On Finite Semigroup Cross-Sections and Complete Rewriting Systems

Antonio´ Malheiro Centro de Algebra´ da Universidade de Lisboa Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal and Faculdade de Cienciasˆ e Tecnologia da UNL, Quinta da Torre, 2829-516 Monte de Caparica, Portugal E-mail: [email protected]

Abstract solvable word problem is invariant for any finite presentation defining the same semigroup. Thus we can refer to the In this paper we obtain a [finite] complete rewriting sys- word problem of a given semigroup and not only to some tem defining a semigroup/monoid S, from a given finite of its finite presentations. right cross-section of a subsemigroup/submonoid defined Finite and complete (that is, noetherian and confluent) by a [finite] complete presentation. In the semigroup case rewriting systems are used to solve word problems among the subsemigroup must have a right identity element which other algebraic decision problems (see [3, 8, 11] for exam- must also be part of the cross-section. In the monoid case ples). The word problem is solved using the ‘normal form the submonoid and the cross-section must include the iden- algorithm’: given two words u and v, we can calculate irre- tity of the semigroup. The result on semigroups allow us ∗ ∗ ducible elements u0 and v0 such that u→R u0 and v→R v0, to show that if G is a group defined by a [finite] com- ∗ and we conclude that u↔R v if and only if u0 and v0 are plete rewriting system then the completely simple semi- identical words. This application reveals the importance of group M[G;I,J;P] is also defined by a [finite] complete such rewriting systems. rewriting system. In Group Theory we say that a subgroup H has finite index in a group G if there are finitely many cosets Hg, for any g ∈ G. In that case we get G as a disjoint union of fi- 1. Introduction nitely many cosets H, Hg1, ..., Hgn. The set {1,g1,...,gn} is said to be a cross-section for H in G. This means that One of the most important common topics of Computer each element in G is uniquely representable as the product Science and Mathematics is the word problem. This prob- of an element of H by an element of {1,g1,...,gn}. Groves lem was initially formulated in the group theory by M. and Smith [5] proved that if H is a subgroup of finite index Dehn in 1912 [4]. However, it can be introduced in a nat- in G and H has a [finite] complete rewriting system then G ural way in the semigroup theory. A semigroup presenta- has a [finite] complete rewriting system. The main goal of tion P is a pair hX | Ri where X is an alphabet and R is this paper is to extend this result first for the monoid case a rewriting system on X, that is, a binary relation in the and second to the semigroup case. free semigroup on X. Each semigroup presentation ori- Let S be a semigroup and let M be a subsemigroup of gins a semigroup S(P) resultant of the quotient of the free S. A [finite] subset T is said to be a [finite] (right) cross- + ∗ section for M in S if any element of S can be uniquely fac- semigroup X by the Thue congruence ↔R generated by R. This congruence is the symmetric, reflexive and tran- torized in the form mt, with m ∈ M and t ∈ T. + The first result can be stated without any other extra con- sitive closure of the binary relation →R on X given by: ∗ dition and as we will see the proof can be depicted from the for any (r+1,r−1) ∈ R and any words w1 and w2 on X , we group case. have w1r+1w2 →R w1r−1w2. Now, the word problem can + be formulated as follows: given words u and v in X , de- Theorem 1.1 Let S be a monoid with identity element 1S. cide whether or not u and v represent the same element of Let M be a submonoid of S and let T be finite right cross- ∗ S(P), that is, decide if u↔R v. If there exists an algorithm section for M in S. Suppose that 1S ∈ M ∩T. If M is defined which solves the word problem for any two words then the by a [finite] complete rewriting system, then S is also de- word problem is said to be solvable. The property of having fined by a [finite] complete rewriting system. Our second result requires an extra condition on the sub- 2. Preliminaries semigroup. We say the an element e of a semigroup M is a right identity element if me = m, for any m ∈ M. Let X be an alphabet. We denote by X∗ the free monoid on X