An uncountable J\'{o}nsson algebra in a minimal variety

TL;DR

This paper constructs a Jónsson algebra of size  in the specific variety of Jf3nsson-Tarski algebras, demonstrating a new example within algebraic structures.

Contribution

It provides the first known example of an uncountable Jf3nsson algebra in a minimal variety, expanding understanding of algebraic diversity.

Findings

Constructed a -sized Jf3nsson algebra in Jf3nsson-Tarski variety

Demonstrated existence of uncountable Jf3nsson algebra in minimal variety

Extended algebraic theory of Jf3nsson algebras

Abstract

We construct a J\'{o}nsson algebra of cardinality $\omega_1$ in the variety of J\'{o}nsson-Tarski algebras.

Tables

Table 1. Table 1. Partition of odd ordinals into the sets L λ + n subscript 𝐿 𝜆 𝑛 L_{\lambda+n} .

AA$L_{\lambda }$AA	$L_{\lambda +1}$	$L_{\lambda +2}$	$L_{\lambda +3}$	$\cdots$
$\lambda +1$	$\lambda +3$
$\lambda +5$	$\lambda +7$	$\lambda +9$
$\lambda +11$	$\lambda +13$	$\lambda +15$	$\lambda +17$
$\lambda +19$	$\lambda +21$	$\lambda +23$	$\lambda +25$	$\cdots$

Equations

$$|J/\theta |=|⟨a/\theta ⟩|=|⟨a⟩/\theta |\le |⟨a⟩|\le n.$$

$$\prod _{A\in \mathcal{V},|A|<\omega }J/{\theta }_A,$$

$$|J|\le \left|\prod _{A\in \mathcal{V},|A|<\omega }J/{\theta }_A\right|\le {\omega }^{2^{\omega }}=2^{2^{\omega }}$$

$$A\times A\to A:(x,y)\mapsto x\cdot y,\text{and}A\to A\times A:z\mapsto (\ell (z),r(z))$$

$$rP(x_{\ell })=rP\ell (x_{\ell }\cdot x_r)=r(x_{\ell }\cdot x_r)=x_r.$$

$$p\cdot q=\left(\genfrac{}{}{0ex}{}{p+q+1}{2}\right)+q,\mathrm{if}p+q>1,$$

Full text

This material is based upon work supported by the National Science Foundation grant no. DMS 1500254.

1. Introduction

A Jónsson algebra is an infinite algebra $J$ in a countable algebraic language, which has no proper subalgebra of the same cardinality as $J$. For example, the natural numbers, $\langle\mathbb{N};S,0\rangle$, in the language of the successor function and [math] is a Jónsson algebra. Indeed, any infinite algebra that has no proper subalgebras at all is a Jónsson algebra. Since Jónsson algebras have countable languages, those Jónsson algebras that have no proper subalgebras must be countable.

Uncountable Jónsson algebras are more difficult to construct and are more interesting. Keisler and Rowbottom announced in [9] that if $V=L$, then there is a Jónsson algebra of every infinite cardinality. Erdős and Hajnal showed in [2] that if GCH holds, then there is a Jónsson algebra of cardinality $\kappa^{+}$ for every infinite cardinal $\kappa$. Without GCH, they proved that there is a Jónsson algebra of cardinality $\omega_{n}$ for every finite $n$.

Jónsson algebras in specific varieties have been investigated. Some such results are surveyed in [1], but we list a few of the results here. Scott classified the Jónsson algebras in the variety of commutative groups in [12]: they are the Prüfer $p^{\infty}$-groups, $\mathbb{Z}_{p^{\infty}}$, hence all have cardinality $\omega_{0}$. Shelah constructed Jónsson groups of cardinality $\omega_{1}$ and of any cardinality $\lambda$ for which $\lambda^{+}=2^{\lambda}$ in [13]. Ol’shanskii constructed Tarski Monsters in [11], which are special kinds of countable, noncommutative, Jónsson groups. McKenzie showed in [10] that, under GCH, any Jónsson algebra in the variety of semigroups must be the underlying semigroup of a group. Hanf announced in [4] that if there is a Jónsson algebra of cardinality $\kappa$, then a Jónsson algebra of cardinality $\kappa$ can be found in the variety of commutative loops. Whaley proved in [19] that there are no Jónsson algebras of regular cardinality in the variety of lattices.

