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Syntactic and Rees Indices of Subsemigroups JOURNAL OF ALGEBRA 205, 435]450Ž. 1998 ARTICLE NO. JA977392 Syntactic and Rees Indices of Subsemigroups N. RuskucÏ U Mathematical Institute, Uni¨ersity of St Andrews, St Andrews, KY16 9SS, Scotland and View metadata, citation and similar papers at core.ac.uk brought to you by CORE R. M. Thomas² provided by Elsevier - Publisher Connector Department of Mathematics and Computer Science, Uni¨ersity of Leicester, Leicester, LE1 7RH, England Communicated by T. E. Hall Received August 28, 1996 We define two different notions of index for subsemigroups of semigroups: the Ž.right syntactic index and the Rees index. We investigate the relationships be- tween them and with the group index. In particular, we show that the syntactic index is a generalisation of both the group index and the Rees index. We use this fact to prove further similarities between the group index and the Rees index. It turns out, however, that very few of the nice properties of the group index are inherited by the syntactic index. Q 1998 Academic Press 1. INTRODUCTION The notion of the index of a subgroup in a group is one of the most basic concepts of group theory. It provides one with a sort of ``measure'' of how much smaller the subgroup is when compared to the group. This is particularly reflected in the fact that subgroups of finite index share many properties with the group. A notion of the indexŽ. which in this paper will be called the Rees index for subsemigroups of semigroups was introduced by Jurawx 11 , and is considered in more detail inwxwx 2, 4, 6, 5, 24 . In 24 it is shown that this index has many nice properties similar to those of the group index. *E-mail: [email protected], URL: http:rr www-groups.dcs.st-and.ac.ukr; nikr. ² E-mail: [email protected], URL: http:rr www.mcs.le.ac.ukr; rthomasr. 435 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. 436 RUSKUCÏ AND THOMAS However, the Rees index is not a generalisation of the group index. For instance, an infinite group cannot have any proper subgroups of finite Rees index. This opens up the possibility that the group index and the Rees index are particular instances of a more general index. In this paper we define one such general index, which we call the syntactic index. It turns out that some properties of the group index have their analogues for the syntactic index, but that some others fail, and fail rather badlyŽ see Section 3. The ``good'' properties of the syntactic index, however, prove to be good enough to yield some corollaries about the Rees index, thus strengthening the analogy between the Rees index and the group index established inwx 24 . The search for a better, but still natural, generalisation of the Rees index and the group index is an ongoing project. 2. PRELIMINARIES: SEMIGROUPS, RELATIONS, CONGRUENCES AND ACTIONS In this section we introduce the notation for the rest of the paper, and we also lay the foundations of the theory of syntactic congruences. For the basic notions of semigroup theory we refer the reader to the monograph wx9 . Slightly non-standardly, for a semigroup S, we shall use the notation S1 to mean the semigroup obtained from S by adjoining an identity element Ž.regardless of whether or not S already has one . Also, if p is a relation on 1 1 S, we denote by p the relation p j ÄŽ.1, 14 on S . Let X be a set, and let p be an equivalence relation on X. The equivalence class of x g X is denoted by xrp. The set of all equivalence classes is denoted by Xrp. The number of equivalence classes is called the index of p in X and is denoted by wxX : p . The full relation X = X is denoted by FX , and the diagonal relation ÄŽ.x, x : x g X 4is denoted by D X . Let S be a semigroup, and let p : S = S be any equivalence relation on S. We introduce three relations related to p as follows: 1 SrŽ.p s Ä4 Žx, y .g S = S : Žxs, ys .g p for all s g S 1 SlŽ.psÄ4 Žx,y .gS=S: Žsx, sy .g p for all s g S 1 SŽ.psÄ4 Žx,y .gS=S: Žsxs12,sys 12 .gpfor all s12, s g S . We remark that the three sets defined above are actually the sets p R, p L, and p C fromwxw 7, p. 176 . Thus 7, Lemma 10.3 x can, in our notation, be SYNTACTIC AND REES INDICES 437 stated as follows: PROPOSITION 2.1. Let S be a semigroup and let p be an arbitrary equi¨alence relation on S. Ž.i Sr Žp .is the maximal right congruence of S contained in p. Ž.ii Sl Žp .is the maximal left congruence of S contained in p. Ž.iii S Ž.p is the maximal congruence of S contained in p. Živ .S Žp .s SSrl Ž Žp ..sSS lr Ž Žp ... Let S be a semigroup. For an element s g S and a subset X : S define 1 QsSŽ.,XsÄ xgS:sx g X 4 . Now we have as an immediate consequence of the definition of SrŽ.p : PROPOSITION 2.2. Let S be a semigroup, let p be any equi¨alence relation on S, and let Ci Ž. i g I be all the equi¨alence classes of p in S. Then for arbitrary x, y g Sweha¨e Ž.x,y gSrSiSi Ž.p mQx Ž,C .sQy Ž,C . for all i g I. Closely related to right congruences on semigroups are actions of semigroups. For a semigroup S, aŽ. right action of S on a set X is a mapping a: X = S ª X, Ž.x, s ¬ xs, such that Ž.xs12 s s xss Ž 12 .for all xgXand all s12, s g S. An element x0g X is a source if for every x g X there exists s g S such that xs0 sx. An element y g X is said to be indecomposable if xs / y for all x g X and all s g S. Two actions a on X and b on Y are said to be isomorphic if there exists a bijection f: X ª Y such that Ž.xf s s Ž.xs f for all x g X and all s g S. The following proposition is well known: PROPOSITION 2.3. Let S be a semigroup, let r be a right congruence on S, and let a be an action of S on a set X with an indecomposable source x0. 11 1 1 Ž.iS acts on the factor set S rr byŽ xrr , s.Ž.¬ xs rr ; let us 1 denote this action by AŽ.r . The element 1rr is an indecomposable source of this action. Ž.ii The relation C Ž X, a, x00 .s ÄŽ.s, t g S = S : xssx0 t4 is a right congruence on S. 11 1 Ž.iii CS Ž rr,AŽ.r,1rr .ŽŽ..sr and A C X, a, x0 ( a. We can use the above proposition to prove the following two theorems, which we will need in the subsequent sections. 438 RUSKUCÏ AND THOMAS THEOREM 2.4. Let S be a semigroup. If r is a right congruence of finite index in S, then the congruence SŽ.r also has finite index in S. In particular, for any equi¨alence relation p on S, wS : SrŽ.p xis finite if and only if wS:SŽ.pxwis finite if and only if S : SlŽ.p xis finite. 11 Proof. Consider the action AŽ.r of S on the Ž finite . set S rr given in Proposition 2.3Ž. i . Define a new relation h on S by 1 1 h s Ä4Ž.x, y g S = S : Cx s Cy for all C g S rr . It is straightforward to check that h is a congruence on S, that it is contained in r, and that it has finite index. Hence h : SŽ.r by Proposi- tion 2.1Ž. i , and so S Žr .has finite index. Now let p be any equivalence relation. By Proposition 2.1Ž. i and Ž iii . it follows that SŽ.p : Sr Ž.p . Thus, if S Ž.p has finite index, then so does SrrŽ.p. Conversely, if S Ž.p has finite index, then so does SSŽr Ž..p by the first part of the theorem. By Proposition 2.1Ž. iv we have SSŽ.rlrrlrŽ.p sSSSŽ. Ž.Ž.Ž.p sSS Ž.p sS Ž.p. The remaining assertion holds by symmetry. THEOREM 2.5. A finitely generated semigroup S has only finitely many right congruences of any gi¨en finite index n. Proof. Let X be a set with n q 1 elements. Any action of S on X is fully determined by the actions of generators of S on X. Since S is finitely generated, we conclude that there are only finitely many actions of S on X. Each of these actions having an indecomposable source yields a right congruence, and every right congruence can be obtained in this way by Proposition 2.3Ž. ii and Ž iii . This completes the proof of the theorem. 3. SYNTACTIC INDEX Let G be a group, and let H be a subgroup of G.
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