and by X+, the free semigroup on X. For an element Theorem 1.2 Let S be a semigroup and let M be a sub- of X∗, a word w, we denote the length of w by |w|. semigroup of S having a right identity element e. Suppose A rewriting system is a pair (X,R), where X is an al- that there is a finite right cross-section for M in S includ- phabet and R is a binary relation in X∗. The elements of ing the element e. If M is defined by a [finite] complete R are referred to as rewriting rules. Usually, a rewriting rewriting system, then S is also defined by a [finite] com- rule r ∈ R is written in the form r = (r+1,r−1) or, simply, plete rewriting system. r+1 → r−1. In the following the rewriting system will be simply denoted by R. We say that a rewriting system R is This last result has a very important consequence in finite if both R and X are finite. Semigroup Theory. A non-empty set A of a semigroup In X∗ we define a binary relation, → , denoted as single- S is called a left ideal if SA ⊆ A, a right ideal if AS ⊆ A R step reduction, in the following way: and an ideal if it is both a left and a right ideal. Com- pletely simple semigroups, among other characterizations u→ v ⇔ u = w1r+1w2 and v = w1r−1w2 (see [6]), are those semigroups with no proper ideals and R ∗ with minimal left and right ideals. for some (r+1,r−1) ∈ R and w1,w2 ∈ X . The transitive In the 1920’s Suschkewitsch [12] has obtained a descrip- and reflexive closure of → is denoted by −→∗ . By −→+ R R R tion of the completely simple semigroups in the following ∗ we denote the transitive closure of →R . A word u ∈ X terms: let G be a group, let I and J be non-empty sets and is said to be R-reducible, if there is a word v ∈ X∗ such let P = (p ji) be a J × I matrix with entries in G; the set that u→R v. If a word is not R-reducible, it is called R- I × G × J with multiplication irreducible or simply irreducible. By Irr(R) we denote the set of all R-irreducible words. (i1,s1, j1)(i2,s2, j2) = (i1,s1 p j1i2 s2, j2), ∗ By ↔R we denote the equivalence relation induced by → which is a congruence on the free monoid X∗. The is a completely simple semigroup; conversely, every com- R ∗ pletely simple semigroup is isomorphic to a semigroup quotient of the free monoid X by what is called the Thue ∗ constructed in this way. The semigroup constructed as congruence ↔R generated by R gives the monoid defined above is denoted by M[G;I,J;P] and it is called the I × J by R and it is denoted by M(X;R). The set X is called the Rees matrix semigroup over the group G with the sandwich generating set and R the set of defining relations. More matrix P. generally, a monoid is said to be defined by the rewriting This description will allow us to use the above theorem system R if M =∼ M(X;R). Thus, the elements of M are and conclude the following about completely simple semi- identified with congruence classes of words from X∗. We groups: will sometimes identify words and elements they represent. We say that a rewriting system R on X is noetherian if Theorem 1.3 Let G be a group, let I and J be non-empty the relation →R is well-founded, in other words, if there is finite sets and let P = (p ji) be a J ×I matrix with entries in no infinite descending chains G. If the group G is defined by a [finite] complete rewriting system then the completely simple semigroup M[G;I,J;P] w1 →R w2 →R w3 →R ···→R wn →R ··· . is also defined by a [finite] complete rewriting system. ∗ We say that R is confluent if whenever we have u−→R v ∗ 0 ∗ ∗ To understand the finiteness condition on the sets I and and u−→R v there is a word w ∈ X such that v−→R w and 0 ∗ J on the above result it is essential to have in mind the re- v −→R w. If R is simultaneously noetherian and confluent sult [2] by H. Ayik and N. Ruskuc,ˇ where the authors con- we say that R is complete. sider finite presentability of Rees matrix semigroups. They It is easy to verify that, if R is a noetherian rewriting have conclude that the Rees matrix semigroup M[G;I,J;P] system, each congruence class of M(X;R) contains at least is finitely presented (i.e., defined by a finite rewriting sys- one irreducible element. Assuming R noetherian, then R tem over a finite alphabet) if and only if the group G is is a complete rewriting system if and only if each con- finitely presented and the