The question we consider is: Is there a Jónsson algebra in a minimal variety? This question is motivated by the observation that, if $J$ is one of the known Jónsson algebras, then some cyclic (= $1$-generated) subalgebras of $J$ seem to satisfy more identities than $J$ itself does.

In Section 2 of this paper we prove that there is no residually finite Jónsson algebra in a minimal variety (Corollary 2.4). In Section 3 we prove that the variety of Jónsson-Tarski algebras is minimal (Corollary 3.3), and in Section 4 we describe an uncountable Jónsson algebra in the variety of Jónsson-Tarski algebras (Theorem 4.4).

2. Residually finite Jónsson algebras

We started our investigation into whether there is a Jónsson algebra in a minimal variety by examining minimal locally finite varieties, where much is known. For example, the minimal, locally finite varieties containing a nontrivial solvable member have been classified, [6, 14, 16], and none contain a Jónsson algebra. The minimal, locally finite, idempotent varieties have been classified [17, 18], and none contain a Jónsson algebra. In this section we will prove that there is no residually finite Jónsson algebra in a minimal variety, but first let us explain the connection to Jónsson algebras in minimal locally finite varieties.

It is known that any minimal locally finite variety $\mathcal{V}$ possesses a “localization functor” $e:{\mathcal{V}}\to e({\mathcal{V}})$, where the target variety $e({\mathcal{V}})$ is a minimal, locally finite, term minimal variety (see [7]). Moreover, the target varieties have been classified into 20 types of minimal, locally finite, term minimal varieties, [15]. This suggests a path to proving that there is no Jónsson algebra in a minimal, locally finite variety. First one should examine the 20 types of minimal, locally finite varieties of the form $e({\mathcal{V}})$, and prove that none of them contains a Jónsson algebra. Then one should show that the localization functor reflects the absence of Jónsson algebras.

We were able to complete the first step for 19 of the 20 types, but we were unable to complete this step for varieties $e({\mathcal{V}})$ of the type generated by strictly simple, $G^{0}$-algebras of type ${\bf 5}$, whose complex nature is examined in [8]. The method we used was to recognize that 19 of the 20 types of minimal, locally finite, term minimal varieties consist of residually finite algebras. Then, we observed that there is no residually finite Jónsson algebra in a minimal variety. We record the proof of this last statement in this section.

The 20th type of minimal, locally finite, term minimal variety need not consist of residually finite algebras, and we were not able to determine whether there are varieties of this type which contain Jónsson algebras. We were also unable to prove that the localization functor reflects the nonexistence of uncountable Jónsson algebras. So, while it seems this circle of ideas might still represent a viable path toward proving the nonexistence of uncountable Jónsson algebras in minimal, locally finite varieties, in this section we shall only explain why there are no residually finite Jónsson algebras in minimal varieties.

Theorem 2.1**.**

Let $\theta$ be a congruence on a Jónsson algebra $J$.

1. $(1)$

If $|J/\theta|=|J|$, then $J/\theta$ is also a Jónsson algebra. 2. $(2)$

If $|J/\theta|<\textrm{\rm cf}(|J|)$, then $J/\theta$ is cyclic. 3. $(2)^{\prime}$

If $\theta$ is a uniform congruence (all $\theta$-classes have the same size) and $|J/\theta|<|J|$, then $J/\theta$ is a cyclic algebra that is generated by any one of its elements.

Proof.

Let $\kappa=|J|$ and let $\nu:J\to J/\theta$ be the natural map.

To prove the contrapositive of Item (1), assume $S\leq J/\theta$ is a proper subalgebra of $J/\theta$ of size $\kappa$. Then $\nu^{-1}(S)$ is a proper subalgebra of size $\kappa$ of $J$, contradicting the assumption that $J$ is Jónsson.

To prove Item (2), note that the $\theta$-classes partition $J$ into $(<\textrm{cf}(\kappa))$-many subsets. Necessarily one of these classes, say $j/\theta$, has size $\kappa$. Since $J$ is Jónsson, the subalgebra $\langle j/\theta\rangle$ generated by this class is $J$ itself. The Second Isomorphism Theorem implies that the quotient $J/\theta$ is generated by the single $\theta$-class $j/\theta$.

Item $(2)^{\prime}$ is like Item (2) with a minor difference. Let $\lambda=|J/\theta|<|J|=\kappa$ and let $\mu$ be the class size for $\theta$. Necessarily, $\mu\cdot\lambda=\kappa>\lambda$, so $\mu=\kappa$ and every $\theta$-class $j/\theta$ has size $\kappa$. As in the proof of (2), this implies that every element of $J/\theta$ generates $J/\theta$. ∎

Corollary 2.2**.**

Let $J$ be a Jónsson algebra which has a finite bound $n$ on the size of its cyclic subalgebras. $J$ is not residually finite.

