{1}\in P_{1}$ is a prefix of some $r\in\Pi$; since $P_{1}$ and $\Pi$ are given by explicit lists, this takes just polynomial time. For $P_{2}$ it is similar.    $\Box$

Lemma 4.10

*The following input-output problem has a deterministic polynomial-time algorithm.

$\bullet$ Input: A finite set $S\subset A^{*}$, and $m$ right-ideal morphisms $\psi_{j}$ for $j=1,\ldots,m$, where $S$ is given by an explicit list of words, and each $\psi_{j}$ is given explicitly by the list of pairs of words $\{(x,\psi_{j}(x)):x\in{\rm domC}(\psi_{j})\}$.

$\bullet$ Output: The finite set $\,\psi_{m}\ldots\psi_{1}(S)$, given explicitly by a list of words.*

Proof. Let $\Psi=\,$ $\psi_{m}\circ\ \ldots\ \circ\psi_{1}\circ{\sf id}_{S}$. Then $\psi_{m}\ldots\psi_{1}(S)=\Psi({\rm domC}(\Psi))$. By Theorem 4.5(3) and (5), $\ell(\Psi({\rm domC}(\Psi)))$ $\leq$ $\ell(S)+\sum_{i=1}^{m}\ell(\psi_{i}({\rm domC}(\psi_{i})))\,$ and $\,|\Psi({\rm domC}(\Psi))|\leq$ $|S|+\sum_{i=1}^{m}|\psi_{i}({\rm domC}(\psi_{i}))|$. So the size of $\psi_{m}\ldots\psi_{1}(S)$, in terms of the number of words and their lengths, is polynomially bounded by the size of the input.

We now compute $\psi_{m}\ldots\psi_{1}(S)$ by applying $\psi_{j}$ to $\psi_{j-1}\ldots\psi_{1}(S)$ for increasing $j$. Since the sizes of the sets remain polynomially bounded, this algorithm takes polynomial time.    $\Box$

Corollary 4.11

*The following input-output problems have deterministic polynomial-time algorithms.

$\bullet$ Input: A list of $n$ right-ideal morphisms $\varphi_{i}$ for $i=1,\ldots,n$, given explicitly by finite tables.

$\bullet$ Output 1: A finite set, as an explicit list of words, that contains $\,\varphi_{n}\ldots$ $\varphi_{1}({\rm domC}(\varphi_{n}\ldots\varphi_{1}))$.

$\bullet$ Output 2: The finite set $\,{\rm imC}(\varphi_{n}\ldots\varphi_{1})$, as an explicit list of words.*

Proof. (1) By Theorem 4.5(5) we have $\,\varphi_{n}\ldots\varphi_{1}({\rm domC}(\varphi_{n}\ldots\varphi_{1}))$ $\ \subseteq\ \bigcup_{i=1}^{n}$ $\varphi_{n}\ldots\varphi_{i}({\rm domC}(\varphi_{i}))$. By Lemma 4.10, each set $\varphi_{n}\ldots\varphi_{i}({\rm domC}(\varphi_{i}))$, as well as their union, is computable in polynomial time (as an explicit list of words).

(2) Let $\Phi=\varphi_{n}\ldots\varphi_{1}$, $P_{i}={\rm domC}(\varphi_{i})$, and $Q_{i}={\rm imC}(\varphi_{i})$. As in the proof of Theorem 4.5(1),  ${\rm Dom}(\varphi_{2}\circ\varphi_{1})=$ $\varphi_{1}^{-1}(Q_{1}A^{*}\cap P_{2}A^{*})$,  ${\rm Im}(\varphi_{2}\circ\varphi_{1})=\varphi_{2}(Q_{1}A^{*}\cap P_{2}A^{*})$, and the maps $\varphi_{1}^{-1}(Q_{1}A^{*}\cap P_{2}A^{*})\ \stackrel{{\scriptstyle\varphi_{1}}}{{\longrightarrow}}$ \ Q_{1}A^{*}\cap P_{2}A^{*}\stackrel{{\scriptstyle\varphi_{2}}}{{\longrightarrow}}\ $\varphi_{2}(Q_{1}A^{*}\cap P_{2}A^{*})$  are total and onto. By Lemma 3.3 of [4] (mentioned before Theorem 4.5) we have $Q_{1}A^{*}\cap P_{2}A^{*}=S_{1}A^{*}$ for some finite prefix code $S_{1}$ with $S_{1}\subseteq Q_{1}\cup P_{2}$. Moreover, by Lemma 4.3, $\varphi_{2}(S_{1}A^{*})=R_{2}A^{*}$, where $\,{\rm imC}(\varphi_{2}\varphi_{1})=R_{2}\subseteq\varphi_{2}(S_{1})$.

