Chinese syzygies by insertions
Nohra Hage, Philippe Malbos
TL;DR
This paper develops a finite, convergent algebraic presentation for the Chinese monoid using insertion algorithms, enabling new representations and actions on categories.
Contribution
It introduces a novel semi-quadratic presentation with column generators and extends it to a coherent presentation based on insertion algorithms.
Findings
Constructed a finite convergent presentation for Chinese monoids.
Extended the presentation to a coherent one with relations among insertion algorithms.
Facilitated new categorical representations of Chinese monoids.
Abstract
We construct a finite convergent semi-quadratic presentation for the Chinese monoid by adding column generators and using combinatorial properties of insertion algorithms on Chinese staircases. We extend this presentation into a coherent one whose generators are columns, rewriting rules are defined by insertion algorithms, and whose syzygies are defined as relations among insertion algorithms. Such a coherent presentation is used for representations of Chinese monoids, in particular, it is a way to describe actions of Chinese monoids on categories.
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Chinese syzygies by insertions
Nohra Hage Philippe Malbos
Abstract – We construct a finite convergent semi-quadratic presentation for the Chinese monoid by adding column generators and using combinatorial properties of insertion algorithms on Chinese staircases. We extend this presentation into a coherent one whose generators are columns, rewriting rules are defined by insertion algorithms, and whose syzygies are defined as relations among insertion algorithms. Such a coherent presentation is used for representations of Chinese monoids, in particular, it is a way to describe actions of Chinese monoids on categories.
Keywords – Chinese monoids, syzygies, Chinese staircases, insertion algorithms, string rewriting.
M.S.C. 2010 – Primary: 20M05, 16S15. Secondary: 68Q42, 20M35.
1. Introduction
The structure of Chinese monoids appeared in the classification of monoids with the growth function coinciding with that of plactic monoids, [7]. The latter monoids emerged from the works of Schensted [26] and Knuth [19] on the combinatorial study of Young tableaux and they have found several applications in algebraic combinatorics, representation theory and probabilistic combinatorics, [23, 8, 6]. One of the motivations for studying Chinese monoids is that they are also related to Young tableaux, and therefore they might play a similar role as plactic monoids in the same application areas. Representations of the Chinese monoid of finite rank are studied in [29] by constructing all its irreducible representations. More generally, we are interested in the study of the actions of Chinese monoids on categories. One approach is to make explicit coherent presentations whose relations are described by insertion algorithms and whose syzygies are defined as relations among these relations. In this article, we compute coherent presentations for the Chinese monoid by rewriting methods using combinatorial properties of Chinese staircases.
This work is a part of a broader project that consists of studying, by a rewriting approach, families of monoids defined from combinatorial objects constructed using insertion algorithms. For instance, plactic monoids of type A are related to Young tableaux [20], plactic monoids of classical types to symplectic and orthogonal tableaux, [21, 22], Chinese monoids to Chinese staircases, [7, 4], hypoplactic monoids to quasi-ribbon tableaux, [24], left and right patience sorting monoids to left and right patience sorting tableaux, [28, 3], and stalactic monoids to stalactic tableaux [17, 25]. Moreover, binary search trees, binary search trees with multiplicities and pairs of twin binary search trees are used to describe normal forms for sylvester monoids, [16], taiga monoids, [25], and Baxter monoids, [10]. We are interested in the study of the syzygies for the presentations of theses monoids by computing finite coherent convergent presentations. Such coherent presentations are constructed for Artin monoids in [9] and for plactic monoids of type A in [15]. The study of the syzygies in a monoid produces in higher dimensions free objects that are homotopically equivalent to the original monoid and then allows computation of its homological invariants. Indeed, this study provides the first two steps in the computation of a polygraphic resolution of the monoid, that is, a categorical cofibrant replacement of the monoid in a free -category, whose acyclicity is proved by an iterative construction of a normalization reduction strategy, [11, 12]. Moreover, coherent presentations are also useful to describe the notion of an action of the monoid on categories, [9].