Proof.

Suppose to the contrary that $J$ is residually finite and has a finite bound $n$ on the size of its cyclic subalgebras. Choose $n+1$ elements $a_{i}\in J$, $i=0,\ldots,n$. For each $i<j$ select a congruence $\theta_{ij}$ on $J$ that has finite index and satisfies $(a_{i},a_{j})\notin\theta_{ij}$. If $\theta=\cap_{i<j}\theta_{ij}$, then $J/\theta$ is finite. By Theorem 2.1 (2), there is an element $a\in J$ such that $a/\theta$ generates $J/\theta$. Hence

[TABLE]

But $a_{i}/\theta\neq a_{j}/\theta$ for any $i<j$, so $|J/\theta|\geq n+1$, which is a contradiction. ∎

The hypothesis of the previous corollary asserting that $J$ has a finite bound on the size of its cyclic subalgebras will hold if $J$ belongs to a variety whose free algebra on one generator, $F_{\mathcal{V}}(1)$, is finite. Call such a variety $1$-finite.

Corollary 2.3**.**

A $1$-finite variety contains no residually finite Jónsson algebra. Hence a locally finite variety contains no residually finite Jónsson algebra. $\Box$

Corollary 2.4**.**

There is no residually finite Jónsson algebra in a minimal variety.

Proof.

Suppose that $J$ is a residually finite Jónsson algebra in a minimal variety $\mathcal{V}$. Then $J$ has a nontrivial finite quotient, which necessarily generates $\mathcal{V}$, hence $\mathcal{V}$ is a finitely generated variety. Finitely generated varieties are locally finite, so this corollary follows from Corollary 2.3. ∎

In fact, we do not know if there are any uncountable, residually finite Jónsson algebras at all.111$\langle\mathbb{N};S,0\rangle$ and the unital ring $\langle\mathbb{Z};\cdot,+,-,0,1\rangle$ are examples of countable, residually finite Jónsson algebras. The results above do place a cardinality bound on the size of residually finite Jónsson algebras, which we record.

Theorem 2.5**.**

If $J$ is a residually finite Jónsson algebra, then $|J|\leq 2^{2^{\omega}}$. If $J$ is defined in a finite language, then $|J|\leq 2^{\omega}$.

Proof.

Let $\mathcal{V}={\mathcal{V}}(J)$ be the variety generated by $J$, and let $A\in{\mathcal{V}}$ be any finite member. All cyclic algebras in the subvariety ${\mathcal{V}}(A)\leq{\mathcal{V}}$ have size bounded by $|F_{{\mathcal{V}}(A)}(1)|\leq|A|^{|A|}$, which is finite. Since the finite quotients of $J$ are cyclic, $J$ has a maximal finite quotient that lies in this subvariety. That is, $J$ has a least congruence $\theta_{A}$ such that $J/\theta_{A}\in{\mathcal{V}}(A)$ (and $|J/\theta_{A}|\leq|A|^{|A|}$ will hold). Now $J$ is embeddable in

[TABLE]

since $J$ is residually finite. The factors in this product are finite, while the number of factors in this displayed product equals the number of finite algebras in ${\mathcal{V}}$. A variety in a countable language has at most $2^{\omega}$ finite members, leading to the cardinality bound

[TABLE]

when the language is countable. A variety in a finite language has at most countably many finite members, so the same calculation produces the bound $|J|\leq 2^{\omega}$ when the language is finite. ∎

We shall leave open the following questions.

Question 1. Is there an uncountable, residually finite Jónsson algebra?

Question 2. Is there an uncountable Jónsson algebra in a minimal, locally finite variety?

Henceforth we concentrate on constructing a non-locally finite, minimal variety that contains an uncountable Jónsson algebra. Specifically, we construct a Jónsson algebra of cardinality $\omega_{1}$ in the variety of Jónsson–Tarski algebras.

3. The variety of Jónsson-Tarski algebras is minimal

A Jónsson-Tarski algebra is an algebra $\langle A;\cdot,\ell,r\rangle$ in the language $\mathcal{L}$ of one binary operation $\cdot$ and two unary operations $\ell$ and $r$, which satisfies identities expressing that

[TABLE]

are inverse bijections. These identities are

1. $\varepsilon({\ell})$:

$\ell(x\cdot y)=x$, 2. $\varepsilon({r})$:

$r(x\cdot y)=y$, and 3. $\varepsilon({\cdot})$:

$\ell(z)\cdot r(z)=z$.

Let $\Sigma=\{\varepsilon({\ell}),\varepsilon({r}),\varepsilon({\cdot})\}$ denote the set of these identities.

Jónsson and Tarski introduced the variety axiomatized by $\Sigma$ in [5]. Jónsson and Tarski were interested in this variety because it has the property that the algebra $F_{\mathcal{V}}(n)$ freely generated by the $n$-element set $X=\{x_{1},\ldots,x_{n-1},x_{n}\}$, $n>1$, is also freely generated by the $(n-1)$-element set $\{x_{1},\ldots,(x_{n-1}\cdot x_{n})\}$, where the last two elements of $X$ are replaced by their product. Consequently $F_{\mathcal{V}}(n)\cong F_{\mathcal{V}}(n-1)$ in this variety, and in fact any two free algebras with a finite, positive number of generators are isomorphic. We mention this for information only. We shall not use anything from [5] other than the definition of the variety of Jónsson-Tarski algebras.

We call the language of the binary symbol $\cdot$ the “multiplication sublanguage”, $\mathcal{L}_{m}$, and the language of the two unary symbols only, $\ell$ and $r$, the “unary sublanguage”, $\mathcal{L}_{u}$. Call an $\mathcal{L}$-term an “$m,u$-term” if it has the form $w(u_{1}(x_{i_{1}}),\ldots,u_{k}(x_{i_{k}}))$ where $w(x_{1},\ldots,x_{k})$ is an $\mathcal{L}_{m}$-term and the $u_{j}(x_{i_{j}})$ are $\mathcal{L}_{u}$-terms. Let $\mathcal{MU}$ denote the set of $m,u$-terms in the language $\mathcal{L}$.

Lemma 3.1**.**

Every $\mathcal{L}$-term is $\Sigma$-equivalent to a term in $\mathcal{MU}$.

Proof.

The set of $\mathcal{L}$-terms is the smallest set containing the variables and closed under multiplication and $\ell$ and $r$. $\mathcal{MU}$ contains the variables and is closed under multiplication, so we only need to show that, for any $t\in\mathcal{MU}$, $\ell(t)$ and $r(t)$ are $\Sigma$-equivalent to terms in $\mathcal{MU}$.

If $t$ contains at least one occurrence of the multiplication symbol, then $t=p\cdot q$ where $p,q\in\mathcal{MU}$. Then $\ell(t)$ and $r(t)$ are $\Sigma$-equivalent to $p$ and $q$ respectively, which belong to $\mathcal{MU}$. If $t$ does not contain an occurrence of the multiplication symbol, then $t$ is a term in the sublanguage $\mathcal{L}_{u}$. In this case $\ell(t)$ and $r(t)$ are also terms in the sublanguage $\mathcal{L}_{u}$, so they are in $\mathcal{MU}$. ∎

Lemma 3.2**.**

If $s$ and $t$ are $\mathcal{L}$-terms, then either

1. (1)

$\Sigma$* entails the identity $s=t$, or* 2. (2)

$\Sigma\cup\{s=t\}$* entails $x=y$.*

Proof.

We argue by induction on the total number of multiplication symbols occurring in $s=t$.

Replace $s$ and $t$ with $\Sigma$-equivalent $m,u$-terms, each involving a minimal number of multiplication symbols. This does not affect the assumptions nor the conclusions of the theorem. Now we consider cases.

Case 1**.**

At least one of $s$ or $t$ has at least one multiplication symbol.

Proof of this case.  In this case, the identities $\ell(s)=\ell(t)$ and $r(s)=r(t)$, which are both derivable from $s=t$, each have fewer total multiplication symbols than $s=t$ after reducing modulo $\Sigma$. By induction we have either

1. (I)

$\Sigma$ entails $\ell(s)=\ell(t)$ and $r(s)=r(t)$, or 2. (II)

$\Sigma\cup\{\ell(s)=\ell(t)\}$ entails $x=y$ or $\Sigma\cup\{r(s)=r(t)\}$ entails $x=y$.

Under Item (I), $\Sigma$ entails $s=\ell(s)\cdot r(s)=\ell(t)\cdot r(t)=t$, and we get Item (1) of the theorem statement. Under Item (II), $\Sigma\cup\{s=t\}$ entails both $\ell(s)=\ell(t)$ and $r(s)=r(t)$, hence entails $x=y$. This gives us Item (2) of the theorem statement.

For the rest of the proof we may restrict attention to the case where $s$ and $t$ are $\mathcal{L}_{u}$-terms. We write a term like $\ell(\ell(r(\ell(r(r(z))))))$ as $\ell\ell r\ell rr(z)$ and refer to $z$ as the variable of the term. We refer to the sequence of $\ell$’s and $r$’s (which is $\ell\ell r\ell rr$ in this example) as the prefix. We refer to the number of symbols in the prefix (which is six in this example) as the length of the term. We refer to the underlined symbol $\ell\ell r\ell r\underline{r}(z)$ as the rightmost operation symbol of the term. Without loss of generality, we assume that the length of $s$ is at least that of $t$.

Case 2**.**

$s$* and $t$ have different variables and the same prefix.*

Proof of this case.  In this case we show by induction on the combined lengths of $s$ and $t$ that $\Sigma\cup\{s=t\}$ entails $x=y$. If the common prefix of $s$ and $t$ is empty, then $s$ and $t$ are already distinct variables, so $\Sigma\cup\{s=t\}$ entails $x=y$. Otherwise we may assume that $s$ is $PQ(x)$ and $t$ is $PQ(y)$ where $PQ$ is the prefix, $P$ is a possibly empty string, and $Q$ is either $\ell$ or $r$. Now, by substituting the terms $x_{\ell}\cdot x_{r}$ and $y_{\ell}\cdot y_{r}$ for $x$ and $y$, where $x_{\ell},x_{r},y_{\ell},y_{r}$ are distinct new variables, we find that $\Sigma\cup\{s=t\}$ entails $PQ(x_{\ell}\cdot x_{r})=PQ(y_{\ell}\cdot y_{r})$, which reduces modulo $\Sigma$ to $P(x_{Q})=P(y_{Q})$. By induction, $\Sigma\cup\{P(x_{Q})=P(y_{Q})\}$ entails $x=y$, so we conclude that $\Sigma\cup\{s=t\}$ entails $x=y$.

Case 3**.**

$s$* and $t$ have different variables and different prefixes.*

Proof of this case.  In this case we also show that $\Sigma\cup\{s=t\}$ entails $x=y$. We may assume that $s=t$ is expressible as $P(x)=P^{\prime}(y)$, where $P$ is nonempty and is the prefix of $s$, while $P^{\prime}$ may be an empty string and is the prefix of $t$. Substituting $z$ for $y$ we derive $P(x)=P^{\prime}(z)$ from $P(x)=P^{\prime}(y)$. From $P^{\prime}(y)=P(x)=P^{\prime}(z)$ we derive the identity $P^{\prime}(y)=P^{\prime}(z)$. Citing Case 2 we conclude that $\Sigma\cup\{s=t\}$ entails $x=y$.

Case 4**.**

$s$* and $t$ have the same variables, but different rightmost operation symbols.*

Proof of this case.  In this case we express $s=t$ as $PQ(x)=P^{\prime}Q^{\prime}(x)$ where $Q$ is a single operation symbol, $\ell$ or $r$, $Q\neq Q^{\prime}$, the length of $PQ(x)$ is at least that of $P^{\prime}Q^{\prime}(x)$, and we allow $P^{\prime}Q^{\prime}$ to be empty.

We first treat the case where $P^{\prime}Q^{\prime}$ is not empty, so $\{Q,Q^{\prime}\}=\{\ell,r\}$. Substituting $x_{\ell}\cdot x_{r}$ for $x$, where $x_{\ell}$ and $x_{r}$ are new variables, we derive from $PQ(x)=P^{\prime}Q^{\prime}(x)$ the identity $P(x_{Q})=P^{\prime}(x_{Q^{\prime}})$. This is an identity of the type handled in either Case 2 or Case 3. Hence $\Sigma\cup\{s=t\}$ entails $x=y$.

Next we treat the case where $P^{\prime}Q^{\prime}$ is empty, so $s=t$ is $PQ(x)=x$. We assume that $Q=\ell$, and omit the argument for the similar case where $Q=r$. Thus, it is our aim to show that $\Sigma\cup\{P\ell(x)=x\}$ entails $x=y$. Substitute $x_{\ell}\cdot x_{r}$ for $x$ in $P\ell(x)=x$ and apply $r$ to both sides to obtain the second equality in

[TABLE]

Thus, from $\Sigma\cup\{P\ell(x)=x\}$ we have derived $rP(x_{\ell})=x_{r}$, which is an identity of the type considered in Case 3. From the Case 3 argument we derive $x=y$.

Case 5**.**

$s$* and $t$ have the same variables, and the same rightmost operation symbols.*

Proof of this case.  If the prefixes of $s$ and $t$ are nonempty, then $s=t$ is expressible as $PQ(x)=P^{\prime}Q(x)$ where $Q$ is a single operation symbol, $\ell$ or $r$. By substituting $x_{\ell}\cdot x_{r}$ for $x$ and reducing modulo $\Sigma$, we obtain $P(x_{Q})=P^{\prime}(x_{Q})$. Now either we are in Case 4, from which it follows that $\Sigma\cup\{s=t\}$ entails $x=y$, or else we are back in Case 5 but with terms of strictly shorter length. If we are back in Case 5, we repeat the argument, and we either eventually reach Case 4, or we reach an expression of the form $x=x$ and conclude that $s$ and $t$ had exactly the same prefixes and the same variables, hence $\Sigma$ entails $s=t$. Thus the assertion of the lemma holds.

∎

Corollary 3.3**.**

The variety of Jónsson-Tarski algebras is minimal.

Proof.

If the variety $\mathcal{J}$ axiomatized by $\Sigma$ were not minimal, then there would exist an identity $s=t$ that does not hold throughout $\mathcal{J}$, but which holds in some minimal subvariety of $\mathcal{J}$. For this identity we would have $\Sigma\not\models s=t$ and $\Sigma\cup\{s=t\}\not\models x=y$, contrary to Lemma 3.2. ∎

4. Jónsson Jónsson-Tarski algebras

Our goal in this section is to construct a Jónsson algebra on the set $\omega_{1}$ in the variety of Jónsson-Tarski algebras. We start with the easier project of constructing a Jónsson Jónsson-Tarski algebra structure on $\omega$.

To construct a Jónsson-Tarski algebra on a set $J$, it suffices to describe the table for the multiplication operation, since it is possible to read the tables for $\ell$ and $r$ off of the multiplication table. Moreover, the only condition that a multiplication table must satisfy for it to be a table for a Jónsson-Tarski algebra is that it be the table of a bijection $J\times J\to J$, which means that every element of $J$ occurs in one and only one cell of the multiplication table.

Theorem 4.1**.**

There exists a Jónsson Jónsson-Tarski algebra, $J_{\omega}$, with universe $\omega$.

Proof.

We define our Jónsson Jónsson-Tarski algebra with the following multiplication table:

[TABLE]

This table was created by first placing the numbers $0,1$, and $2$ in the upper left corner in a certain pattern. The rest of the table was filled by placing the remaining natural numbers in the remaining empty cells in increasing order moving diagonally up and to the right.

In this table, the product $m\cdot n$ is placed in the cell located in the $m$-th row and $n$-th column. For example, $3\cdot 4=32$, by definition, so $32$ is placed in the cell in the $3$rd row and $4$th column. To read off the $\ell$ and $r$ operations from this multiplication table, recall that any $n\in\omega$ occurs in one and only one cell of the table. The value of $\ell(n)$ is the row header for that cell containing $n$, while $r(n)$ is the column header for that cell. For example, $\ell(13)=1$ and $r(23)=2$.

An important feature of this algebra, $J_{\omega}$, is that the functions $\ell$ and $r$ are regressive. That is, $\ell(n)<n$ and $r(n)<n$ when $n\neq 0$. This can be checked by hand for $n=1$ and $2$, and then proved for larger $n$ using the following formula for the multiplication

[TABLE]

which is not hard to establish.

An immediate consequence of regressiveness is:

Claim 4.2**.**

If $n\in\omega$, then [math] is in the subalgebra of $J_{\omega}$ generated by $\{n\}$.

Proof of Claim.  Since $\ell$ is regressive, the sequence $n,\ell(n),\ell(\ell(n)),\dots$ must eventually include [math].

Claim 4.3**.**

$J_{\omega}$* is generated by $\{0\}$.*

Proof of Claim.  Suppose otherwise, and let $n\in\omega$ be the least natural number not in the subalgebra $S=\langle\{0\}\rangle$. Clearly $n>0$, so $\ell(n),r(n)<n$, which implies that $\ell(n),r(n)\in S$. But now $n=\ell(n)\cdot r(n)\in S$, contradicting the choice of $n$.

The claims show that any $n\in\omega$ generates $J_{\omega}$, so $J_{\omega}$ has no proper (nonempty) subalgebras. This establishes that $J_{\omega}$ is Jónsson. ∎

It is impossible to construct a Jónsson Jónsson-Tarski algebra $J_{\omega_{1}}$ on $\omega_{1}$ that has the same properties as $J_{\omega}$, namely the properties that $\ell$ and $r$ are regressive functions, and that each element of $J_{\omega}$ generates the whole algebra. In the first place, single elements can only generate countable subalgebras, so the uncountable algebra $J_{\omega_{1}}$ cannot be $1$-generated (or even countably generated). In the second place, as pointed out to us by Don Monk, Fodor’s Lemma prevents the existence of a Jónsson Jónsson-Tarski algebra on $\omega_{1}$ which has both $\ell$ and $r$ regressive. To see this, note that if $\ell$ is regressive on $\omega_{1}$, then it is constant on a set $S$ stationary in $\omega_{1}$. If $r$ is also regressive on $\omega_{1}$, so regressive on $S$, then $r$ is constant on a stationary subset $T\subseteq S$. If $\ell$ and $r$ are both constant on $T$, then the map $\omega_{1}\to\omega_{1}\times\omega_{1}:\alpha\mapsto(\ell(\alpha),r(\alpha))$ is constant on $T$. But in a Jónsson-Tarski algebra this map is a bijection.

Nevertheless, we do have:

Theorem 4.4**.**

There exists a Jónsson Jónsson-Tarski algebra, $J_{\omega_{1}}$, with universe $\omega_{1}$.

Proof.

We will construct the multiplication table for $J_{\omega_{1}}$ by transfinite recursion. The main part of the argument will involve explaining, for each countable limit ordinal $\lambda$, how to extend a given Jónsson-Tarski multiplication on $\lambda$ to a Jónsson-Tarski multiplication on $\lambda+\omega$, thereby enlarging a Jónsson-Tarski algebra $J_{\lambda}$ to a Jónsson-Tarski superalgebra $J_{\lambda+\omega}$. Our construction will be guided by the desire to maintain the following Property $\Omega$:

For each pair of countable limit ordinals $\alpha<\beta$ the subalgebra of $J_{\beta}$ generated by any element of the form $\alpha+n$, $n\in\omega$, is $\alpha+\omega$.

Theorem 4.1 provides a Jónsson-Tarski multiplication defined on $\omega$ which satisfies Property $\Omega$.

Now assume that $\lambda$ is an infinite countable limit ordinal, and that we have a Jónsson-Tarski multiplication with Property $\Omega$ defined on $\lambda$. We explain how to extend the $\lambda\times\lambda$ Jónsson-Tarski multiplication table that has Property $\Omega$ to a $(\lambda+\omega)\times(\lambda+\omega)$ multiplication table that has Property $\Omega$.

Call $\lambda+n$ “even” if $n$ is even, and “odd” if $n$ is odd. Say that $\lambda+n$ is “divisible by $4$” or is “$0\pmod{4}$” if $n\equiv 0\pmod{4}$. ETC.

We first explain where to place the value $\lambda+n$ in the multiplication table of $J_{\lambda+\omega}$ when $\lambda+n$ is divisible by $4$. Place this ordinal in the cell whose column header is $r(\lambda+n)=\lambda+n+2$, and whose row header is specified as follows. Choose $\ell$ to be any bijection from the countably infinite set of ordinals in $\{\lambda+k\mid k\equiv 0\pmod{4}\}$ onto the set $\lambda$, and then the row header for the cell containing $\lambda+n$ will be $\ell(\lambda+n)$.

For the ordinals $\lambda+n$ that are $2\pmod{4}$, we again define $r(\lambda+n)=\lambda+n+2$. Then we let $\ell$ be the map defined by $\ell(\lambda+n)=\lambda+\frac{n-2}{4}$. The function $\ell$ maps the set of ordinals that are $2\pmod{4}$ bijectively onto the set $\{\lambda+k\mid k\in\omega\}$.

So far, we have defined $\ell$ so that it is a bijection from the set of even ordinals $\geq\lambda$ onto the set $\lambda+\omega$, while $r(\lambda+n)=\lambda+n+2$ holds whenever $\lambda+n$ is even. The multiplication table as described so far is indicated in Figure 1. The placement in the figure of the ordinals that are divisible by $4$ is only suggestive: they are shown in the correct columns, but all we know about which rows they occupy is that there is one ordinal of the form $\lambda+n$, $n\equiv 0\pmod{4}$, in each row whose row header is $<\lambda$.

Finally, we use the odd ordinals to populate the rest of this $(\lambda+\omega)\times(\lambda+\omega)$ table. We begin by partitioning the set of odd ordinals above $\lambda$ into countably many countable sets labelled $L_{\lambda+n}$ for $n\in\omega$, as indicated by the columns of Table 1.

This way of partitioning of the odd ordinals above $\lambda$ has the property that $\lambda+n\in L_{\lambda+m}$ implies $m<n$.

Let each set $L_{\lambda+n}$ fill the L-shaped region of the table in Figure 2 corresponding to $\lambda+n$. By that, we mean the region containing all cells at addresses $(\lambda+n,\alpha)$ and $(\alpha,\lambda+n)$ for all $\alpha\leq\lambda+n$. We can do this since the set of these cells is countably infinite, and so is $L_{\lambda+n}$. It could be, from our placement of the even ordinals, that some of the cells in this region are already occupied. However, it can be easily seen that at most two cells in each L-shaped region will be occupied by even ordinals, since no two even ordinals were assigned the same $\ell$-value and no two even ordinals were assigned the same $r$-value. Thus, each L-shaped region will still have infinitely many cells in which to place the elements of $L_{\lambda+n}$. See Figure 2.

Once the elements of the sets $L_{\lambda+n}$ have been placed in the table, we will have filled the square $(\lambda+\omega)\times(\lambda+\omega)$. Moreover, we have arranged that, whenever $\lambda+n\in L_{\lambda+m}$ we have $m<n$, and we either have $\ell(\lambda+n)=\lambda+m$ or $r(\lambda+n)=\lambda+m$. We now argue that this construction extends the original $\lambda\times\lambda$ table in a way that preserves Property $\Omega$.

We assume that Property $\Omega$ holds for the subalgebra $J_{\lambda}$ of $J_{\lambda+\omega}$, so to show that Property $\Omega$ holds for $J_{\lambda+\omega}$ it suffices to show that any $\lambda+n$ generates the set $\lambda+\omega$. As a first step, we argue that any $\lambda+n$ generates $\lambda$. Toward this end, we argue that any $\lambda+n$, $n>0$, generates some $\lambda+m$, where $m<n$. If $n$ is odd, then $\lambda+n$ belongs to $L_{\lambda+m}$ for some $m<n$, hence either $\ell(\lambda+n)=\lambda+m$ or $r(\lambda+n)=\lambda+m$. If $n$ is $2\pmod{4}$, then $\ell(\lambda+n)=\lambda+\frac{n-2}{4}$, and $\frac{n-2}{4}<n$. Finally, if $n$ is $0\pmod{4}$, then $\ell(r(\lambda+n))=\ell(\lambda+n+2)=\lambda+\frac{(n+2)-2}{4}=\lambda+\frac{n}{4}$, and $\frac{n}{4}<n$. This completes the proof that any $\lambda+n$ generates some smaller $\lambda+m$, hence generates $\lambda$.

Now, $\lambda$ generates the set of all even ordinals between $\lambda$ and $\lambda+\omega$, since $r$ adds $2$ to any even ordinal. Next, $\ell$ is a bijection from the set of even ordinals between $\lambda$ and $\lambda+\omega$ onto the set $\lambda+\omega$. So, once we have generated all of the even ordinals, we can generate all of $\lambda+\omega$ by applying $\ell$.

This concludes the argument that Property $\Omega$ extends from $\lambda$ to $\lambda+\omega$.

Since we have defined a Jónsson-Tarski algebra $J_{\lambda+\omega}$ on $\lambda+\omega$ for any countable limit ordinal $\lambda$, and done it in a way so that the table on $\lambda+\omega$ extends the table on $\lambda$, and Property $\Omega$ is preserved at each extension stage, we get a Jónsson-Tarski algebra $J_{\omega_{1}}$ on $\omega_{1}$ satisfying Property $\Omega$. It follows that the subalgebras of $J_{\omega_{1}}$ are exactly the unions of sets of the form $\lambda+\omega$. That is, they are exactly the countable limit ordinals. This is enough to prove that $J_{\omega_{1}}$ is Jónsson. ∎