By induction, for $j\geq 2$ suppose $\,{\rm imC}(\varphi_{j}\ldots\varphi_{1})=R_{j}$ $\subseteq$ $\varphi_{j}(S_{j-1})$, where $R_{j}$ and $S_{j-1}$ are finite prefix codes such that $S_{j-1}\subseteq R_{j-1}\cup P_{j}$,  $S_{j-1}A^{*}=R_{j-1}A^{*}\cap P_{j}A^{*}$,  $R_{j}A^{*}=$ ${\rm Im}(\varphi_{j}\ldots\varphi_{1})=\varphi_{j}(S_{j-1}A^{*})$, and the maps $\varphi_{j}^{-1}(R_{j}A^{*}\cap P_{j+1}A^{*})$ $\ \stackrel{{\scriptstyle\varphi_{j}}}{{\longrightarrow}}$ \ R_{j}A^{*}\cap P_{j+1}A^{*}\stackrel{{\scriptstyle\varphi_{j+1}}}{{\longrightarrow}}\ $\varphi_{j+1}(R_{j}A^{*}\cap P_{j+1}A^{*})$  are total and onto. Then by Lemma 3.3 of [4] we again have $R_{j}A^{*}\cap P_{j+1}A^{*}=S_{j}A^{*}$ for some finite prefix code $S_{j}$ with $S_{j}\subseteq R_{j}\cup P_{j+1}$; and by Lemma 4.3, $\varphi_{j+1}(S_{j}A^{*})=R_{j+1}A^{*}$ for some finite prefix code $R_{j+1}$ such that ${\rm imC}(\varphi_{j+1}\varphi_{j}\ldots\varphi_{1})=$ $R_{j+1}\subseteq\varphi_{j+1}(S_{j})$. Applying Theorem 4.5 to $\,R_{i}={\rm imC}(\varphi_{i}\ \ldots\ \varphi_{1})$ for any $i\geq 2$ we have

\,|R_{i}|\,\leq\,|\varphi_{i}(P_{i})|+\ \ldots\ $+$ $|\varphi_{2}(P_{2})|+|{\rm imC}(\varphi_{1})|$,  and

\,\ell(R_{i})\,\leq\,\ell(\varphi_{i}(P_{i}))+\ \ldots\ $+$ $\ell(\varphi_{2}(P_{2}))+\ell({\rm imC}(\varphi_{1}))$.

Since $S_{j}\subseteq P_{j}\cup R_{j-1}$, we have $|S_{j}|\leq|P_{j}|+|R_{j-1}|$ $\,\leq\,$ |P_{j}|+|\varphi_{j-1}(P_{j-1})|+\ \ldots\ $+$ $|\varphi_{2}(P_{2})|+|{\rm imC}(\varphi_{1})|$,  and $\ell(S_{j})\leq\ell(P_{j})+\ell(R_{j-1})$ $\,\leq\,$ \ell(P_{j})+\ell(\varphi_{j-1}(P_{j-1}))+\ \ldots\ $+$ $\ell(\varphi_{2}(P_{2}))+\ell({\rm imC}(\varphi_{1}))$. Thus, the size of each $R_{i}$ and $S_{j}$ is less than the input size; by input size we mean the total length of all the words in the input lists.

By Lemma 4.9, the prefix code $S_{j}$ is computed from $R_{j}$ and $P_{j+1}$, as an explicit list, in time $\leq T_{j}(|P_{j}|+\ell(P_{j})+|R_{j-1}|+\ell(R_{j-1}))$, for some polynomial $T_{j}(.)$. And $R_{j+1}$ is computed from $S_{j}$ by applying $\varphi_{j+1}$ to $S_{j}$, and then keeping the elements that do not have a prefix in $\varphi_{j+1}(S_{j})$. Computing $\varphi_{j+1}(S_{j})$ takes at most quadratic time, and finding the prefix code in $\varphi_{j+1}(S_{j})$ also takes at most quadratic time.

In the end we obtain $R_{n}={\rm imC}(\varphi_{n}\ldots\varphi_{1})$ as an explicit list of words.    $\Box$

When we consider the word problem of $M_{k,1}$ over a finite generating set, we measure the input size by the length of input word (with each generator having length 1). But for the word problem of $M_{k,1}$ over the infinite generating set $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$ we count the length of the position transpositions $\tau_{i-1,i}$ as $i$, in the definition of the input size of the word problem. Indeed, at least $\log_{2}i$ bits are needed to describe the subscript $i$ of $\tau_{i-1,i}$. Moreover, in the connection between $M_{k,1}$ (over $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$) and circuits, $\tau_{i-1,i}$ is interpreted as the wire-crossing operation of wire number $i$ and wire number $i-1$; this suggests that viewing the size of $\tau_{i-1,i}$ as $i$ is more natural than $\log_{2}i$. In any case, we will see next that the word problem of $M_{k,1}$ over $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$ is coNP-complete, even if the size of $\tau_{i-1,i}$ is more generously measured as $i$; this is a stronger result than if $\log_{2}i$ were used.

Theorem 4.12

(coNP-complete word problem).*  The word problem of $M_{k,1}$ over the infinite generating set $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$ is coNP-complete, where $\Gamma_{k,1}$ is any finite generating set of $M_{k,1}$.*

Proof. In [5] (see also [3]) it was shown that the word problem of the Thompson-Higman group $G_{k,1}$ over $\Gamma_{G_{k,1}}\cup\{\tau_{i-1,i}:i>1\}$  is coNP-complete, where $\Gamma_{G_{k,1}}$ is any finite generating set of $G_{k,1}$. Hence, since the elements of the finite set $\Gamma_{G_{k,1}}$ can be expressed by a finite set of words over $\Gamma_{k,1}$, it follows that the word problem of $M_{k,1}$ over $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$ is coNP-hard.