The Chinese monoid of rank , denoted by , is generated by and submitted to the relations for all . These relations generate the Chinese congruence, denoted by , and interpreted in [4] using the notion of Chinese staircases. A Chinese staircase is a collection of boxes in right-justified rows, filled with non-negative integers, whose rows and columns are indexed with from top to bottom and from right to left respectively and where the -th row contains boxes, for . We will denote by the set of Chinese staircases over and by the map on that reads a Chinese staircase row by row from right to left and from top to bottom as defined in Subsection 3.1. A Schensted-like insertion algorithm, denoted by \;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;, is introduced in [4], and consists in inserting an element of into a Chinese staircase from the right, yielding to a new Chinese staircase. From a word on , we associate a Chinese staircase obtained by insertion of in the empty staircase by application of \;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\; step by step from left to right:
[TABLE]
Similarly, a Chinese staircase denoted by is computed by inserting the elements of from right to left in the empty staircase by application of the left insertion introduced in [2] and that inserts an element of into a Chinese staircase from the left. The set of Chinese staircases satisfies the cross-section property for the Chinese congruence , that is, for all words on , if and only if the insertion algorithm yields the same Chinese staircase: , [4]. So the elements of the Chinese monoid can be identified with the Chinese staircases, which therefore also form a monoid. Moreover, the right and left insertion algorithms allow one to define two internal products on by setting t\star_{r}t^{\prime}=(t\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;\operatorname{R}(t^{\prime})) and , for all in . Following the cross-section property, the compositions and are associative and the following equality
[TABLE]
holds, for all in and in . In particular, the following equality holds, for any word on . In this way, the products and equip the set with two monoid structures that are anti-isomorphic.
We construct in Section 4 a finite semi-quadratic convergent presentation for the monoid , denoted by , whose set of generators is made of columns over of length at most and square generators and whose rules are
[TABLE]
for all in such that does not form a Chinese staircase and is equal to the Chinese staircase composed by the columns and . We show that this rewriting system can be obtained from the Knuth-like presentation of by applying Tietze transformations that consist in adding or removing definable generators and in adding or removing derivable relations on a presentation of a monoid in such a way that they do not change the presented monoid, see [9]. Moreover, we show that the confluence of the rewriting system is a direct consequence of the associativity of the product . We deduce that the monoid has finite derivation type and finite homological type . Note that the finite convergent presentations of Chinese monoids already obtained in [5, 14], by completion of Chinese relations, and in [2] by adding column generators, are not semi-quadratic, and thus it is difficult to extend them into coherent ones.
We extend in Section 5 the rewriting system into a finite coherent convergent presentation of the Chinese monoid with an explicit description of the Chinese syzygies. We show in Theorem 5.6 that extends into a finite convergent coherent presentation of the monoid by adjunction of generating syzygies with the following decagonal form
[TABLE]
for all in such that and are not normal forms with respect to , and where the -cells denote either a rewriting rule of or an identity. We show in Subsection 5.7 how the generating syzygy of the coherent presentation of the Chinese monoid can be interpreted in terms of the right and left insertion algorithms. Finally, we use in Subsection 5.8 this coherent presentation in order to describe the actions of Chinese monoids on categories.
2. Preliminaries on rewriting
This preliminary section recalls the basic notions of rewriting we use in this article. For a fuller account of the theory, we refer the reader to [1]. We will also recall from [9, 13] the notion of coherent presentation of a monoid that extends the notion of a presentation by syzygies taking into account all the relations amongst the relations. We will denote by the free monoid of words written in the alphabet , the product being concatenation of words, and the identity being the empty word, denoted by . We will denote by a word in of length , where belong to , and by its length.
2.1. String rewriting systems
A (string) rewriting system on is a subset of . An element of is called a rule with source and target , and denoted by . We will denote respectively by and the source and target of . A one step reduction is defined by for all words in and rule , and will be denoted by . One step reductions form the reduction relation on denoted by . A rewriting path with respect to is a finite or infinite sequence . This corresponds to the reflexive and transitive closure of the relation , that we denote by . A word in is reduced if there is no reduction with source . A normal form for a word in is a reduced word such that reduces into . The rewriting system terminates if it has no infinite rewriting path, and it is (weakly) normalizing if every word in reduces to some normal form. A rewriting system is reduced if, for every rule in , the source is -reduced and the target is reduced. The reflexive, symmetric and transitive closure of is the congruence on generated by , that we denote by . The monoid presented by is the quotient of the free monoid by the congruence . Two rewriting systems are Tietze equivalent if they present isomorphic monoids. Recall that a Tietze transformation between two rewriting systems is a sequence of elementary Tietze transformations, defined on a rewriting system on by the following operations:
- i)
adjunction or elimination of an element in and of a rule , where is an element in that does not contain , 2. ii)
adjunction or elimination of a rule such that and are equivalent by the congruence generated by .
