Generating sets of finite singular transformation semigroups

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1 Semigroup Forum DOI /s RESEARCH ARTICLE Generating sets of finite singular transformation semigroups Gonca Ayık Hayrullah Ayık Leyla Bugay Osman Kelekci Dedicated to the memory of John Mackintosh Howie ( Received: 1 August 011 / Accepted: 17 February 01 Springer Science+Business Media, LLC 01 Abstract J.M. Howie proved that Sing n, the semigroup of all singular mappings of {1,...,n} into itself, is generated by its idempotents of defect 1 (in J. London Math. Soc. 41, , He also proved that if n 3 then a minimal generating set for Sing n contains n(n 1/ transformations of defect 1 (in Gomes and Howie, Math. Proc. Camb. Philos. Soc , In this paper we find necessary and sufficient conditions for any set for transformations of defect 1 in Sing n to be a (minimal generating set for Sing n. Keywords Singular transformation semigroups Idempotents (Minimal generating set 1 Introduction The full transformation semigroup T X on a set X and the semigroup analogue of the symmetric group S X has been much studied over the last fifty years, in both the finite and the infinite cases. Among recent contributions are [1 3, 8]. Here we are concerned solely with the case where X X n {1,...,n}, and we write respectively T n and S n rather than T Xn and S Xn. Moreover, the semigroup T n \ S n of all singular self-maps of X n is denoted by Sing n. (For unexplained terms in semigroup theory, see [7]. Let S be any semigroup, and let A be any nonempty subset of S. Then the subsemigroup generated by A, that is the smallest subsemigroup of S containing A, denoted by A. It is well known that the rank of Sing n, defined by rank(sing n min { A : A Sing n }, Communicated by Steve Pride. G. Ayık ( H. Ayık L. Bugay O. Kelekci Department of Mathematics, Çukurova University, Adana, Turkey agonca@cu.edu.tr

2 G. Ayık et al. is equal to n(n 1/ forn 3(see[4]. The idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of Sing n is also equal to n(n 1/ (see [6]. Recently, it has been proved in [3] that, for r m n,the(m, r-rank, defined as the cardinality of a minimal generating set of (m, r-path-cycles, of Sing n is once again n(n 1/. The main goal of this study is to investigate whether a given subset A of transformations of defect 1 in Sing n is a (minimal generating set for Sing n. Let α T n.thedefect set, defect, kernel and Fix of α are defined by Def(α X n \ im(α def(α Def(α ker(α { (x, y X n X n : xα yα } Fix(α {x X n : xα x}, respectively. For any α, β T n, it is well known that ker(α ker(αβ, im(αβ im(β, and that (α, β D im(α im(β def(α def(β (α, β H ker(α ker(β and im(α im(β. We denote the D-Green class of all singular self maps of defect r by D n r (1 r n 1. It is clear that α D n 1 if and only if there exist i, j X n with i j such that ker(α is the equivalence relation on X n generated by {(i, j}, or equivalently, generated by {(j, i}. In this case we define the set Ker(α by Ker(α {i, j}. (Notice that ker(α denotes an equivalence relation on X n, and that Ker(α denotes a subset of X n. Equivalently, α D n 1 if and only if there exists k X n such that Def(α {k}. It is well known that α T n is an idempotent element if and only if the restriction of α to im(α is the identity map on im(α. We denote the set of all idempotents in a subset A of T n by E(A. Letα be in E(D n 1 with Def(α {i}. Then there exists unique j X n such that i j and iα j. In this case we write ( i α ζ i,j. j A digraph is called complete if, for all pairs of vertices u v in V(, either the directed edge (u, v E( or the directed edge (v, u E(. For two vertices u, v V(, if there exist a directed path from u to v then we say u is connected to v in. Wesay is strongly connected if, for any two vertices u, v V(, u is connected to v in. Let A be a subset of E(D n 1 {ζ i,j : i, j X n and i j}. Then we define the digraph A as follows:

3 Generating sets of finite singular transformation semigroups the vertex set of A, denoted by V V( A,isX n ; and the directed edge set of A, denoted by E E( A,is E { (j, i V V : ζ i,j A }. For any subset A of E(D n 1 it is shown in [6] that A is a minimal generating set for Sing n if and only if the digraph A is complete and strongly connected. Now let A be a nonempty subset of D n 1. Then we define another digraph Ɣ A as follows: the vertex set of Ɣ A, denoted by V V(Ɣ A,isA; and the directed edge set of Ɣ A, denoted by E E(Ɣ A,is E { (α, β V V : Def(α Ker(β }. The main goal of this paper is to prove that any subset A of D n 1 is a (minimal generating set for Sing n if and only if, for each idempotent ζ i,j E(D n 1, there exist α, β A such that (i Ker(α {i, j}, (ii Def(β {i}, and (iii α is connected to β in the digraph Ɣ A. Moreover, for A E(D n 1, we show that the directed graph A is complete and strongly connected if and only if, for each idempotent ζ i,j E(D n 1, there exist α, β A such that Ker(α {i, j}, Def(β {i} and that α is connected to β in the digraph Ɣ A. Therefore, the main result of this paper includes the result of [6, Theorem 1]. Preliminaries For an α D n 1 with Ker(α {i, j} and Def(α {k} we define the idempotents α e and α f, and the permutation α p on X n as follows: First we define α e as ( ( either α e i or α e j. j i In the first case we define ( α f k iα and in the second case we define ( α f k jα { k x i and xα p xα x i, and xα p { k x j xα x j.

4 Notice that in both cases and we have For example, if α ( or ( α α e α p 1 ( ( α f k k, iα jα G. Ayık et al. α α e α p α p α f. (1 then, since Ker(α {1, } and Def(α {1},wehave ( ( 1 α ( p α f 3 ( ( ( α α e α p 1 α ( p α f. 3 It is well known that each element in Sing n can be written as a product of idempotents from E(D n 1. Now we state a very easy lemma which will be useful throughout this paper. Lemma 1 For i j,k we have ( i ( i ( j k i j. Lemma For n 3 let α, β D n 1. Then αβ D n 1 if and only if Def(α Ker(β. Proof With the notation in (1, since αβ (α p α f (β e β p, it follows that αβ D n 1 if and only if α f β e D n 1. ( Suppose that αβ D n 1, and that Def(α {i} and Ker(β {k,l}. Then or β e ( l k, and there exists i j Xn such that α f ( i j either β e ( k l cases i must be equal to either k or l since α f β e D n 1.. In both ( Suppose that Def(α {i} Ker(β. Then there exists i k X n such that Ker(β {i, k}. Since we can take β e ( i k, it follows from Lemma 1 and (1 that αβ ( α p α f ( β e β p α p( α f β e β p ( α p α f β p ( α e α p β p α e( α p β p. Since α e E(D n 1 and α p β p S n, it follows that αβ α e (α p β p D n 1,as required. Denote the H-Green class containing the idempotent ( i j in Dn 1 by H i,j. Then it is clear that α H i,j if and only if Def(α {i} and Ker(α {i, j}. Lemma 3 Let α D n 1. Then α H i,j if and only if there exists a permutation β S n such that α ( i j β and i Fix(β.

5 Generating sets of finite singular transformation semigroups Proof ( Suppose that α H i,j. Since Def(α {i} and Ker(α {i, j}, we can take α e ( i j. Moreover, it follows from the definition of α p that i Fix(α p. Thus, from (1, we have α α e α p, as required. ( Suppose that there exists a permutation β S n such that α ( i j β and i Fix(β. Since β S n and i Fix(β, it is clear that (( i im(α im X j n \{i}, and that Ker(α {i, j}. Hence, α H i,j, as required. 3 Generating Set Theorem 4 Let A be a subset of the D-Green class D n 1. Then A is a generating set of Sing n if and only if, for each idempotent ( i j E(Dn 1, there exist α, β A such that (i Ker(α {i, j}, (ii Def(β {i}, and (iii α is connected to β in the digraph Ɣ A. Proof ( Suppose that A is a generating set for Sing n. Then, for each ζ i,j ( i j E(D n 1, there exist α 1,...,α r A such that α 1 α r ζ i,j. Since α 1,...,α r,ζ i,j D n 1,ker(α 1 ker(ζ i,j and im(ζ i,j im(α r, it follows that Ker(α 1 Ker(ζ i,j {i, j}, Def(α r Def(ζ i,j X n \ im(ζ i,j {i} and that, for each 1 k r 1, we have α k α k+1 D n 1. Then, from Lemma, we have Def(α k Ker(α k+1. Hence there exists a directed path from α 1 to α r, that is α 1 is connected to α r in Ɣ A, as required. ( Since each element in Sing n can be written as a product of idempotents in D n 1, it is enough to show that E(D n 1 A. Let ζ i,j E(D n 1. From (i and (ii there exist α, β A such that Ker(α {i, j} and Def(β {i}. Then it follows from (iii that there exists a directed path from α to β, say such that, for each 1 t r 1, α α 1 α α r 1 α r β Def(α t Ker(α t+1.