We will prove now that the word problem of $M_{k,1}$ over $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$ belongs to coNP. The input of the problem consists of two words $(\rho_{m},\ldots,\rho_{1})$ and $(\sigma_{n},\ldots,\sigma_{1})$ over $\Gamma_{k,1}\cup\{\tau_{i-1,i}:i>1\}$. The input size is the weighted length of the words $(\rho_{m},\ldots,\rho_{1})$ and $(\sigma_{n},\ldots,\sigma_{1})$, where each generator in $\Gamma_{k,1}$ has weight 1, and each generator of the form $\tau_{i-1,i}$ has weight $i$. For every right-ideal morphism $\varphi$ we abbreviate $\ell({\rm domC}(\varphi)\cup\varphi({\rm domC}(\varphi)))$ by $\ell(\varphi)$; recall that for a finite set $X\subset A^{*}$, $\ell(X)$ denotes the length of a longest word in $X$.

Since $\Gamma_{k,1}$ is finite there is a constant $c>0$ such that $c\geq\ell(\gamma)$ for all $\gamma\in\Gamma_{k,1}$; also, for each $\tau_{i-1,i}$ we have $\ell(\tau_{i-1,i})=i$. By Theorem 4.5, the table of $\sigma_{n}\circ\ldots\circ\sigma_{1}$ (and more generally, the table of $\sigma_{j}\circ\ldots\circ\sigma_{1}$ for any $j$ with $n\geq j\geq 1$) contains only words of length $\leq\sum_{j=1}^{n}\ell(\sigma_{j})$, and similarly for $\rho_{m}\circ\ldots\circ\rho_{1}$ (and for $\rho_{i}\circ\ldots\circ\rho_{1}$, $m\geq i\geq 1$). So all the words in the tables for any $\sigma_{j}\circ\ldots\circ\sigma_{1}$ and any $\rho_{i}\circ\ldots\circ\rho_{1}$ have lengths that are linearly bounded by the size of the input \big{(}(\rho_{m},\ldots,\rho_{1}), (\sigma_{n},\ldots,\sigma_{1})\big{)}.

Claim. *Let N=\max\{\sum_{i=1}^{m}\ell(\rho_{i}),\ $\sum_{j=1}^{n}\ell(\sigma_{j})\}$. Then  $\rho_{m}\cdot\ldots\cdot\rho_{1}\neq\sigma_{n}\cdot\ldots\cdot\sigma_{1}$  in $M_{k,1}$  iff  there exists $x\in A^{N}$ such that  $\rho_{m}\circ\ldots\circ\rho_{1}(x)$ $\neq$ $\sigma_{n}\circ\ldots\circ\sigma_{1}(x)$. *

Proof of the Claim: As we saw above, the tables of $\rho_{m}\circ\ldots\circ\rho_{1}$ and $\sigma_{n}\circ\ldots\circ\sigma_{1}$ only contain words of length $\leq N$. Thus, restricting $\rho_{m}\circ\ldots\circ\rho_{1}$ and $\sigma_{n}\circ\ldots\circ\sigma_{1}$ to $A^{N}A^{*}$ is an essential restriction, and the resulting tables have domain codes in $A^{N}$. Therefore, $\rho_{m}\cdot\ldots\cdot\rho_{1}$ and $\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ are equal (as elements of $M_{k,1}$)   iff   $\rho_{m}\circ\ldots\circ\rho_{1}$ and $\sigma_{n}\circ\ldots\circ\sigma_{1}$ are equal on $A^{N}$.      [End, Proof of Claim]

The number $N$ in the Claim is immediately obtained form the input. Based on the Claim, we obtain a nondeterministic polynomial-time algorithm which decides (nondeterministically) whether there exists $x\in A^{N}$ such that  $\rho_{m}\circ\ldots\circ\rho_{1}(x)$ $\neq\sigma_{n}\circ\ldots\circ\sigma_{1}(x)$, as follows:

The algorithm guesses $x\in A^{N}$, computes $\rho_{m}\circ\ldots\circ\rho_{1}(x)$ and $\sigma_{n}\circ\ldots\circ\sigma_{1}(x)$, and checks that they are different words ($\in A^{*}$) or that one is undefined and the other is a word. Applying Theorem 4.5 to $\,\rho_{m}\circ\ldots\circ\rho_{1}\circ{\sf id}_{A^{N}}\,$ and to $\,\sigma_{n}\circ\ldots\circ\sigma_{1}\circ{\sf id}_{A^{N}}\,$ shows that  $|\rho_{m}\circ\ldots\circ\rho_{1}(x)|\leq 2N$  and  $|\sigma_{n}\circ\ldots\circ\sigma_{1}(x)|\leq 2N$; here $|\rho_{m}\circ\ldots\circ\rho_{1}(x)|$ denotes the length of the word $\rho_{m}\circ\ldots\circ\rho_{1}(x)\in A^{*}$, and similarly for $\sigma_{n}\circ\ldots\circ\sigma_{1}(x)$. Also by Theorem 4.5, all intermediate results (as we successively apply $\rho_{i}$ for $i=1,\ldots,m$, or $\sigma_{j}$ for $j=1,\ldots,n$) are words of length $\leq 2N$. These successive words are computed by applying the table of $\rho_{i}$ or $\sigma_{j}$ (when $\rho_{i}$ or $\sigma_{j}$ belong to $\Gamma_{k,1}$), or by directly applying the position permutation $\tau_{h,h-1}$ (if $\rho_{i}$ or $\sigma_{j}$ is $\tau_{h,h-1}$). Thus, the output $\rho_{m}\circ\ldots\circ\rho_{1}(x)$ (and similarly, $\sigma_{n}\circ\ldots\circ\sigma_{1}(x)$) can be computed in polynomial time.    $\Box$

4.3 The word problem of $M_{k,1}$ is in P

We now move ahead with the the proof of our main result.