Two rewriting systems are Tietze equivalent if, and only if, there exists a Tietze transformation between them, see [9] for more details.
2.2. Confluence
A branching (resp. local branching) of a rewriting system on is a non ordered pair of reductions (resp. one step reductions) of on the same word. A branching is aspherical if it is of the form , for a one step reduction and Peiffer when it is of the form for one step reductions and with source and respectively. The overlapping branchings are the remaining local branchings. An overlapping local branching is critical when it is minimal for the order generated by the relations (f,g)\>\sqsubseteq\>\big{(}wfw^{\prime},wgw^{\prime}), given for all local branching and words in . A branching is confluent if there exist reductions and reducing to the same word:
[TABLE]
The rewriting system is confluent if all of its branchings are confluent, and convergent if it is both confluent and terminating. If is convergent, then every word in has a unique normal form.
2.3. Normalization strategies
Recall that a reduction strategy for a rewriting system on specifies a way to apply the rules in a deterministic way. It is defined as a mapping of every word in to a one step reduction with source . When is normalizing, a normalization strategy is a mapping of every word to a rewriting path with source and target a chosen normal form of . For a reduced rewriting system, we distinguish two canonical reduction strategies to reduce words: the leftmost one and the rightmost one, according to the way we apply first the rewriting rule that reduces the leftmost or the rightmost subword. They are defined as follows. For every word of , the set of one step reductions with source can be ordered from left to right by setting , for one step reductions and such that . If is finite, then the order is total and the set of one step reductions of source is finite. Hence this set contains a smallest element and a greatest element , respectively called the leftmost and the rightmost one step reductions on . If, moreover, the rewriting system terminates, the iteration of (resp. ) yields a normalization strategy for called the leftmost (resp. rightmost) normalization strategy of :
[TABLE]
The leftmost (resp. rightmost) rewriting path on a word is the rewriting path obtained by applying the leftmost (resp. rightmost) normalization strategy (resp. ). We refer the reader to [11] for more details on rewriting normalization strategies.
A rewriting system on is semi-quadratic if for all in we have and . The sources of the critical branchings of a semi-quadratic rewriting system are of length . When is reduced, there are at most two rewriting paths with respect to with source a word of length . We will denote by (resp. ) the length of the leftmost (resp. rightmost) rewriting path from to its normal form.
2.4. Coherent presentations
We recall the notion of coherent presentation of monoids formulated in terms of polygraphs in [9], see also [13]. Rewriting systems can be interpreted as -polygraphs with only one [math]-cell. Such a -polygraph is a data , where is a set and is a globular extension of the free monoid seen as a -category. The elements of are generating -cells relating -cells in , with source and target , denoted respectively by and . A rewriting system on can be described by such a -polygraph where the generating -cells are the rules of . Recall that a -category is a category enriched in groupoids. We will denote by the -category freely generated by the -polygraph , see [13] for expanded definitions.
A pair of -cells of such that and is called a -sphere of . A -polygraph is a data made of a -polygraph and a globular extension of the -category , that is a set of -cells , where is a -sphere of . The -cell (resp. ) is called the source (resp. target) of , and denoted by (resp. ). Such a -cell can be represented with the following globular shape:
[TABLE]
where denotes the unique [math]-cell of . We will denote by the free -category generated by the -polygraph . An extended presentation of a monoid is a -polygraph whose underlying -polygraph is a presentation of . A coherent presentation of is an extended presentation of such that the cellular extension of the -category is acyclic, that is, for every -sphere of , there exists a -cell in the -category such that and . The elements in are called syzygies of the presentation .
Recall Squier’s coherence theorem from [27], see also [13], that states that, any convergent rewriting system on presenting a monoid can be extended into a coherent presentation of having a generating syzygy
[TABLE]
for every critical branching of , where and are chosen confluent rewriting paths.