sets I and J are finite. gruence class of M(X;R) contains exactly one irreducible After some preliminaries on Section 2, we turn, in Sec- element [3]. Hence, a complete rewriting system fixes a tion 3, to the proof of Theorem 1.2. It follows a section unique normal form for each of its congruence classes. dedicated to the proof of Theorem 1.1 and we finish with A sufficient condition for a rewriting system R be Section 4 where we prove Theorem 1.3. noetherian is given in the following lemma. Lemma 2.1 Let R be a rewriting system on an alphabet Let R be a [finite] complete rewriting system on an al- X and let > be a binary relation irreflexive, transitive and phabet X. Denote by e the word in X+ representing the well-founded in X∗. right identity of M. Let Y be an alphabet disjoint from X in one-to-one cor-

1. If > satisﬁes u→R v implies u > v, then R is respondence with T. Denote by u the letter in Y represent- noetherian. ing the right identity of M which as we have supposed it

also belongs to T. So we have u =S e, that is, u is equal to 2. If > satisfies (u,v) ∈ R implies u > v and u > v implies e in S. ∗ that w1uw2 > w1vw2, for all w1,w2 ∈ X , then R is If we define a correspondence from X ∪Y to S by associ- noetherian. ating any element in X to the corresponding representant in M (and thus in S) and to any element of Y the correspondent Let S be a monoid. Given a surjective morphism element of T, and we extend it to (X ∪Y)+ we will clearly ϕ : X∗ → S and a noetherian rewriting system R on X, the obtain a surjective morphism. Along this section we will following proposition gives us sufficient conditions to guar- identify the elements of (X ∪Y)+ with the corresponding antee that there exist a complete rewriting system for S. elements of S. It might be now clear to the reader that we will use Proposition 2.2 Let S be a monoid and R a noetherian Proposition 2.2 to prove that S is defined by a [finite] com- rewriting system on X. Let : X∗ → S be a surjective mor- ϕ plete rewriting system. phism such that Let us consider the following sets of rewriting rules on the alphabet X ∪Y: 1. ϕ(r+1) = ϕ(r−1), for each relation (r+1,r−1) ∈ R, R = {wz → xv : w,z,v ∈ Y,x ∈ Irr(R),wz = xv} 2. the restriction of ϕ to Irr(R) is one-to-one. 1 S R2 = {wx → yv : w,v ∈ Y,x ∈ X,y ∈ Irr(R),wx =S yv} Then R is a complete rewriting system that defines S. R3 = {yw → w : w ∈ Y,y ∈ Irr(R),yw =S w} R = {u → e}. In the above concepts is clear that if we replace monoid by 4 ∗ + semigroup and X by X we obtain similar definitions and 0 Now, denote by R the set R∪R1 ∪R2 ∪R3 ∪R4. Notice that we still with valid results. if T is finite and hX | Ri is finite then R0 is also finite. It is important to notice that for a noetherian rewriting We have to see if these rules are valid in S. We observe system R on a set X there only exists a finite number of de- that each element of S can factorized in the form mt, with ∗ scending chains starting at a word u of X . Thus there exists m ∈ M and t ∈ T. Also, each element t ∈ T is of the form a maximum on the length of any word appearing on all de- m t, for some m ∈ M. Together with these observations we ∗ t t scending chains starting at a word u of X . That maximum see that the rewriting rules of R0 are obtained when they are is called the stretch of u and it is denoted by stR(u). Ob- equalities in S, so we may conclude that they are valid in S. serve that if stR(u) > stR(v) then stR(w1uw2) > stR(w1vw2), We claim that the set of irreducible elements of the ∗ for any words w1,w2 ∈ X , that is, the stretch is compatible rewriting system R0 on the alphabet X ∪Y is (Irr(R)\{y ∈ with the concatenation product. Q})(Y\{u}) ∪ Y\{u} ∪ Irr(R), where Q is the set {y ∈ Further information on rewriting systems can be found Irr(R) : yw =S w,w ∈ Y}. We begin by observing that if in [3]. we start with a word in (X ∪Y)+ with at least one letter from Y we can: first apply rewriting rules from R2 to ob- 3. Cross-sections - the semigroup case tain a word in the form zv, with z ∈ X+ and v ∈ Y +; then we can apply rewriting rules from R1 to v to obtain a word of This section is dedicated to the proof of Theorem 1.2. the form z0v0, with z0 ∈ X+ and v0 ∈ Y; now, if v0 = u we ap- Let S be a semigroup and let M be a subsemigroup of S ply the rewriting rule from R4 