Theorem 4.13

(Word problem in P).*  The word problem of the Thompson-Higman monoids $M_{k,1}$, over any finite generating set, can be decided in deterministic polynomial time.*

We assume that a fixed finite generating set $\Gamma_{k,1}$ of $M_{k,1}$ has been chosen. The input consists of two sequences $(\rho_{m},\ldots,\rho_{1})$ and $(\sigma_{n},\ldots,\sigma_{1})$ over $\Gamma_{k,1}$, and the input size is $m+n$; since $\Gamma_{k,1}$ is finite and fixed, it does not matter whether we choose $m+n$ as input size, or the sum of the lengths of all the words in the tables of the elements of $\Gamma_{k,1}$. We want to decide in deterministic polynomial time whether, as elements of $M_{k,1}$, the products $\rho_{m}\cdot\ldots\cdot\rho_{1}$ and $\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ are equal.

Overview of the proof:

$\bullet$  We compute the finite sets $\,{\rm imC}(\rho_{m}\circ\ldots\circ\rho_{1})$, $\,{\rm imC}(\sigma_{n}\circ\ldots\circ\sigma_{1})\subset A^{*}$, explicitly described by lists of words. By Corollary 4.11 (Output 2) this can be done in polynomial time, and these sets have polynomial size. (Note however that by Proposition 4.2, the table sizes of $\rho_{m}\circ\ldots\circ\rho_{1}$ or $\sigma_{n}\circ\ldots\circ\sigma_{1}$ could be exponential in $m$ or $n$.)

$\bullet$  We check whether ${\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ $\cap$ ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$ is essential in ${\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ and in ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$. By Lemma 4.9 (Question 2) this can be done in polynomial time. If the answer is “no” then $\rho_{m}\cdot\ldots\cdot\rho_{1}$ $\neq$ $\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ in $M_{k,1}$, since they don’t have a common maximum essential extension. Otherwise, the computation continues.

$\bullet$  We compute the finite prefix code $\Pi\subset A^{*}$ such that $\Pi A^{*}={\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ $\cap$ ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$. By Lemma 4.9 (Output 1) this can be done in polynomial time, and $\Pi$ has polynomial size. Hence, the table of ${\sf id}_{\Pi A^{*}}$ can be computed in polynomial time.

$\bullet$  We restrict $\rho_{m}\circ\ldots\circ\rho_{1}$ and $\sigma_{n}\circ\ldots\circ\sigma_{1}$ in such a way that their images are in $\Pi A^{*}$. In other words, we replace them by $\rho=$ ${\sf id}_{\Pi A^{*}}\circ\rho_{m}\circ\ldots\circ\rho_{1}$, respectively $\sigma=$ ${\sf id}_{\Pi A^{*}}\circ\sigma_{n}\circ\ldots\circ\sigma_{1}$. Since $\Pi A^{*}$ is essential in ${\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ and in ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$, we have $\rho=\rho_{m}\cdot\ldots\cdot\rho_{1}$ in $M_{k,1}$, and $\sigma=\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ in $M_{k,1}$. So, $\rho_{m}\cdot\ldots\cdot\rho_{1}=\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ in $M_{k,1}$ iff $\rho=\sigma$ in $M_{k,1}$.

$\bullet$  We compute finite sets $R_{1},R_{2}\subset A^{*}$, such that $\rho({\rm domC}(\rho))\subseteq R_{1}$ and $\sigma({\rm domC}(\sigma))\subseteq R_{2}$. Since $\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))\subseteq\Pi A^{*}$, we can pick $R_{1},R_{2}$ so that $R_{1}\cup R_{2}\subseteq\Pi A^{*}$. By Corollary 4.11 (Output 1), the sets $R_{1},R_{2}$ can be computed as explicit lists in polynomial time. Let $R=R_{1}\cup R_{2}$.

$\bullet$  We note that $\rho=\sigma$ in $M_{k,1}$ iff for all $r\in\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))$: $\,\rho^{-1}(r)=\sigma^{-1}(r)$. This holds iff for all $r\in R$: $\,\rho^{-1}(r)=\sigma^{-1}(r)$.

$\bullet$  For every $r\in R$ we construct a deterministic finite automaton (DFA) accepting the finite set $\rho^{-1}(r)\subset A^{*}$, and a DFA accepting the finite set $\sigma^{-1}(r)\subset A^{*}$. By Corollary 4.15 this can be done in polynomial time, and the DFAs have polynomial size. (The finite sets $\rho^{-1}(r)$ and $\sigma^{-1}(r)$ themselves could have exponential size.) Note that ${\rm domC}(\rho)\subseteq\rho^{-1}(\rho({\rm domC}(\rho)))$ $\subseteq\rho^{-1}(R)$, and similarly for $\sigma$. Note that usually, ${\rm domC}(\rho)\not\subseteq\rho^{-1}({\rm imC}(\rho))\,$ (since $\rho$ is not normal in general), and similarly for $\sigma$; so we need to use $\rho({\rm domC}(\rho))$, and not just ${\rm imC}(\rho)$.

$\bullet$  For every $r\in R$ we check whether the DFA for $\rho^{-1}(r)$ and the DFA for $\sigma^{-1}(r)$ are equivalent. By classical automata theory, equivalence of DFAs can be checked in polynomial time.

[End of Overview.]