3. Insertions on Chinese staircases
In this section, we recall the structure of Chinese staircase and the right and left insertion algorithms on Chinese staircases. We also recall the structure of Chinese monoid and the cross-section property for this monoid and we deduce properties of the insertions products on Chinese staircases.
3.1. Chinese staircases
A mirror Young diagram of shape is a collection of boxes in right-justified rows, whose rows (resp. columns) are indexed with the totally ordered set , for in , from top to bottom (resp. from right to left) and where the -th row contains boxes for . A (Chinese) staircase over is a mirror Young diagram of shape filled with non-negative integers. Denote by (resp. ) the contents of the box in row and column for (resp. ). A box filled by [math] is called empty. Denote by the set of staircases over and by the map that reads a staircase row by row, from right to left and from top to bottom, and where the -th row is read as follows , for . For instance, for the following staircase over :
t21
we have . By removing the bottom row of a staircase over , we obtain a staircase over , denoted by , as on the following picture:
\scriptstyle 1$$\scriptstyle n-1$$\scriptstyle\vdots$$\scriptstyle n$$\scriptstyle t_{n}$$\scriptstyle t_{n1}$$\scriptstyle n$$\scriptstyle 1$$\ldots$$\ldots$$t^{\prime}
According to this, such a staircase can be denoted by , where is the bottom row of .
3.2. The right insertion algorithm
Recall the right insertion map \;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;:\operatorname{Ch}_{n}\times[n]\rightarrow\operatorname{Ch}_{n} introduced in [4]. Let be a staircase and an element in . If , then t\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;x=(t^{\prime},R^{\prime}_{1}), where is obtained from by adding to . If , let be maximal such that the entry in column of is non-zero or if such a does not exist, set . Three cases appear:
- i)
If , then t\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;x=(t^{\prime}\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;x,R_{1}), 2. ii)
If , then t\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;x=(t^{\prime}\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;y_{1},R^{\prime}_{1}), where is obtained from by subtracting from and adding to , 3. iii)
If , then t\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;x=(t^{\prime},R^{\prime}_{1}), where is obtained from by subtracting from and adding to .
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0.39993pt}\nullfont \crcr}}\kern 35.53343pt}}}\;\raisebox{1.00006pt}{{\rotatebox[origin={c}]{180.0}{\rightsquigarrow}}}_{r}\;1\big{)} in three steps:
[TABLE]
3.3. The left insertion algorithm
A left insertion map that inserts an element in into a staircase , is defined in [2] in two steps as follows. Let be an element in , initially set to .
Step 1. For , iterate the following. If every entry in the -th row is empty, do nothing. Otherwise, let be minimal such that is non-zero. There are two cases according to the values of :
- i)
Suppose . If , decrement by , increment by , and set . If , decrement by , and set . 2. ii)
Suppose . If , decrement by , increment by , and set . If , do nothing.
Step 2. For , if , then increment by . Otherwise, decrement by .
For example, we compute \big{(}4\rightsquigarrow_{l}\scalebox{1.0}{\raisebox{11.38092pt}{ \leavevmode\hbox{\vtop{ \offinterlineskip\openup-0.39993pt\halign{&\opttoksa@YT={\font@YT}\getcolor@YT{\save@YT{\opttoksb@YT}}\nil@YT\getcolor@YT{\startbox@@YT\the\opttoksa@YT\the\opttoksb@YT}#\endbox@YT\cr\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss$$\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss$$\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox 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width=8.79993pt,height=0.39993pt}&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss\scriptstyle\scriptstyle 0\hss}\vss}\kern-8.4pt\vrule height=8.4pt,width=0.39993pt\kern 8.00006pt\vrule height=8.4pt,width=0.39993pt}\kern-0.19997pt\kern-8.4pt\hrule width=8.79993pt,height=0.39993pt\kern 8.00006pt\hrule width=8.79993pt,height=0.39993pt}&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss\scriptstyle 4\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont\cr\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss\scriptstyle 4\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss\scriptstyle 3\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss\scriptstyle 2\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont&\lower 0.39993pt\vbox{\kern 0.19997pt\hbox{\kern 0.39993pt\vbox to8.4pt{\vss\hbox to8.00006pt{\hss\scriptstyle 1\hss}\vss}\kern-8.4pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt\kern 8.00006pt\vrule width=0.0pt,height=8.4pt\kern 0.39993pt}\kern-0.19997pt\kern-8.4pt\kern 0.39993pt\kern 8.00006pt\kern 0.39993pt}\nullfont \crcr}}\kern 35.53343pt}} }\big{)} in three steps:
[TABLE]
3.4. Insertion products on Chinese staircases
For any word , denote by (resp. ) the staircase obtained from by inserting its letters iteratively from left to right (resp. right to left) using the right (resp. left) insertion starting from the empty staircase:
[TABLE]
Define now an internal product (resp. ) on by setting
[TABLE]
for all in . By definition the relations (resp. ) and (resp. ) hold, showing that the product (resp. ) is unitary with respect to .
3.5. The cross-section property
The Chinese monoid of rank , denoted by , is presented by the rewriting system on , whose rules are the Chinese relations, [7]:
[TABLE]
These relations generate the Chinese congruence, denoted by , which can be also interpreted in terms of Chinese staircases as follows. The set of Chinese staircases satisfies the cross-section property for the monoid , that is, for all words on , if and only if , [4, Theorem 2.1]. As a consequence of the cross-section property, we deduce the following result.
3.6. Corollary.
The composition is associative and the following equality
[TABLE]
holds in , for all in and in . In particular, the composition is associative and the following relation
[TABLE]
holds for all in .
4. Column presentation of the Chinese monoid
We construct a finite semi-quadratic convergent presentation of the Chinese monoid by adding the columns over of length at most and square generators to the presentation (4) and by using the combinatorial properties of the insertion algorithms on the Chinese staircases.
4.1. Column generators
We consider one column generator of length for all , one column generator of length for any , and one square generator for any , corresponding to the following three staircases:
[TABLE]
where the shaded areas represent empty boxes. We will denote by the set defined by
[TABLE]
Let us define the map that reads a staircase row by row, from right to left and from top to bottom, and where the reading of the -th row, for , is the following word in :
[TABLE]
For instance, consider the following staircase over :
[TABLE]
4.2. Reduced column presentation
We denote by the rewriting system on whose rules are
[TABLE]
for all and in such that . Normal forms with respect to this rewriting system are called Chinese normal forms. Note that the leftmost and rightmost reductions are the only reductions on a word in with respect to . There will be denoted respectively by
[TABLE]
4.3. Theorem.
The rewriting system is a finite semi-quadratic convergent presentation of the Chinese monoid .
Theorem 4.3 will be proved later in the section. First, we deduce the following corollary:
4.4. Corollary.
The following properties hold:
- i)
The monoid has finite derivation type . 2. ii)
The monoid has finite homological type .
Proof.
In [11] the authors showed that if a monoid admits a finite convergent presentation, then it is of finite derivation type , and the property of finite derivation type implies the property of finite homological type . Thus, Conditions i) and ii) are consequences of Theorem 4.3. ∎
The rest of this section is devoted to the proof of Theorem 4.3. First, prove that is a semi-quadratic presentation of the monoid . We add in Subsection 4.5 the columns generators of length and the square generators with their defining rules. This forms a non-confluent rewriting system that we complete into a presentation of , that we call the precolumn presentation. Then we show in Subsection 4.7 that the rules of are obtained from the precolumn presentation by applying one step of Knuth-Bendix’s completion, [18], on the precolumn presentation. Hence is a presentation of the monoid . Finally, we show in Proposition 4.8 that is terminating and is confluent using the associativity of the product .
4.5. Precolumn presentation
Consider the rewriting system on and whose rules are given by the following four families
[TABLE]
corresponding to the Chinese relations (4), hence is a presentation of the monoid . We add to the set of rules (8) the following set of rules
[TABLE]
making a rewriting system on that presents the monoid .
4.6. Lemma.
For , the rewriting system on , whose set of rules is , where
[TABLE]
is a finite semi-quadratic presentation of the Chinese monoid .
Proof.
We make explicit a Tietze equivalence between the rewriting systems and . For , consider the following critical branching
[TABLE]
of the rewriting system . We consider the Tietze transformation that substitutes therule for the rule , for every . Similarly, we substitute the rules , , , , and respectively for the rules , , , , and using the following critical branchings of the rewriting system :
[TABLE]
[TABLE]
The set of rules obtained in this way is equal to . This proves that the rewriting systems and are Tietze equivalent. ∎
4.7. Completion of the precolumn presentation
The rewriting system is not confluent, it has the following non-confluent critical branchings, that can be completed by Knuth-Bendix completion, [18], by the dotted arrows as follows:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The rules of together with the family of the dotted rules defined by i)-vii) form the set
[TABLE]
That is, the set of rules of . Finally, by this construction, we prove that is at most of length in , showing the semi-quadraticity of the presentation.
4.8. Proposition.
The rewriting system is convergent.
Proof.
Prove that is terminating. Consider the total order defined on by
[TABLE]
[TABLE]
where denotes the lexicographic order on induced by the natural order on . Consider the map sending a word in to its number of columns. Define the length-lexicographic order on with respect to by setting, for all and in :
[TABLE]
where denotes the lexicographic order on induced by . Any reduction with respect to decrease a word in either with respect to or with respect to , showing that the rewriting system is terminating.
Prove that is confluent. Any critical pair of has the form , for in . Note that, by associativity of , the rewriting path (resp. ) reduces (resp. ) to (resp. ), for all in and in . Hence, every critical pair of has the following reduction diagram:
[TABLE]
which is confluent by the associativity of the product . This proves that the rewriting system is locally confluent and thus confluent by termination hypothesis. ∎
5. Chinese syzygies by insertions
In this section we extend the rewriting system into a finite coherent convergent presentation of the Chinese monoid with an explicit description of the generating syzygies. By semi-quadraticity of , every rewriting path with source is an alternated composition of reductions of the form (7). Moreover, every rewriting rule of can be written
[TABLE]
where , , is a permutation on , and denotes the column generator for any .
5.1. Remark
Note that when is not a square generator, then takes value only if rule (9) is one of the commutation rules of the form
[TABLE]
for . When is a square generator, with , then takes value only if rule (9) is one of the form
[TABLE]
We obtain the following bounds for the rewriting paths with source a critical branching of .
5.2. Proposition.
For all in such that and are not Chinese normal forms, the two following inequalities hold:
[TABLE]
The proof of this result is based on the two following lemmata 5.3 and 5.4. Let be in such that and are not Chinese normal forms. The Chinese normal form of the word can be obtained by applying one, two or three steps of reductions of the leftmost normalization strategy of . In this case, we have . Otherwise, the following lemma shows that .
5.3. Lemma.
Let be in such that and are not Chinese normal forms. Suppose that the word obtained after three steps of reductions of the leftmost normalization strategy of with source is not a Chinese normal form. Then, the Chinese normal form of this word is obtained by applying at most two steps of reductions, that consist only on the commutation rules (10).
Proof.
Let , be in such that and are not Chinese normal forms. By definition of , we have
[TABLE]
with , , , and where , , are permutations on , and , , are Chinese normal forms.
Suppose that is not a Chinese normal form. Following Remark 5.1, its only possible reductions are of form (10) or (11). Let us prove that the rules (11) cannot be applied. Suppose the contrary. Then . Since is a Chinese normal form, we obtain that and . Since , this proves that is not a Chinese normal form, which yields a contradiction.
Then we can only apply a commutation rule on , with , and we rewrite the word into . Suppose that is not a Chinese normal form, then we can apply on it a rule of type (10) or (11). As in the previous step, let us prove that the rules (11) cannot be applied. Suppose the contrary. Then . Since is a Chinese normal form, we obtain that , which yields a contradiction. Then we can only apply a commutation rule on .
We have thus proved that the Chinese normal form of the word is obtained by applying at most two steps of reductions that consist only on the commutation rules. ∎
5.4. Lemma.
For all in such that is a square generator and the words and are not Chinese normal forms, the inequality holds.
Proof.
By hypotheses, the word has the following forms: and , for all , , , and , for all , and , for all , , for all , and for all . For all these forms, one can check that . ∎
5.5. Proof of Proposition 5.2
Let , , be in such that and are not Chinese normal forms. Let us prove that . Suppose that the word obtained after two steps of reductions of the leftmost normalization strategy of with source is not a Chinese normal form. Consider a reduction as in (13), and suppose that is not a Chinese normal form. By Lemma 5.3 its only possible reductions are commutation rules, hence there is a reduction . Suppose that is not a Chinese normal form, then by the same argument there is a reduction , where and are Chinese normal forms. Since is a Chinese normal form, we obtain that is a Chinese normal form. This proves the first inequality in (12).
Let us prove that . Suppose that the word obtained after three steps of reductions of the rightmost normalization strategy of with source is not a Chinese normal form. By definition of , we have the following reductions
[TABLE]
with , , , , , , , and where are permutations on , and , , and are Chinese normal forms.
Suppose that the word obtained after applying four steps of reductions of the rightmost normalization strategy with source is not a Chinese normal form. Then and the second reduction of (14) is . Following Remark 5.1, the rule is of form (10) or (11). Let us prove that it cannot be of form (10). Suppose the contrary. Since is a Chinese normal form, we obtain . Moreover, since is not a Chinese normal form, the inequality holds, hence . In this way, the first reduction of (14) is , where is a Chinese normal form, and its second reduction is . Since the word obtained after three steps of reductions of the rightmost normalization strategy of with source is not a Chinese normal form, the word is not a Chinese normal form, which yields a contradiction.
Thus, the rule is of form (11) and is a square generator such that and are not Chinese normal forms. Hence by Lemma 5.4 we obtain . This proves the second inequality in (12).
5.6. Theorem.
The rewriting system extends into a finite coherent convergent presentation of the Chinese monoid by adjunction of a generating syzygy
[TABLE]
for all in such that and are not Chinese normal forms, and where the -cells denote either a rewriting rule of or an identity.
Proof.
Any critical branching of has the form
[TABLE]
for all in such that and are not Chinese normal forms, that is confluent by Theorem 4.3. Moreover by Proposition 5.2, and . We conclude with Squier’s coherence theorem recalled in Subsection 2.4. ∎
Note that some -cells in the boundary of the generating syzygy can be identity. However, following construction given in the proof of Proposition 5.2, if the source (resp. target) of is of length , then its target (resp. source) is of length at most .
5.7. Relations among the insertion algorithms
Note that the generating syzygies of the coherent presentation of the monoid obtained in Theorem 5.6 can be interpreted in terms of the right and left insertion algorithms as follows. Consider the rewriting system on , whose rules are
[TABLE]
for all in such that . By Corollary 3.6, the equality holds for all in , and thus this rewriting system coincides with . Hence, the generating syzygy of the coherent presentation of Theorem 5.6 has the following form
[TABLE]
for all in such that and , where the application of the leftmost (resp. rightmost) normalization strategy (resp. ) on the word corresponds to the application of the right (resp. left) insertion
[TABLE]
5.8. Actions of Chinese monoids on categories
A monoid can be seen as a -category with exactly one [math]-cell , with the elements of the monoid as -cells and with identity -cells only. The category of actions of on categories is the category of -representations of in the category of categories. The full subcategory of whose objects are the -functors is denoted by . We refer the reader to [9] for a full introduction on the category of -representations of -categories. More explicitly, an action of the monoid is specified by a category , an endofunctor for every in , a natural isomorphism for every elements and of , and a natural isomorphism such that:
- i)
for every triple of elements of the monoid , the following diagram commutes
[TABLE] 2. ii)
for every element of the monoid , the following diagrams commute
[TABLE]
Let be a monoid and let be an extended presentation of . The -polygraph is a coherent presentation of if, and only if, for every -category , there is an equivalence of categories between and , that is natural in , [9]. In this way, up to equivalence, the actions of a monoid on categories are the same as the -functors from to .
Using this description, Theorem 5.6 allows us to present actions of Chinese monoids on categories as follows:
5.9. Theorem.
The category of actions of the Chinese monoid on categories is equivalent to the category of -functors from the free -category generated by the rewriting system to the category of categories, that sends any generating syzygy to commutative diagrams in the category .
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