The p n sequences of semigroup varieties generated by combinatorial 0-simple semigroups

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1 The p n sequences of semigroup varieties generated by combinatorial 0-simple semigroups Kamilla Kátai-Urbán and Csaba Szabó Abstract. We analyze the free spectra of semigroup varieties generated by finite combinatorial 0-simple semigroups. We give estimates for all of them. 1. Introduction The study of p n sequences was initiated by Grätzer in his book on universal algebra [5]. Let t = t(x 1,..., x n ) be an n-ary term. A term operation t A is said to be essentially n-ary, if it depends on all of its variables. For n 1, denote the number of essentially n-ary term operations over A by p n (A). Let A be a k-element finite algebra and let V denote the variety generated by A. It is known that the size of the free algebra F V (n) in V freely generated by n elements is at most k kn. If k 2, then this number is at least n. The free spectrum of a variety V is the sequence of cardinalities F V (n), n = 0, 1, 2,..., and the equalities F V (n) = n ( n k=0 k) pk (A) hold. For example, for the free spectrum of Boolean algebras we have F V (n) = 2 2n. The first important question about free spectra is the following: Within the above bounds what are the possible numbers? For finite algebras there may be a strong connection between structural properties of the algebra and the free spectra. If G is a finite group, then the size of the n-generated relatively free group in the variety generated by G is exponential in n if G is nilpotent and at least doubly exponential if G is not nilpotent ([6] and [9]). Another theorem on free spectra is Theorem 12.3 in [7]: If V is a nontrivial locally finite congruence distributive variety, then for every c such that 0 < c < 1, and for every large n, the free spectrum of V is bounded below by 2 2cn. We also have some results on the free spectra called gap theorems. For example, Theorem 12.2 in [7] says: Let V be a variety generated by a k-element algebra. Then either F V (n) cn k for some constant c, or else F V (n) 2 n k for all n. For simple algebras, the following characterization of possible free spectra was proved by Joel Berman in [1]. The tame congruence types are denoted by 1, 2, 3, 4 or 5 and called unary, affine, Boolean, lattice or semilattice type, respectively. Date: December 27, Key words and phrases: free spectra, semigroup. 1

2 2 KAMILLA KÁTAI-URBÁN AND CSABA SZABÓ Theorem 1.1. For each k 2 there exist positive constants d 1,..., d 5 and c 1, c 2, c 4 such that if A is a k-element simple algebra and V is the variety generated by A, then for sufficiently large n, (1) if typ(a) =1, then d 1 n F V (n) c 1 n log 2 k ; (2) if typ(a) =2, then d 2 k n F V (n) c 2 k (k 1)n ; (3) if typ(a) =3, then k d3kn F V (n) k kn ; (4) if typ(a) =4, then k d4kn / n F V (n) k c4kn / n ; (5) if typ(a) =5, then d 5 k n F V (n) k σ(n) for σ(n) = ( ) nk n n k(k 1) 3 (k 1) 3 (k 1) n (k 1)3. As we see, for type 5 there is a huge gap between the two bounds. A very interesting class of type 5 algebras is the class of finite 0-simple semigroups. For a basic reference on the general properties of p n sequences for semigroups see [2]. A full description of finite semigroups for which the p n sequence is bounded by a polynomial, is presented in [4]. The aim of paper [2] is to prove that p n sequences of finite semigroups satisfying S 2 = S are eventually strictly increasing, except in a few well-known cases, when they are bounded (by a constant). Semigroups with bounded p n sequences are described first in terms of identities and then structurally as nilpotent extensions of semilattices, Boolean groups and rectangular bands [3]. We continue the investigation of p n sequences from the other direction. We investigate the p n sequences and free spectra of varieties generated by finite 0-simple semigroups. In [14] and [15] p n sequences of band varieties are examined. Here we have to comment on