Automata – notation and facts: In the following, DFA stands for deterministic finite automaton. The language accepted by a DFA ${\cal A}$ is denoted by ${\cal L}({\cal A})$. A DFA is a structure $(S,A,\delta,s_{0},F)$ where $S$ is the set of states, $A$ is the input alphabet, $s_{0}\in S$ is the start state, $F\subseteq S$ is the set of accept states, and $\delta:S\times A\to S$ is the next-state function; in general, $\delta$ is a partial function (by “function” we always mean partial function). We extend the definition of $\delta$ to a function $S\times A^{*}\to S$ by defining $\delta(s,w)$ to be the state that the DFA reaches from $s$ after reading $w$ (for any $w\in A^{*}$ and $s\in S$). See [21, 24] for background on finite automata. A DFA is called acyclic iff its underlying directed graph has no directed cycle. It is easy to prove that a language $L\subseteq A^{*}$ is finite iff $L$ is accepted by an acyclic DFA. Moreover, $L$ is a finite prefix code iff $L$ is accepted by an acyclic DFA that has a single accept state (take the prefix tree of the prefix code, with the leaves as accept states, then glue all the leaves together into a single accept state). By the size of a DFA ${\cal A}$ we mean the number of states, $|S|$; we denote this by ${\sf size}({\cal A})$. For a finite set $P\subseteq A^{*}$ we denote the length of the longest words in $P$ by $\ell(P)$, and we define the total length of $P$ by $\,\Sigma(P)=\sum_{x\in P}|x|$; obviously, $\Sigma(P)\leq|P|\cdot\ell(P)$.

For a language $L\subseteq A^{*}$ and a partial function $\Phi:A^{*}\to A^{*}$, we define the inverse image of $L$ under $\Phi$ by   $\Phi^{-1}(L)=\{x\in A^{*}:\Phi(x)\in L\}$.

For $L\subseteq A^{*}$ we denote the set of all strict prefixes of the words in $L$ by spref$(L)$; precisely, ${\sf spref}(L)=\{x\in A^{*}:(\exists w\in L)[\,x\leq_{\rm pref}w$ and $x\neq w\,]\}$.

The reason why we use acyclic DFAs to describe finite sets is that a finite set can be exponentially larger than the number of states of a DFA that accepts it; e.g., $A^{n}$ is accepted by an acyclic DFA with $n+1$ states. This conciseness plays a crucial role in our polynomial-time algorithm for the word problem of $M_{k,1}$.

Lemma 4.14

Let ${\cal A}$ be an acyclic DFA with a single accept state. Let $\varphi$ be a normal right-ideal morphism, with $\,{\rm domC}(\varphi)\neq\{\varepsilon\}\,$ and $\,{\rm imC}(\varphi)\neq\{\varepsilon\}$.

Then $\varphi^{-1}({\cal L}({\cal A}))$ is accepted by a one-accept-state acyclic DFA $\varphi^{-1}({\cal A})$ whose number of states is $\,{\sf size}(\varphi^{-1}({\cal A}))$ $<$ ${\sf size}({\cal A})+\Sigma({\rm domC}(\varphi))$. The transition table of the DFA $\varphi^{-1}({\cal A})$ can be constructed deterministically in polynomial time, based on the transition table of ${\cal A}$ and the table of $\varphi$.

Proof. If $\,\varphi^{-1}({\cal L}({\cal A}))=\varnothing$ then ${\sf size}(\varphi^{-1}({\cal A}))=0$, so the result is trivial. Let us assume now that $\,\varphi^{-1}({\cal L}({\cal A}))\neq\varnothing$. Let ${\cal A}=(S,A,\delta,s_{0},\{s_{A}\})$ where $s_{A}$ is the single accept state; $s_{A}$ has no out-going edges (they would be useless). For any set $X\subseteq A^{*}$ and any state $s\in S$ we denote $\{\delta(s,x):x\in X\}$ by $\delta(s,X)$. Let $P={\rm domC}(\varphi)$ and $Q={\rm imC}(\varphi)$. Since ${\cal A}$ is acyclic, its state set $S$ can be partitioned into $\delta(s_{0},{\sf spref}(Q))$ and $\delta(s_{0},QA^{*})$. Since $Q\neq\{\varepsilon\}$, the block $\delta(s_{0},{\sf spref}(Q))$ contains $s_{0}$, so the block is non-empty. The block $\delta(s_{0},QA^{*})$ is non-empty because of the assumption $\varphi^{-1}({\cal L}({\cal A}))\neq\varnothing$, which implies ${\cal L}({\cal A})\cap QA^{*}\neq\varnothing$.

Since ${\cal L}({\cal A})$ is a prefix code and $\varphi$ is a right-ideal morphism, $\varphi^{-1}({\cal L}({\cal A}))$ is a prefix code. To accept $\varphi^{-1}({\cal L}({\cal A}))$ we define an acyclic DFA, called $\varphi^{-1}({\cal A})$, as follows:

$\bullet$  State set of $\varphi^{-1}({\cal A})$:  ${\sf spref}(P)\ \cup\ \delta(s_{0},QA^{*})$;

start state: $\varepsilon$, i.e., the root of the prefix tree of $P$ (since $P\neq\{\varepsilon\}$, $\,\varepsilon\in{\sf spref}(P)$);

accept state: the accept state $s_{A}$ of ${\cal A}$.

$\bullet$  State-transition function $\delta_{1}$ of $\varphi^{-1}({\cal A})$:

For every $r\in{\sf spref}(P)$ and $a\in A$ such that $ra\in{\sf spref}(P)$:   $\delta_{1}(r,a)=ra$.