and we end up with a word having a right identity e. Suppose that there exists a finite in X+; in any case to the remaining word of X+ we ap- right cross-section T for M in S. Also, suppose that e be- ply the rewriting rules from R to get an element in Irr(R). longs to T. If we get a factorization of the form yw, with y ∈ Q and We observe that any element t of the cross-section is an w ∈ Y\{u}, we apply a rewriting rule from R3 to obtain a element in S and hence it is factorized in the form mt0, for word in Y\{u}. Finally, if we have a word in X+ we apply some m ∈ M and t0 ∈ T. Thus we have an element et = emt0 rewriting rules from R to get a word in Irr(R). The final set in S. It follows from the uniqueness of the decomposition of irreducible elements will be the set claimed above. of the elements in S that t = t0. Concluding, any element t As already mentioned each element of S is uniquely rep- of the cross-section is equal to mtt, for some mt ∈ M. resentable as the product of an element of M and an element of T. Clearly the set M(T\{e}) is in one-to-one cor- In this case we are working on the free monoids over respondence with (Irr(R)\{y ∈ Q})(Y\{u}) ∪Y\{u} and X and over X ∪Y and therefore including the empty word. Me = M is in one-to-one correspondence with Irr(R). Thus Thus we don’t need to distinguish a word representing the we conclude that S is one-to-one correspondence with our identity of M which is also the identity of S. Also the set Y 0 set of irreducible elements of R . will be in one-to-one correspondence with T\{1S}. According to Proposition 2.2 it remains to show that The factorization of the elements in T is trivial in this the rewriting system is noetherian. In order to follow that case, since each t ∈ T is uniquely decomposed in the form propose we will introduce a binary relation > on the set 1St, with 1S ∈ M and t ∈ T. Thus we will not have the set + (X ∪Y) that it is irreflexive, transitive and well-founded. R3 in our final rewriting system. Moreover this relation is also compatible with the concate- We consider the sets R1 and R2 of rewriting rules nation product. Hence we will use Lemma 2.1(2) to prove that R is noetherian. {wz → xv : w,z ∈ Y,v ∈ Y ∪ {1},x ∈ Irr(R),wz =S xv} A word z in (X ∪Y)∗ has the form {wx → yv : w ∈ Y,v ∈ Y ∪{1},x ∈ X,y ∈ Irr(R),wx =S yv}, 0 xnunxn−1 ···x1u1x0, respectively, on the alphabet X ∪Y. Let R be the set R ∪ R ∪ R . If T is finite and hX | Ri is finite then R0 is also ∗ 1 2 with n ∈ N0, ui ∈ Y, and x j ∈ X . Let us define maps finite. In this case it is clear that the set of irreducible elements ψ (z) = |u ···u | = n, 0 n 1 is Irr(R) ∪ Irr(R)Y, that it is in one-to-one correspondence ψ2i−1(z) = stR(xi−1), with S. ψ2i(z) = xi−1, Keeping the same relation ordering we also conclude that u > v whenever (u,v) ∈ R0. The slightly changes on the for i ∈ {1,...,n + 1}. We define > by: given z1,z2 ∈ (X ∪ sets R1 and R2 (notice that now v may be the empty word) + Y) , we have z1 > z2 if it exists k ∈ N0 such that ψi(z1) = can only lead us to the conclusion that ψ0(u) > ψ0(v). ψi(z2), for i < k, and ψk(z1) > ψk(z2) (if k odd or 0) and Hence we can also say that R0 is noetherian. Therefore, by ψ (z )−→+ ψ (z ) (if k is even). Proposition 2.2, the rewriting system R0 is [finite] complete k 1 R k 2 Finally, we claim that u > v for each (u,v) ∈ R0. Let us and it defines the monoid S. see case by case: 5. Completely simple semigroups • if (u,v) ∈ R then ψ0(u) = ψ0(v), ψ1(u) ≥ ψ1(v) and ψ (u)−→+ ψ (v); 2 R 2 The isomorphism mentioned in the Introduction be- tween a completely simple semigroup and a I ×J Rees ma- • if (u,v) ∈ R then ψ (u) > ψ (v); 1 0 0 trix semigroup over the group G with the sandwich matrix • if (u,v) ∈ R2 then ψ0(u) = ψ0(v) and ψ1(u) > ψ1(v); P can be, in a certain sense, normalized. It is possible to find a sandwich matrix P, which it will be called normal, • if (u,v) ∈ R3 then ψi(u) = ψi(v), for i = 0,1,2, and having more appropriate entries. ψ3(u) > ψ3(v); and Let G be a group, let I and J be non-empty finite sets and let P = (p ji) be a J × I matrix with entries in G. With- • if (u,v) ∈ R4 then ψ0(u) > ψ0(v). out lost of generality we can suppose that both sets I and J Thus R is noetherian. It follows from Proposition 2.2 that have an element named 1 and that the first column has ele- 0 + R is a [finite] complete rewriting system on (X ∪Y) that ments pi1’s and the first row p1 j’s. Let e denote the identity defines the semigroup S. element of the group G. The sandwich matrix P is called normal if every entry in the first row and in the first column 4. Cross-sections - the monoid case of P is equal to e. The normalization theorem says that if S is a completely In this section we will prove Theorem 1.1. So let S be a simple semigroup then S is isomorphic to a Rees matrix monoid and let M be a submonoid of S with 1S ∈ M. Also, semigroup M[G;I,J;P] in which the matrix P is normal. suppose that there is a finite right cross-section T for M in The isomorphism from S = M[G;I,J;Q] to M[G;I,J;P], S with 1S ∈ T. with Q = [q ji] not necessarily normal, is obtained from −1 The proof of Theorem 1.1 follows the same idea of the the correspondence (i,a, j) 7→ (i,q11 q1iaq j1, j), and where −1 −1 previous case, the semigroup case. In fact we can say that P = [p ji = q j1 q jiq1i q11] is normal (see [6] for more de- it is a simplified version of it. We will only sketch the proof tails). From now on we will assume that the sandwich ma- pointing out the differences. trix P is normal. The subsemigroup {1} × G × {1} has an identity ele- [6] J. M. Howie. Fundamentals of Semigroup Theory. Claren- ment (1,e,1). The set {1} × {e} × J is a finite right cross- don Press, Oxford, 1995. section for {1} × G × {1} in {1} × G × J since each word [7] M. Kambites. Presentations for semigroups and semi- (1,a, j) is uniquely decomposed as (1,a,1)(1,e, j), with groupoids. Int. J. Algebra Comput., 15(2):291–308, 2005. (1,a,1) ∈ {1} × G × {1} and (1,e, j) ∈ {1} × {e} × J. [8] N. Kuhn and K. Madlener. A method for enumerating cosets of a group presented by a canonical sistem. In G. Gonnet, Suppose that G is defined by a [finite] complete rewrit- editor, Proceedings, ISSAC’89, pages 338–350. ACM, New ing system. By Theorem 1.2 we conclude that {1} × G × J York, 1989. is defined by a [finite] complete rewriting system. [9] M. V. Lawson. Rees matrix semigroups over semigroupoids Now, {1} × G × J has a left identity (1,e,1). Also, the and the structure of a class of abundant semigroups. Acta set I ×{e}×{1} is a finite left cross-section for {1}×G×J Sci. Math., 66(3-4):517–540, 2000. in M[G;I,J;P]. Thus, by applying the dual of Theorem 1.2 [10] A. Malheiro. Finite derivation type for Rees matrix semi- for left cross-sections we get the intended result, that is, groups. Theor. Comput. Sci., 355(3):274–290, 2006. M[G;I,J;P] is defined by a [finite] complete rewriting sys- [11] F. Otto. Conjugacy in monoids with a special Church- tem. Rosser presentation is decidable. Semigr. Forum, 29:223– 240, 1984. [12] A. Suschkewitsch. Uber¨ die endlichen gruppen ohne das 6. Conclusion gesetz der eindeutigen umkehrbarkeit. Math. Ann., 99:30– 40, 1928. Completely simple semigroups are an important class of semigroups. The problem of relating the properties of the group G with the properties of the semigroup M[G;I,J;P] was studied in several papers (see for example [1, 2, 7, 9, 10]). Concerning complete rewriting systems we have proved that if G is defined by a [finite] complete rewriting system and both I and J are finite then M[G;I,J;P] is also defined by a [finite] complete rewriting system. We end with the following natural question: Open Problem: If a finitely presented completely simple semigroup S = M[G;I,J;P] is defined by a complete rewriting system is then G also defined by a complete rewriting system.

Acknowledgements

This work was developed within the project POCTI- ISFL-1-143 of CAUL, ﬁnanced by FCT and FEDER. It is also part of the project ”Semigroups and Languages - PTDC/MAT/69514/2006”, ﬁnanced by FCT.

References

[1] I. M. Araujo.´ Finite presentability of semigroup construc- tions. Int. J. Algebra Comput., 12(1-2):19–31, 2002. [2] H. Ayık and N. Ruskuc.ˇ Generators and relations of Rees matrix semigroups. Proc. Edinb. Math. Soc., II. Ser., 42(3):481–495, 1999. [3] R.V. Book and F. Otto. String-Rewriting Systems. Springer- Verlag, New York, 1993. [4] M. Dehn. Uber unendliche diskontinuierliche gruppen. Math. Ann., 72:116–144, 1912. [5] J. R. J. Groves and G. C. Smith. Soluble groups with a ﬁ- nite rewriting system. Proc. Edinb. Math. Soc., 36:283–288, 1993.