terminology. A semigroup is called 0-simple if it has a zero element 0, is different from the two-element semigroup with zero multiplication, and its only ideals are {0} and the whole semigroup. Note that the term simple has different meanings in universal algebra and semigroup theory. In universal algebra, an algebra is called simple if it has no nontrivial congruences. In semigroup theory, a semigroup of this kind is called congruence-free, and a semigroup is termed simple if it has no proper ideals. The notion of a 0-simple semigroup is the analogue of the latter notion within the class of semigroups with zero. In the present paper, we use only the term 0-simple, so no confusion can occur. Notice that a group considered as a semigroup has no proper ideal, hence every group with 0 adjoined is 0-simple, but not all 0-simple semigroups arise in this way. Let Λ and I be nonempty sets, G a group and let M = (m λ,i ) be a Λ by I matrix with entries in G {0} such that each row and column contains at least one nonzero element. The Rees matrix semigroup over G 0 is M 0 (I, G, Λ; M) = (I G Λ) {0} with the operation { (i, gm λ,j h, µ) if m λ,j 0; (i, g, λ)(j, h, µ) = 0 if m λ,j = 0, and 0a = a0 = 0 for every a M 0 (I, G, Λ; M). One can also consider this semigroup as the semigroup of I Λ matrices with one entry from G and all other

3 p n SEQUENCES OF SEMIGROUPS 3 entries 0, with the operation, where A B = AMB. The matrix M is called a sandwich matrix. The importance of Rees matrix semigroups for us is the following: every finite Rees matrix semigroup over a group with 0 is 0-simple, and conversely, each finite 0-simple semigroup is isomorphic to a Rees matrix semigroup over a group with 0. In what follows, we shall consider Rees matrix semigroups only over groups with 0, and we shall call them, for short, Rees matrix semigroups. A semigroup is called combinatorial if it does not contain a nontrivial group as a subsemigroup. For a Rees matrix semigroup this means that the group G is trivial, G = {1}. In this case we omit the group element component from the notation and write (i, λ) for (i, 1, λ). The matrix M will turn into a 0-1 matrix and the operation is simplified to { (i, µ) if m λ,j = 1; (i, λ)(j, µ) = 0 if m λ,j = 0. In [16] Reilly described the lattice of varieties generated by completely 0-simple semigroups. In particular, he proved that there are exactly 10 combinatorial varieties among them. They are the trivial variety and the varieties in the first column of Table 1. Each of these varieties can be generated by one finite 0-simple semigroup (the choice of which is in general not unique). The second column of Table 1 contains a combinatorial Rees matrix semigroup generating the given variety, and in the third column we find the sandwich matrix for the given Rees matrix semigroup. The main goal of our paper is to prove the last column of Table 1, namely to present the size of the n-generated free algebra in these varieties. We use the following notation: F V (n) f(n) means lim F V (n) /f(n) = 1 and F V (n) log f(n) means that log F V (n) log f(n). The semigroups Y, L, R, B 2 and A 2 can be presented by defining relations as follows, where the superscript 0 indicates that this is a presentation of a semigroup with zero. Y = e e 2 = e 0 ; L = e, f e 2 = ef = e, f 2 = fe = f 0 ; R = e, f e 2 = fe = e, f 2 = ef = f 0 ; B 2 = a, b aba = a, bab = b, a 2 = b 2 = 0 0 ; A 2 = a, b aba = a, bab = b, a 2 = a, b 2 = 0 0. Thus Y is the two-element semilattice, L [R] is the two-element left [right] zero semigroup with zero adjoined, and B 2 is the five-element combinatorial Brandt semigroup. The varieties in the first four rows of Table 1 are varieties of bands (i.e. of semigroups in which every element is idempotent), their names are: semilattices, left normal bands, right normal bands, and normal bands, respectively. Moreover, we have N B = LN B RN B, and X 2 = B X for X = LN B, RN B, N B. 2. The free spectra of the nine varieties The authors proved in [10] that