For every $r\in{\sf spref}(P)$ and $a\in A$ such that $ra\in P$:   $\delta_{1}(r,a)=\delta(s_{0},\varphi(ra))$.

For every $s\in\delta(s_{0},QA^{*})$:   $\delta_{1}(s,a)=\delta(s,a)$.

It follows immediately from this definition that for all $p\in P$:  $\delta_{1}(\varepsilon,p)=\delta(s_{0},\varphi(p))$. The construction of $\varphi^{-1}({\cal A})$ assumes that $\varphi$ maps $P$ onto $Q$, i.e., it uses the assumption that $\varphi$ is normal. As usual, “function” means partial function, so $\delta(.,.)$ and $\delta_{1}(.,.)$ need not be defined on every state-letter pair.

The DFA $\varphi^{-1}({\cal A})$ can be pictured as being constructed as follows: The DFA has two parts. The first part is the prefix tree of $P$, but with the leaves left out (and with edges to leaves left dangling). The second part is the DFA ${\cal A}$ restricted to the state subset $\delta(s_{0},QA^{*})$. The two parts are glued together by connecting any dangling edge, originally pointing to a leaf $p\in P$, to the state $\,\delta(s_{0},\varphi(p))\in\delta(s_{0},QA^{*})$.

The description of $\varphi^{-1}({\cal A})$ constitutes a deterministic polynomial time algorithm for constructing the transition table of $\varphi^{-1}({\cal A})$, based on the transition table of ${\cal A}$ and on the table of $\varphi$. By the construction, the number of states of $\varphi^{-1}({\cal A})$ is $\,<{\sf size}({\cal A})+\Sigma(P)$ We will prove now that the DFA $\varphi^{-1}({\cal A})$ accepts exactly $\varphi^{-1}({\cal L}({\cal A}))$; i.e.,  $\varphi^{-1}({\cal L}({\cal A}))={\cal L}(\varphi^{-1}({\cal A}))$.

$[\subseteq]$  Consider any $y\in{\cal L}({\cal A})$ such that $\varphi^{-1}(y)\neq\varnothing$. We want to show that $\varphi^{-1}({\cal A})$ accepts all the words in $\varphi^{-1}(y)$. Since $\varphi^{-1}(y)\neq\varnothing$ we have $y\in{\rm Im}(\varphi)$, hence $y=qw$ for some strings $q\in Q={\rm imC}(\varphi)$ and $w\in A^{*}$. Since $Q$ is a prefix code, $q$ and $w$ are uniquely determined by $y$. Moreover, since $y\in{\cal L}({\cal A})$ it follows that $y$ has an accepting path in ${\cal A}$ of the form

s_{0}\ \stackrel{{\scriptstyle q}}{{\longrightarrow}}\ \delta(s_{0},q)\ $\stackrel{{\scriptstyle w}}{{\longrightarrow}}\ s_{A}$.

For every $x\in\varphi^{-1}(y)$ we have $x\in{\rm Dom}(\varphi)=PA^{*}$, hence $x=pv$ for some strings $p\in P$ and $v\in A^{*}$. So $\varphi(x)=\varphi(p)\ v$. We also have $\varphi(x)=y=qw$, hence $\varphi(p)$ and $q$ are prefix-comparable. Therefore, $\varphi(p)=q$, since $Q$ is a prefix code and since $\varphi(p)\in Q$ (by normality of $\varphi$); hence $v=w$. Thus every $x\in\varphi^{-1}(y)$ has the form $pw$ for some string $p\in\varphi^{-1}(q)$. Now in $\varphi^{-1}({\cal A})$ there is the following accepting path on input $x=pw\in\varphi^{-1}(y)$:

$\varepsilon\ \stackrel{{\scriptstyle p}}{{\longrightarrow}}\ \delta_{1}(\varepsilon,p)=$ $\delta(s_{0},\varphi(p))\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ s_{A}$.

Thus $\varphi^{-1}({\cal A})$ accepts $x=pw=pv$.

$[\supseteq]$  Suppose $\varphi^{-1}({\cal A})$ accepts $x$. Then, because of the prefix tree of $P$ at the beginning of $\varphi^{-1}({\cal A})$, $x$ has the form $x=pw$ for some strings $p\in P$ and $w\in A^{*}$. The accepting path in $\varphi^{-1}({\cal A})$ on input $pw$ has the form

$s_{0}\ \stackrel{{\scriptstyle p}}{{\longrightarrow}}\ \delta_{1}(\varepsilon,p)=$ $\delta(s_{0},\varphi(p))\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ s_{A}$.

Also, $\varphi(x)=qw$ where $q=\varphi(p)\in Q$ (here we use normality of $\varphi$). Hence ${\cal A}$ has the following computation path on input $qw$:

$s_{0}\ \stackrel{{\scriptstyle q}}{{\longrightarrow}}\ \delta(s_{0},q)=$ $\delta(s_{0},\varphi(p))\ \stackrel{{\scriptstyle w}}{{\longrightarrow}}\ s_{A}$.

So, $\varphi(x)=\varphi(p)\,w=qw\in{\cal L}({\cal A})$. Hence, $x\in\varphi^{-1}(qw)$ $\subseteq\varphi^{-1}({\cal L}({\cal A}))$. Thus ${\cal L}(\varphi^{-1}({\cal A}))$ $\subseteq$ $\varphi^{-1}({\cal L}({\cal A}))$.    $\Box$

Corollary 4.15

Let ${\cal A}$ be an acyclic DFA with a single accept state. For $i=1,\ldots,n$, let $P_{i},Q_{i}\subset A^{*}$ be finite prefix codes, and let $\varphi_{i}:P_{i}A^{*}\to Q_{i}A^{*}$ be normal right-ideal morphisms. We assume that $\,P_{i}\neq\{\varepsilon\}\,$ and $\,Q_{i}\neq\{\varepsilon\}$.