6 G. Ayık et al. Since Def(α t Ker(α t+1 for each 1 t r 1, it follows from Lemma inductively that α 1 α r D n 1, and so Ker(α Ker(α 1 α r and im(β im(α 1 α r. Therefore, α 1 α r H i,j. Since H i,j is a finite group with the identity ζ i,j, it follows that there exists a positive integer m such that ζ i,j (α 1 α r m A, as required. Since rank(sing n n(n 1, a generating set of Sing n with n(n 1 elements is a minimal generating set. Thus we have the following corollary from Theorem 4: Corollary 5 Let A be a subset of D n 1 with n(n 1 elements. Then A is a minimal generating set of Sing n if and only if, for each idempotent ( i j E(Dn 1, there exist α, β A such that (i Ker(α {i, j}, (ii Def(β {i}, and (iii α is connected to β in the digraph Ɣ A. 4 Remarks From Theorem 4 and [6] we have the following result with the notation above: Corollary 6 Let A be a subset of E(D n 1 with n(n 1 elements. Then the directed graph A is complete and strongly connected if and only if, for each idempotent ( i j E(Dn 1, there exist α, β A such that (i Ker(α {i, j}, (ii Def(β {i}, and (iii α is connected to β in the digraph Ɣ A. In other words, the main result of J.M. Howie in [6] coincides with our Theorem 4 when A is a subset of E(D n 1 with n(n 1 elements. For example, let A {α 1,α,α 3,α 4,α 5,α 6 } where α 1 α 3 α 5 (, α (, α (, α (, (, ( With the notation above, for ζ 1, E(D 4 1, we have Ker(α 1 {1, } and Def(α 3 {1}. Similarly, for all other elements of E(D 4 1, it can be shown that conditions (i and (ii of Theorem 4 are satisfied.

7 Generating sets of finite singular transformation semigroups Moreover, Ɣ A : is a Hamiltonian digraph since the cycle α 5 α 3 α α 4 α 6 α 1 α 5 is a Hamiltonian cycle. Hence, Ɣ A is strongly connected, and so condition (iii of Theorem 4 is satisfied. Therefore, A is a (minimal generating set of Sing 4. Indeed, since ζ 1, (α 1 α 3 3, ζ 1,3 α α 6 α 5 α 3, ζ 1,4 α3 3 ζ,1 α 1 α 6, ζ,3 (α 4 α 6, ζ,4 (α 5 α 6 ζ 3,1 α 3, ζ 3, α 4, ζ 3,4 (α 6 α 4 ζ 4,1 (α 3 α 1 3, ζ 4, α 3 5, ζ 4,3 (α 6 α 5, it follows that E(D 4 1 A, and so A is a (minimal generating set for Sing 4. Notice that, to write each ζ i,j E(D 4 1 as a product of elements of A above, we use the paths in Ɣ A. Therefore, Ɣ A is also useful for writing the idempotents of defect 1 as a product of elements of A. Acknowledgements The authors would like to thank Nesin Mathematics Village (Şirince-Izmir, Turkey and its supporters for providing a peaceful environment where some parts of this research was carried out. References 1. André, J.M.: Semigroups that contain all singular transformations. Semigroup Forum 68, (004. Ayık, G., Ayık, H., Howie, J.M.: On factorisations and generators in transformation semigroup. Semigroup Forum 70, 5 37 ( Ayık, G., Ayık, H., Howie, J.M., Ünlü, Y.: Rank properties of the semigroup of singular transformations on a finite set. Commun. Algebra 36(7, ( Gomes, G.M.S., Howie, J.M.: On the ranks of certain finite semigroups of transformations. Math. Proc. Camb. Philos. Soc. 101, (1987

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