4 4 KAMILLA KÁTAI-URBÁN AND CSABA SZABÓ Variety generating matrix (M) free spectra semigroup of variety SL Y [1] 2 n 1 [ 1 LN B L n2 1] [ ] RN B R 1 1 [ ] 1 1 N B N 1 1 B B 2 [ 1 ] LN B 2 L RN B 2 R 2 [ 1 0 ] N B 2 N A A 2 [ 1 ] Table 1 n2 n 1 n(n + 1)2 n 2 log n 2n log n 2n log n 2n log n 2n n 2 2 n2 Theorem 2.1. log F B (n) 2n log n. For the investigation of the variety A see [11]. Theorem 2.2. F A (n) n 2 2 n2. In this paper we investigate the free spectra of the remaining seven varieties. We do this via counting the number of distinct n-variable terms in each generating semigroup. Now, we go step by step through each variety from Table 1. Although a few free spectra have already been characterized before, for completeness we include the appropriate details here, as well. Since we consider term operations over combinatorial Rees matrix semigroups whose non-zero elements are pairs, throughout the paper we denote the variables x i also as pairs (u i, v i ). Two terms t 1 and t 2 are called equivalent (t 1 (x 1,..., x n ) t 2 (x 1,..., x n ) or shortly t 1 t 2 ) over an algebra A if the term operations t A 1 and t A 2 are equal, i.e., for every a A n, t A 1 (a) = t A 2 (a).

5 p n SEQUENCES OF SEMIGROUPS 5 Since the variety of semilattices is contained in each variety considered in the paper, a term containing k (pairwise distinct) variables necessary determines an essentially k-ary term operation on each semigroup listed in Table 1. By a graph we mean an undirected graph without multiple edges and loops. For a graph G we denote its sets of vertices and of edges by V (G) and E(G), respectively. The connected component of the graph G containing the vertex v is denoted by Comp G (v). Let t = t(x 1, x 2,..., x n ) be an n-ary term. We assign a bipartite graph G(t) to t as follows. Let the top vertices of G(t) be u 1,..., u n and the bottom vertices v 1,..., v n. There is an edge between v i and u j if x j follows x i somewhere in t, that is, if x i x j is a segment of t. A similar construction of directed graphs can be found in [12], paragraph III/A and in [17]. The following result is taken from [17]. Proposition 2.3. Let S be a combinatorial Rees matrix semigroup in Table 1 distinct from A 2, M be the sandwich matrix of S and J be the r l all 1 matrix. Let p = p(x 1,..., x n ) = x i1... x ik and q = q(x 1,..., x n ) = x j1... x jm (i 1,..., i k, j 1,..., j m {1,..., n}) be two terms. (1) If M = J then p q over S if and only if the following conditions are satisfied: (a) the variables occurring in p and q are the same; (b) if r 2, then x i1 = x j1 ; (c) if l 2, then x ik = x jm. (2) If M J then p q over S if and only if the following conditions are satisfied: (a) the components of G(p) and G(q) are the same; (b) Comp G(p) (u i1 ) = Comp G(q) (u j1 ); (c) Comp G(p) (v ik ) = Comp G(q) (v jm ); (d) if there are two equal rows of M, then x i1 = x j1 ; (e) if there are two equal columns of M, then x ik = x jm. Proposition 2.4. The free spectrum of SL is 2 n 1, n = 1, 2,.... Proof. The corresponding matrix is the identity matrix M = [1]. Clearly, the semilattice term operations depend only on the set of variables, hence the number of essentially n-ary term operations is the same as the number of subsets of a set of size n. We have to exclude the empty word. Proposition 2.5. The free spectrum of RN B and LN B is n2 n 1, n = 1, 2,.... Proof. The generating semigroups of RN B and LN B are dually isomorphic, hence the free spectra are the same. Thus it is enough to consider RN B. We calculate p k (R) first. The corresponding matrix is M = [ 1 1 ]. We have two identical columns in M but a single row. Hence, by Proposition 2.3(1), two terms in k variables are equivalent if and only if their last variables are the same. We have k many choices for the last variable, hence there are k many essentially k-ary term operations, i.e. p k (R) = k. Thus F RN B = k ( n k) = n2 n 1. Proposition 2.6. The free spectrum of N B is n(n + 1)2 n 2, n = 1, 2,....