Then $\,(\varphi_{n}\circ\ldots\circ\varphi_{1})^{-1}({\cal L}({\cal A}))\,$ is accepted by an acyclic DFA with size $\,<{\sf size}({\cal A})+\sum_{i=1}^{n}\Sigma(P_{i})$, with one accept state. The transition table of this DFA can be constructed deterministically in polynomial time, based on the transition table of ${\cal A}$ and the tables of $\varphi_{i}$ (for $i=1,\ldots,n$).

Proof. We assume that $(\varphi_{n}\circ\ldots\circ\varphi_{1})^{-1}({\cal L}({\cal A}))$ $\neq$ $\varnothing\,$ (since the empty set is accepted by a DFA of size 0). We use induction on $n$. For $n=1$ the Corollary is just Lemma 4.14.

Let $n\geq 1$, assume the Corollary holds for $n$ normal morphisms, and consider one more normal right-ideal morphism $\varphi_{0}:P_{0}A^{*}\to Q_{0}A^{*}$, where $P_{0},Q_{0}\subset A^{*}$ are finite prefix codes with $P_{0}\neq\{\varepsilon\}\neq Q_{0}$. And assume $\,(\varphi_{n}\circ\ldots\circ\varphi_{1}$ $\circ$ $\varphi_{0})^{-1}({\cal L}({\cal A}))\neq\varnothing$.

Since $(\varphi_{n}\circ\ldots\circ\varphi_{1}\circ$ $\varphi_{0})^{-1}({\cal L}({\cal A}))$ $=$ $\varphi_{0}^{-1}\circ(\varphi_{n}\circ\ldots\circ$ $\varphi_{1})^{-1}({\cal L}({\cal A}))$, let us apply Lemma 4.14 to $\varphi_{0}$ and the acyclic DFA $(\varphi_{n}\circ\ldots\circ\varphi_{1})^{-1}({\cal A})$. We have $\varepsilon\not\in{\rm Dom}(\varphi_{n}\ \ldots\ \varphi_{1}\varphi_{0})$; indeed, $P_{i}\neq\{\varepsilon\}$ is equivalent to $\varepsilon\not\in{\rm Dom}(\varphi_{i})$; moreover we have $\varepsilon\not\in{\rm Dom}(\varphi_{0})$, and ${\rm Dom}(\varphi_{n}\ \ldots\ \varphi_{1}\varphi_{0})$ $\subseteq$ ${\rm Dom}(\varphi_{0})$. Similarly, $Q_{i}\neq\{\varepsilon\}$ is equivalent to $\varepsilon\not\in{\rm Im}(\varphi_{i})$; and $\varepsilon\not\in{\rm Im}(\varphi_{n})$ implies $\varepsilon\not\in{\rm Im}(\varphi_{n}\ \ldots\ \varphi_{1}\varphi_{0})$.

The conclusion of Lemma 4.14 is then that $(\varphi_{n}\circ\ldots\circ\varphi_{1}$ $\circ$ $\varphi_{0})^{-1}({\cal L}({\cal A}))$ is accepted by an acyclic DFA $(\varphi_{n}\circ\ldots\circ\varphi_{1}\circ\varphi_{0})^{-1}({\cal A})$ whose size is $\,<\,$ ${\sf size}((\varphi_{n}\circ\ldots\circ\varphi_{1})^{-1}({\cal A}))$ $+$ $\Sigma(P_{0})$ $\,<\,{\sf size}({\cal A})+\sum_{i=1}^{n}\Sigma(P_{i})+\Sigma(P_{0})$ $=$ ${\sf size}({\cal A})+\sum_{i=0}^{n}\Sigma(P_{i})$.    $\Box$

Proof of Theorem 4.13:

Let $(\rho_{m},\ldots,\rho_{1})$ and $(\sigma_{n},\ldots,\sigma_{1})$ be two sequences of generators from the finite generating set $\Gamma_{k,1}$. The elements of $\Gamma_{k,1}$ can be chosen so that the assumptions of Corollary 4.15 hold; see Section 3 of [1], where such a generating set is given. We want to decide in deterministic polynomial time whether the products $\rho_{m}\cdot\ldots\cdot\rho_{1}$ and $\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ are the same, as elements of $M_{k,1}$.

First, by Corollary 4.11 (Output 2) we can compute the sets ${\rm imC}(\rho_{m}\circ\ldots\circ\rho_{1})$ and ${\rm imC}(\sigma_{n}\circ\ldots\circ\sigma_{1})$, explicitly described by lists of words, in polynomial time. By Lemma 4.9 (Question 2) we can check in polynomial time whether the right ideal $\,{\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ $\cap$ ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})\,$ is essential in ${\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ and in ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$. If it is not essential we immediately conclude that $\rho_{m}\cdot\ldots\cdot\rho_{1}\neq$ $\sigma_{n}\cdot\ldots\cdot\sigma_{1}$. On the other hand, if it is essential, Lemma 4.9 (Output 1) lets us compute a generating set $\Pi$ for the right ideal $\,{\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ $\cap$ ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$, in deterministic polynomial time; the generating set $\Pi$ is a finite prefix code, given explicitly by a list of words. By Corollary 4.7 and because $\,\Pi\subseteq$ ${\rm imC}(\rho_{m}\circ\ldots\circ\rho_{1})$ $\cup$ ${\rm imC}(\sigma_{n}\circ\ldots\circ\sigma_{1})$, $\Pi$ has linearly bounded cardinality and the length of the longest words in $\Pi$ is linearly bounded in terms of $n+m$.