6 6 KAMILLA KÁTAI-URBÁN AND CSABA SZABÓ [ ] 1 1 Proof. The corresponding matrix is M =. We have two identical rows and 1 1 columns in M, hence similary to the previous argument, there are k 2 essentially k-ary term operations over N B. Now F N B (n) = k 2( n k) = n(n + 1)2 n 2. The analysis of the remaining three varieties will be based on Theorem 2.1, where F B (n) denotes the number n-ary term operations over B 2. First, we investigate the p n sequences of these varieties. Proposition 2.7. The number of essentially n-ary term operations for the algebras N 2, L 2, R 2 satisfy the following inequalities: p n (B 2 ) p n (L 2 ) = p n (R 2 ) p n (N 2 ) n 2 p n (B 2 ). Proof. In case of each of these three semigroups a term operation induced by a term t depends on the components of G(t), on the first variable and on the last variable. By symmetry p n (L 2 ) = p n (R 2 ). We have B 2 < L 2 < N 2, hence p n (B 2 ) p n (L 2 ) = p n (R 2 ) p n (N 2 ). By Proposition 2.3(2) the only difference in the equivalence of terms is in the role of the first and last variables. In case of B 2 a graph G(t) belongs to at most Comp G (t) 2 many distinct term operations and in case of N 2 it belongs to at most n 2 many distinct term operations. Hence p n (N 2 ) n 2 p n (B 2 ). Theorem 2.8. Let V denote one of the varieties LN B 2, RN B 2, N B 2. Then log F V (n) 2n log n. Proof. By Proposition 2.7 for every generating semigroup S of the above varieties V, we have p n (B 2 ) p n (S) n 2 p n (B 2 ) and so F B (n) F V (n) n 2 F B (n). Taking the logarithm of each term and applying log F B (n) 2n log n from Theorem 2.1 we obtain log F V (n) 2n log n Thus F V (n) 2n log n. log F B(n) 2n log n 1 and log F V(n) 2n log n log F B(n) 2n log n + 1 n Further work on the varieties containing B 2 One might express some disappointment about the estimates for the free spectra of the varieties B, LN B 2, RN B 2, N B 2. Indeed, we have only presented an asymptotic formula for the logarithm of the appropriate functions and all these logarithms are equal. Here we add a few details about the structure of term operations in these varieties. A partition π of the set {1,..., n} is a collection of pairwise disjoint non-empty subsets of {1,..., n}, called blocks, whose union equals {1,..., n}. Denote by Π n the set of all partitions of the set {1,..., n} and by π the number of the blocks of the partition π.

7 p n SEQUENCES OF SEMIGROUPS 7 Let S(n, k) = {π Π n : π = k}, the number of partitions of size k of an n-element set. The numbers S(n, k) are called the Stirling-numbers of the second kind. The number of the partitions of the set {1,..., n} is called the Bell-number and is denoted by B(n). With our notation, we have B(n) = Π n = n k=0 S(n, k). We shall use the following asymptotic formula for B(n), see [13]: B(n) 1 n r n+ 1 2 e r n 1 (1) where r log r = n. Note that for every ε > 0 there is an l such that for n > l we have n/ log n < r < n(1 + ε)/ log n. Our main tool will be the following theorem from [10]. Lemma 3.1. Let π 1, π 2 be partitions of the sets {u 1,..., u n } and {v 1,..., v n }, respectively, with π 1 = π 2 = k (1 k n). For every pair of such partitions π 1, π 2, there is an n-ary term t = x 1... x 1, such that t B2 is an essentially n-ary term operation and the components of the corresponding bipartite graph G(t) induce the partitions π 1 and π 2 on the sets {u 1,..., u n } and {v 1,..., v n }, respectively. Let us fix some notation. Let t 1 = t 1 (x 1,..., x n ) = x i1 x i2... x ik and t 2 = t 2 (x 1,..., x n ) = x j1 x j2... x jm be two terms, where x ik = x j1. We denote by t 1 t 2 the linking of these two terms, such that we include x ik only once: t 1 t 2 = x i1... x ik x j2... x jm. We will use the notation t k for the initial segment of t until the first appearance of the variable x k (including x k ) and t k for the final segment of t from the last appearance of the variable x k (including x k ). We construct a few more terms modifying Lemma 3.1. Lemma 3.2. Let π 1, π 2 be partitions of size k of the sets {u 1,..., u n } and {v 1,..., v n }, respectively, and let 1 i, j n. There is an n-ary term t = x i... x j, such that t B2 is an essentially n-ary term operation and the components of the corresponding bipartite graph G(t) induce the partitions π 1 and π 2 on the sets {u 1,..., u n } and {v 1,..., v n }, respectively. Proof. Let t = x 1... x 1 be a term inducing π 1 and π 2 on the vertices of G(t). Such a term exists by Lemma 3.1. Let t 1 = t i t t j. The first variable of t 1 is x i, the last variable is x j. The graph G(t 1 ) contains the edges of G(t) as t is a segment of t 1. Since t 1 is built of segments of t, and at the concatenating points no extra edges were constructed, we have G(t 1 ) = G(t). Let us examine the components of G(t), the graph corresponding to the term t = x 1 x 2... x n and the partitions π 1 and π 2 induced. Since x i x i+1 (i = 1,..., n 1) is a segment of t the vertex v i is connected to u i+1 for i = 1,..., n 1. Thus there are at most two isolated (not connected to any other vertex) vertices in G(t): v n and u 1. Every point belongs to a unique component and every block of π 1 distinct from {u 1 } is connected to a unique block of π 2. Hence {u 1 } and {v n } are the only candidates for blocks not connected to any other block. Thus if π 1 has k blocks, then π 2 has k or k 1 or k + 1 blocks. On the other hand, the edges of G(t) determine a matching between the blocks of π 1 and π 2. There are possibly two vertices, u 1 and v n not belonging to the matching.

8 8 KAMILLA KÁTAI-URBÁN AND CSABA SZABÓ Proposition 3.3. The number of essentially n-ary term operations for the algebras B 2, L 2, R 2, N 2 satisfy the following inequalities: n (1) k 2 S(n, k) n 1 n(k 1)S(n 1, k) 2 (2) (3) p n (B 2 ) n k 2 k!s(n, k) n 1 n(k 1)(k 1)!S(n 1, k)s(n, k); n kns(n, k) n 1 n(n 1)S(n 1, k) 2 p n (L 2 ) = p n (R 2 ) n knk!s(n, k) n 1 n(n 1)(k 1)!S(n 1, k)s(n, k); n n 2 S(n, k) n 1 n(n 1)S(n 1, k) 2 p n (N 2 ) n n 2 k!s(n, k) n 1 n(n 1)(k 1)!S(n 1, k)s(n, k). Proof. The first sum in each expression is guaranteed by Lemma 3.2. The number of pairs of partitions of size k is equal to S(n, k) 2. In case of B 2 one can choose both the first and last variables of a term from any of the k partitions. In case of L 2 we have n choices for the first and k choices for the last variable. And in case of N 2 we have n choices for both variables. For the first sum in the upper bounds, the two partitions of size k can be matched in k! many ways, hence the two partitions are induced by at most k 2 k!s(n, k) 2, knk!s(n, k) 2 and n 2 k!s(n, k) 2 many term operations, respectively. The second sum in each expression estimates the number of those term operations where either the first or the last variable occurs exactly once in the expressions but the other at least twice. In that case the appropriate vertex is an isolated point and the two partitions induced by the term are on n and n 1 vertices, respectively. In the lower bounds we choose the first or the last variable in n different ways and then we simply concatenate a k 1-ary term on the remaining variables. For the upper bounds we consider a partition of size k on the n and n 1 vertices and multiplied by the usual (k 1)!, respectively. There is a fair amount of questions arising after this discussion. Where is the truth between the bounds above? We think that the upper bounds are very close to the free spectra, although not all pairs of partitions of the same size are induced by a term on these algebras. For example, it is easy to see that if there is a subset S of the set n = {1, 2,..., n} such that there are no edges between u i and v j or u j and v i for any i S and j n \ S in a graph G, then there is no term t over any of these algebras such that G = G(t). Problem 1. Prove that the upper bounds in Proposition 3.3 (1) (3) are asymptotic values for the p n sequences.