We restrict $\rho_{m}\circ\ldots\circ\rho_{1}$ and $\sigma_{n}\circ\ldots\circ\sigma_{1}$ in such a way that their images are $\Pi A^{*}$; i.e., we replace them by $\rho=$ ${\sf id}_{\Pi A^{*}}\circ\rho_{m}\circ\ldots\circ\rho_{1}$, respectively $\sigma={\sf id}_{\Pi A^{*}}\circ\sigma_{n}\circ\ldots\circ\sigma_{1}$. So, ${\rm Im}(\rho)=\Pi A^{*}={\rm Im}(\sigma)$. Also, since $\Pi A^{*}$ is essential in ${\rm Im}(\rho_{m}\circ\ldots\circ\rho_{1})$ and in ${\rm Im}(\sigma_{n}\circ\ldots\circ\sigma_{1})$ we have: $\rho$ is equal to $\rho_{m}\cdot\ldots\cdot\rho_{1}$ in $M_{k,1}$, and $\sigma$ is equal to $\sigma_{n}\cdot\ldots\cdot\sigma_{1}$ in $M_{k,1}$. So for deciding the word problem it is enough to check whether $\rho=\sigma$ in $M_{k,1}$.

By the next Claim, the sets $\rho({\rm domC}(\rho))$ and $\sigma({\rm domC}(\sigma))$ play a crucial role. However, instead of directly computing $\rho({\rm domC}(\rho))$ and $\sigma({\rm domC}(\sigma))$, we compute finite sets $R_{1},R_{2}\subset A^{*}$ such that $\,\rho({\rm domC}(\rho))\subseteq R_{1}\,$ and $\,\sigma({\rm domC}(\sigma))\subseteq R_{2}\,$. Moreover, since $\,\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))$ $\subseteq$ $\Pi A^{*}$, we can pick $R_{1},R_{2}$ so that $R_{1}\cup R_{2}\subseteq\Pi A^{*}$. By Corollary 4.11 (Output 1), the sets $R_{1},R_{2}$ can be computed in polynomial time as explicit lists of words. Let $R=R_{1}\cup R_{2}$.

Claim.  $\rho=\sigma$* in $M_{k,1}$  iff  $\,\rho^{-1}(r)=\sigma^{-1}(r)\,$ for every $\,r\in\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))$. The latter is equivalent to $\,\rho^{-1}(r)=\sigma^{-1}(r)\,$ for every $r\in R$. *

Proof of the Claim. If $\rho=\sigma$ in $M_{k,1}$ then $\rho^{-1}(r)=\sigma^{-1}(r)$ for every $r\in\Pi A^{*}={\rm Im}(\rho)$ $=$ ${\rm Im}(\sigma)$. Hence this holds in particular for all $r\in\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))$ and for all $r\in R$, since $\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))$ $\subseteq R\subset\Pi A^{*}$. Conversely, if $\rho^{-1}(r)=\sigma^{-1}(r)$ for every $r\in$ $\rho({\rm domC}(\rho))\cup\sigma({\rm domC}(\sigma))$, then for all $x\in\rho^{-1}(r)=\sigma^{-1}(r)$:  $\rho(x)=r=\sigma(x)$. Since ${\rm domC}(\rho)\subseteq\rho^{-1}(\rho({\rm domC}(\rho)))$ and ${\rm domC}(\sigma)\subseteq\sigma^{-1}(\sigma({\rm domC}(\sigma)))$, it follows that $\rho$ and $\sigma$ are equal on ${\rm domC}(\rho)\cup{\rm domC}(\sigma)$, and it follows that ${\rm domC}(\rho)={\rm domC}(\sigma)$. Hence $\rho$ and $\sigma$ are equal as right-ideal morphisms, and hence as elements of $M_{k,1}$.    [This proves the Claim.]

Recall that $|R|$ and $\ell(R)$, and hence $\Sigma(R)$, are polynomially bounded in terms of the input size. To check for each $r\in R$ whether $\,\rho^{-1}(r)=\sigma^{-1}(r)$, we apply Corollary 4.15, which constructs an acyclic DFA ${\cal A}_{\rho}$ for $\rho^{-1}(r)$ from a DFA for $\{r\}$; this is done deterministically in polynomial time. Similarly, an acyclic DFA ${\cal A}_{\sigma}$ for $\sigma^{-1}(r)$ is constructed. Thus, $\rho^{-1}(r)=\sigma^{-1}(r)$ iff ${\cal A}_{\rho}$ and ${\cal A}_{\sigma}$ accept the same language.

Checking whether ${\cal A}_{\rho}$ and ${\cal A}_{\sigma}$ accept the same language is an instance of the equivalence problem for DFAs that are given explicitly by transition tables. It is well known (see e.g., [21], or [24] p. 103) that the equivalence problem for DFAs is decidable deterministically in polynomial time. This proves Theorem 4.13.    $\Box$

Acknowledgement. I would like to thank John Meakin for many discussions over the years concerning the Thompson groups and generalizations to inverse monoids.