9 p n SEQUENCES OF SEMIGROUPS 9 After this, the job remaining is to find asymptotics for the free spectra. The sums in Proposition 3.3 seem hopeless to handle. Problem 2. Find asymptotic formulas for the p n sequences and for the free spectra of the varieties B, LN B 2, RN B 2, N B 2. What happens if we consider a Rees matrix semigroup that is not combinatorial? As an intermediate special case, assume that the sandwich matrix is a 0-1 matrix. The first question to investigate is the following: Problem 3. Find the free spectra of the varieties generated by Rees matrix semigroups M 0 (I, G, Λ; M) where G is an Abelian group and M is a 0-1 matrix over G. We think that the general case needs a new approach. Problem 4. Find the free spectra of all varieties generated by finite Rees matrix semigroups. 4. Acknowledgements We are indebted to Mária Szendrei for her continuous attention and guidance during our research. The research of the authors was supported by the Hungarian National Foundation for Scientific Research, Grant T043671, K67870, N References [1] J. Berman, Free spectra gaps and tame congruence types, Int. J. Algebra Comput. 5 (1995), [2] S. Crvenković, I. Dolinka, N. Ruškuc, The Berman conjecture is true for finite surjective semigroups and their inflations, Semigroup Forum 62 (2001), [3] S. Crvenković, I. Dolinka, N. Ruškuc, Finite semigroups with few term operations, J. Pure Appl. Algebra 157 (2001), [4] S. Crvenković, N. Ruškuc, Log-linear varieties of semigroups, Algebra Universalis 33 (1995), [5] G. Grätzer, Universal algebra, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London [6] G. Higman, The order of relatively free groups, Proc. Internat Conf. Theory of Groups, Canberra 1965, Gordon and Breach Pub., 1967, pp [7] D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics no. 76, Amer. Math. Soc., Providence, [8] J. M. Howie, Fundamental of semigroup theory, London Math. Soc. Monographs, New Series 12, The Clarendon Press, Oxford, [9] P. Neumann, Some indecomposable varieties of groups, Quart. J. Math. Oxford 14 (1963), [10] K. Kátai-Urbán, Cs. Szabó, Free spectrum of the variety generated by the five element combinatorial Brandt semigroup, Semigroup Forum, 73 (2006) [11] K. Kátai-Urbán, Cs. Szabó, Free spectrum of the variety generated by the combinatorial completely 0-simple semigroups, Glasgow Math. J., 49 (2007) [12] G. Mashevitzky, Completely simple and completely 0-simple semigroup identities, Semigroup Forum 37 (1988), [13] L. Moser, M. Wyman, An asymptotic formula for the Bell-numbers, Trans. Royal Soc. Can. 49 (1955)

10 10 KAMILLA KÁTAI-URBÁN AND CSABA SZABÓ [14] G. Pluhár, J. Wood, The free spectra of varieties generated by idempotent semigroups, Algebra Discr. Math., to appear. [15] G. Pluhár, Cs. Szabó, The free spectrum of the variety of bands, Semigroup Forum, to appear. [16] N. Reilly, Varieties generated by completely 0-simple semigroups, manuscript. [17] S. Seif, Cs. Szabó, Computational complexity of checking identities in 0-simple semigroups and matrix semigroups over finite fields, Semigroup Forum 72 (2006), (1) Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary address: (2) Eötvös Loránd University, Department of Algebra and Number Theory, 1117 Budapest, Pázmány Péter sétány 1/c, Hungary address: