Generating Sets in Some Semigroups of Order- Preserving Transformations on a Finite Set

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1 Southeast Asian Bulletin of Mathematics (2014) 38: Southeast Asian Bulletin of Mathematics c SEAMS Generating Sets in Some Semigroups of Order- Preserving Transformations on a Finite Set Hayrullah Ayık and Leyla Bugay Department of Mathematics, Çukurova University, Adana, Turkey hayik@cu.edu.tr; ltanguler@cu.edu.tr Received 10 March 2013 Accepted 5 July 2013 Communicated by Victoria Gould AMS Mathematics Subject Classification(2000): 20M20 Abstract. Let O n denote the order-preserving transformation semigroup on X n = {1,...,n}, under its natural order. For each 1 r n 1, O(n, = {α O n : im(α) r} is a subsemigroup of O n. In this paper we find necessary and sufficient conditions for any subset of O(n, to be a (minimal) generating set of O(n,, for 2 r n 1. Moreover, similar questions related to partial transformations will also be considered. Keywords: (Partial) Order-preserving transformation semigroup; Idempotents; (Minimal) Generating set; Rank. 1. Introduction The partial transformation semigroup P X and the full transformationsemigroup T X on a set X, the semigroups analogue of the symmetric group S X, have been much studied over the last fifty years, for both finite and infinite X. Here we are concerned solely with the case where X = X n = {1,...,n}, and we write respectively P n, T n and S n rather than P Xn, T Xn and S Xn. Among recent contributions are [1, 2, 6, 12, 13]. The domain, image, height and kernel of α P n are defined by dom(α) = {x X n : there exists y X n such that xα = y}, im(α) = {y X n : there exists x X n such that xα = y},

2 164 H. Ayık and L. Bugay h(α) = im(α), ker(α) = {(x,y) X n X n : (x,y dom(α) and xα = yα) or (x,y / dom(α))} respectively. Notice that ker(α) is an equivalence relation on X n and the equivalence classes of ker(α) are all of the pre-image sets of elements in im(α) together with X n dom(α). Then, the set kp(α) = {yα 1 : y im(α)} is called the kernel partition of α and the ordered pair ks(α) = ( kp(α),x n dom(α) ) is called the kernel structure of α. For any α,β P n notice that ks(α) = ks(β) kp(α) = kp(β) ker(α) = ker(β) and dom(α) = dom(β). Moreover, for any α,β in P n it is well known that dom(αβ) dom(α), ker(α) ker(αβ), im(αβ) im(β)andthatα P n isanidempotentifandonlyifxα = x for all x im(α). We denote the set of all idempotents in any subset U of any semigroup by E(U). (See [3, 8] for other terms in semigroup theory which are not explained here.) The singular transformation semigroup Sing n, the order-preserving transformation semigroup O n and the partial order-preserving transformation semigroup PO n on X n, under its natural order, are defined by Sing n = T n S n, O n = {α Sing n : x y xα yα ( x,y X n )}, and PO n = O n {α P n T n : x y xα yα ( x,y dom(α))} respectively. For 1 r n 1, O(n, = {α O n : im(α) r} and PO(n, = {α PO n : im(α) r} are (under usual composition) subsemigroups of O n and PO n, respectively. Notice that O(n,n 1) = O n and PO(n,n 1) = PO n. Let A = {A 1,...,A k } be a partition of a set Y X n. Then A is called an ordered partition, and we write A = (A 1,...,A k ), if x < y for all x A i and y A i+1 (1 i k 1) (the idea of ordering a family of sets appeared on p335 of [10]). Moreover, a set {a 1,...,a k }, such that {a 1,...,a k } A i = 1 for each 1 i k, is called a transversal (or a cross-section) of A. Recall that kernel classes of α O n are convex subsets C of X n in the sense that x,y C and x z y z C (see [4, p187]). A partition {A 1,...,A t } of X n for 1 t n is called a convex partition of X n if, for each 1 i t, A i is a convex subset of X n. Thus, for α O n with height k (1 k n 1), there exist x 1,...,x k 1 X n such that the kernel classes of α are A 1 = {1,...,x 1 }, A 2 = {x 1 + 1,...,x 2 },...,A k =

3 Generating Sets in Some Semigroups 165 {x k 1 + 1,...,n} and A 1 α < < A k α, and hence, (A 1,...,A k ) is a convex ordered partition of X n. Moreover, α can be written in the following tabular form: ( ) A1 A α = 2 A k. A 1 α A 2 α A k α For α PO n with height k, although it is easy to see that the kernel classes of α do not have to be convex subsets of X n, we have the order on the kernel classes A 1,...,A k of kp(α) as defined above, that is, (A 1,...,A k ) is an ordered partition of dom(α). Also we have the order: A 1 α < < A k α. Moreover, ker(α) = k+1 i=1 (A i A i ) where A k+1 = X n dom(α) and α can be written in the following tabular form: α = ( A1 A 2 A k A k+1 A 1 α A 2 α A k α ). (1) ThenitisclearthatαisanidempotentifandonlyifA i α A i foreach1 i k. For any α,β in O(n, (PO(n,), it is easy to show by using the definitions of the Green s equivalences that (α,β) L im(α) = im(β), (α,β) D h(α) = h(β) (α,β) R ks(α) = ks(β), and (α,β) H α = β (see for the definitions of the Green s equivalences [8, pages 45-47]). For each r such that 1 r n 1, we denote Green s D-class of all elements in O(n, (PO(n,) of height k by D k, for 1 k r. Let Π = (V(Π), E(Π)) be a digraph. For two vertices u,v V(Π) we say u is connected to v in Π if there exists a directed path from u to v, that is, either (u,v) E(Π) or (u,w 1 ),...,(w i,w i+1 ),...,(w n,v) E(Π) for some w 1,...,w n V(Π). Moreover, we say Π is strongly connected if, for any two vertices u,v V(Π), u is connected to v in Π. Let X be a non-empty subset of Green s D-class D r of O(n, (PO(n,) for 2 r n 1. Then we define a digraph Γ X as follows: (i) the vertex set of Γ X, denoted by V = V(Γ X ), is X; and (ii) the directed edge set of Γ X, denoted by E = E(Γ X ), is E = {(α,β) V V : αβ Dr }. Let S be any semigroup, and let A be any non-empty subset of S. Then the subsemigroup generated by A, that is, the smallest subsemigroup of S containing A, is denoted by A. The rank of a finitely generated semigroup S, a semigroup generated by finitely many elements, is defined by rank(s) = min{ A : A = S}. It is shown that O n is generated by its idempotents of height n 1 by Howie in [7], and that its rank is n by Gomes and Howie in [5]. Gomes and Howie also

4 166 H. Ayık and L. Bugay proved that the semigroup PO n is generated by its idempotents of height n 1 and its rank is 2n 1. Moreover, for 2 r n 2, Garba proved in [4] that the subsemigroups O(n, and PO(n, are generated by their idempotents of ( n k)( k 1, respectively. height r, and their ranks are ( n n and k=r The main goal of this paper is to find necessary and sufficient conditions for any subset of O(n, (PO(n,) to be a (minimal) generating set of O(n, (PO(n,) for 2 r n Generating Sets of O(n, Since α D k (1 k can not be written as a product of elements with height smallerthan k, and since O(n, is generatedby its idempotents ofheight r (see [7, 4]), it is enough to consider only the subsets of D r as a generating set of O(n, for 2 r n 1. Moreover, a subset X of D r is a generating set of O(n, if and only if E(D r ) X. First we state a useful proposition which can be proved easily. Proposition 2.1. For n,k 2 and 1 r n 1, let α,β,α 1,...,α k D r in O(n, (or PO(n,). Then, (i) αβ D r if and only if im(αβ) = im(β). In other words, im(α) is a transversal of the kernel partition kp(β) of β. (ii) α 1 α k D r if and only if α i α i+1 D r for each 1 i k 1. Theorem 2.2. Let X be a subset of Green s D-class D r in O(n, for 2 r n 1. Then X is a generating set of O(n, if and only if, for each idempotent ξ in D r (or equivalently, for each convex partition A = {A 1,...,A r } of X n, and for each transversal R(A) of A), there exist α,β X such that (i) ker(α) = ker(ξ) (or equivalently kp(α) = A), (ii) im(β) = im(ξ) (or equivalently im(β) = R(A)), and (iii) α is connected to β in the digraph Γ X. Proof. (Sufficiency). Suppose that X is a generating set of O(n, and let ξ be any idempotent in D r. Since X is a generating set of O(n,, there exist α 1,...,α k X such that α 1 α k = ξ. Then, since ker(α 1 ) ker(ξ), im(ξ) im(α k ) and h(α 1 ) = h(α k ) = h(ξ) = r, it follows that ker(α 1 ) = ker(ξ) and im(α k ) = im(ξ). Therefore, the conditions (i) and (ii) hold. Moreover, it follows from Proposition 2.1 (ii) that α i α i+1 D r for each 1 i k 1, and so there exists a directed edge from α i to α i+1 in Γ X for each 1 i k 1. Thus, α 1 is connected to α k in the digraph Γ X. Therefore, the last condition holds as well.

5 Generating Sets in Some Semigroups 167 (Necessity). For any idempotent ξ D r, there exist α,β X such that ker(α) = ker(ξ), im(β) = im(ξ) and α is connected to β in the digraph Γ X. Thus there exists a directed path from α to β, say α = α 1 α 2 α k 1 α k = β, and so α i α i+1 D r for each 1 i k 1. It follows from Proposition 2.1 (ii) that γ = α 1 α k D r. Similarly, we have im(γ) = im(α k ) = im(β) = im(ξ) and ker(γ) = ker(α 1 ) = ker(α) = ker(ξ). Thus both ξ and γ are in the same Green s H-class, that is, ξ = γ X, and so E(D r ) X. Now the result follows from the fact that any element in O(n, can be written as a product of idempotents in D r. Since rank(o n ) = n for n 2 and rank(o(n,) = ( n for 2 r n 2, we have the following corollary: Corollary 2.3. Let X be a subset of Green s D-class D r with cardinality ( n for 2 r n 1. Then X is a minimal generating set of O(n, if and only if, for each idempotent ξ E(D r ), there exist α,β X such that ker(α) = ker(ξ), im(β) = im(ξ) and that α is connected to β in Γ X. For example, consider O(4,2) and X = {α 1,...,α 6 } D 2 where ( ) ( ) ( ) α 1 =, α =, α = ( ) ( ) ( ) α 4 =, α =, α = Notice that ( 4 2) = 6. First of all it is easy to see that X satisfies the first two conditions of Theorem 2.2. Moreover, since im(α i ) is a transversal of the kernel partition kp(α i+1 ), there exists a directed edge from α i to α i+1 for each 1 i 6 (where α 7 = α 1 ) in the digraph Γ X. Thus Γ X is a Hamiltonian digraph. It follows from their definitions that Hamiltonian digraphs are strongly connected (see [11, pages88and 148]), and sofrom Corollary2.3, X is a minimal generating set of O(4, 2). Notice that if X D r is a (minimal) generating set of O(n, then we may use the paths in Γ X to write each idempotent in D r as a product of elements from X. For example, consider the minimal generating set X = {α 1,...,α 6 } of O(4, 2) given above and consider the idempotent ( ) ξ = in D 2. Since ker(ξ) = ker(α 1 ), im(ξ) = im(α 3 ) and α 1 α 2 α 3

6 168 H. Ayık and L. Bugay is a path in Γ X, it follows that ker(α 1 α 2 α 3 ) = ker(ξ) and im(α 1 α 2 α 3 ) = im(ξ), that is, α 1 α 2 α 3 = ξ. We now state the following two lemmas which are useful for determining all minimal generating sets of O(n, for 2 r n 1 easily. Lemma 2.4. Let P 1,...,P m, with ( n 1 = m, be all of the convex partitions of X n into r subsets for 2 r n 1. Then there exist m different subsets I 1,...,I m of X n with cardinality r such that I i is a transversal of P i for each 1 i m. Proof. We shalldenote the claim byp(n, and provethis by adouble induction on n and r by using a kind of Pascal striangleimplication as in [9] and [4, p190]: P(n 1, and P(n 1, P(n, for 2 r n 1. Claim 1: P(n,n 1) holds for n 3. All of the convex partitions of X n into n 1 subsets are P 1 = {{1,2},{3},...,{n}}, P 2 = {{1},{2,3},...,{n}},.. P n 1 = {{1},...,{n 2},{n 1,n}}. Then the subsets I 1,I 2,...,I n 1 of X n defined by I i = X n {i} for each 1 i n 1 satisfy the claim. Claim 2: P(n,2) holds for n 3. All of the convex partitions of X n into 2 subsets are P i = {{1,...,i},{i+1,...,n}} 1 i n 1. Then the subsets I 1,I 2,...,I n 1 of X n defined by I i = {i,n} for each 1 i n 1 satisfy the claim. Claim 3: P(n 1, and P(n 1, P(n, for 5 n and 3 r n 2. Suppose that P(n 1,r 1) and P(n 1, are satisfied for n 5 and 3 r n 2. Let t = ( ) ( n 2 r 1 and k = n 2 r 2). Since P(n 1, is satisfied, for all of the convex partitions U 1,...,U t of X n 1 into r subsets, there exist t different subsets I 1,...,I t ofx n 1 with cardinalityr such that I i is a transversal of U i = {A i1,...,a ir } for each 1 i t. Since P(n 1,r 1) is satisfied, for all of the convex partitions V 1,...,V k of X n 1 into r 1 subsets, there exist k different subsets J 1,...,J k of X n 1 with cardinality r 1 such that J j is a transversal of V j = {B j1,...,b jr 1 } for each 1 j k. For each 1 i t let P i = {A i1,...,a ir {n}},

7 Generating Sets in Some Semigroups 169 and for each 1 j k let P t+j = {B j1,...,b jr 1,{n}}. Then it is clear that all of the convex partitions of X n into r subsets are P 1,...,P t,p t+1,...,p t+k (notice that t+k = ( ) ( n 2 r 1 + n 2 ) ( r 2 = n 1 ). Then the subsets T1,...,T t+k of X n, defined as { Ii for 1 i t, T i = J i t {n} for t+1 i t+k, satisfy the conditions, and so the proof is complete. Lemma 2.5. Let P 1,...,P m, with ( n 1 = m, be all of the convex partitions of X n into r subsets for 2 r n 1. Then there exist α 1,...,α m D r in O(n, such that (i) kp(α 1 ) = P m and kp(α i ) = P i 1, for each 1 < i m; (ii) im(α i ) = im(α j ) if 1 i = j m; and (iii) im(α i ) is a transversal of kp(α i+1 ) = P i for each 1 i m 1 and im(α m ) is a transversal of kp(α 1 ) = P m. Proof. Let P 1,...,P m, with ( n 1 = m, be all of the convex partitions of Xn into r subsets for 2 r n 1. Then it follows from Lemma 2.4 that, for each 1 i m, there exists a subset I i of X n with cardinality r such that I i is a transversal of P i and I i = I j if 1 i = j m. Without loss of generality suppose that P i = {A i1,...,a ir } where (A i1,...,a ir ) is an ordered partition for 1 i m; and that I 1 = {a m1,...,a mr } where a m1 <... < a mr and I i+1 = {a i1,...,a ir } where a i1 <... < a ir for 1 i m 1. Now consider ( ) Am1 A α 1 = mr a m1 a mr and ( ) Ai1 A α i+1 = ir a i1 a ir for each 1 i m 1. Then it is clear that α 1,...,α m D r satisfy the required conditions. Although Lemmas 2.4 and 2.5 are similar to the proposition on pp. 189 and Lemma 2.3 given in [4], we state and prove these lemmas since the transformations α 1,...,α m D r defined in Lemma 2.5 do not need to be idempotent, the convex partitions P 1,...,P m of X n into r subsets given in Lemma 2.5 can be in any order, and im(α i ) does not need to be a transversal of kp(α i ) for each

8 170 H. Ayık and L. Bugay 1 i m unlike in [4]. Thus we are able to determine all minimal generating sets of O(n, for 2 r n 1 easily by using Lemmas 2.4 and 2.5. Indeed, let P 1,...,P m be all ofthe convexpartitions ofx n into r subsets, and let L 1,...,L t be all Green s L-classes in D r where ( ( n 1 = m and n = t. It follows from Lemma 2.5 that there exist α 1,...,α m D r such that (i) kp(α 1 ) = P m and kp(α i ) = P i 1, for each 1 < i m; (ii) im(α i ) = im(α j ) if 1 i = j m; and (iii) im(α i ) is a transversal of kp(α i+1 ) = P i for each 1 i m 1 and im(α m ) is a transversal of kp(α 1 ) = P m. Without loss of generality we may assume that α i L i for 1 i m. If we take an arbitrary element α m+j from L m+j for each 1 j t m, then it follows from Corollary 2.3 that X = {α 1,...,α m,α m+1,...α t } is a (minimal) generating set of O(n,. 3. Generating Sets of PO(n, As in the previous section, it is enough to consider only the subsets of D r as a generating set of PO(n,, and moreover, a subset X of D r is a generating set of PO(n, if and only if E(D r ) X for 2 r n 1. Theorem 3.1. Let X be a subset of Green s D-class D r in PO(n, for 2 r n 1. Then X is a generating set of PO(n, if and only if, for each idempotent ξ in D r, there exist α,β X such that (i) ks(α) = ks(ξ), (ii) im(β) = im(ξ), and (iii) α is connected to β in the digraph Γ X. Proof. The proof is similar to the proof of Theorem 2.2 by considering the fact that dom(αβ) dom(α) for each α,β P n. As mentioned before, rank(po n ) = 2n 1 for n 2 and rank(po(n,) = n ( n k 1 n ( k=r k)( for 2 r n 2. Since n k 1 k=r k)( = 2n 1 for r = n 1, we have the following corollary: Corollary 3.2. For 2 r n 1 let X be a subset of Green s D-class D r with cardinality n ( n k 1 k=r k)(. Then X is a minimal generating set of PO(n, if and only if, for each idempotent ξ E(D r ), there exist α,β X such that ks(α) = ks(ξ), im(β) = im(ξ) and α is connected to β in Γ X.

9 Generating Sets in Some Semigroups 171 Similarly we state and prove the following two lemmas which are useful for determining all minimal generating sets of PO(n, for 2 r n 1 easily. Lemma 3.3. Let I 1,...,I m, with ( n = m, be all of the subsets of Xn with cardinality r, for 2 r n 1. Then, for each 1 i m, there exists a subset A ir+1 of X n and an ordered partition P i = (A i1,...,a ir ) of X n A ir+1 such that I i is a transversal of P i, and P i = P j if 1 i = j m. Proof. Let I 1,...,I m, with ( n = m, be all of the subsets of Xn with cardinality r, for 2 r n 1. Suppose that I i = {a i1,...,a ir } with a i1 <... < a ir for each 1 i m. Then the sets A ir+1 = X n I i and P i = {{a i1 },...,{a ir }} for 1 i m satisfy the required conditions, and the proof is complete. Lemma 3.4. Let I 1,...,I m, with ( n = m, be all of the subsets of Xn with cardinality r, for 2 r n 1. Then there exist α 1,...,α m D r such that (i) im(α i ) = I i for each 1 i m; (ii) ks(α i ) = ks(α j ) if 1 i = j m; and (iii) I i is a transversal of kp(α i+1 ) for each 1 i m 1, and I m is a transversal of kp(α 1 ). Proof. Let I 1,...,I m, with ( n = m, be all of the subsets of Xn with cardinality r, for 2 r n 1. Then it follows from Lemma 3.3 that, for each 1 i m, there exists a subset A ir+1 of X n and an ordered partition P i = (A i1,...,a ir ) of X n A ir+1 such that I i is a transversal of P i and that P i = P j if 1 i = j m. Without loss of generality suppose that I 1 = {a m1,...,a mr } where a m1 <... < a mr and that I i+1 = {a i1,...,a ir } where a i1 <... < a ir for 1 i m 1. Now consider ( ) Am1 A α 1 = mr A mr+1 a m1 a mr and ( Ai1 A α i+1 = ir A ir+1 a i1 a ir in the tabular form (as defined in (1)) for each 1 i m 1. Then it is clear that α 1,...,α m D r satisfy the required conditions. Although Lemmas 3.3 and 3.4 are similar to the proposition on pp. 195 and Lemma 3.3 given in [4], we state and prove these lemmas due to similar reasons stated for O(n,. Analogously, we are able to determine all minimal generating sets of PO(n, for 2 r n 1 easily by using Lemmas 3.3 and 3.4. Indeed, let I 1,...,I m be all of the subsets of X n with cardinality r, and let R 1,...,R t be all Green s R-classes in D r where ( n = m and n ( n k 1 k=r k)( = t. It follows from Lemma 3.4 that there exist α 1,...,α m D r such that ),

10 172 H. Ayık and L. Bugay (i) im(α i ) = I i for each 1 i m; (ii) ks(α i ) = ks(α j ) if 1 i = j m; and (iii) I i is a transversal of kp(α i+1 ) for each 1 i m 1, and I m is a transversal of kp(α 1 ). Without loss of generality we may suppose that α i R i for 1 i m. If we take an arbitrary element α m+j from R m+j for each 1 j t m, then it follows from Corollary 3.2 that X = {α 1,...,α m,α m+1,...α t } is a (minimal) generating set of PO(n,. References [1] G. Ayık, H. Ayık, L. Bugay, O. Kelekci, Generating sets of finite singular transformation semigroups, Semigroup Forum 86 (2013) [2] G. Ayık, H. Ayık, M. Koç, Combinatorial results for order-preserving and orderdecreasing transformations, Turkish J. Math. 35 (2011) [3] O. Ganyushkin, V. Mazorchuk, Classical Finite Transformation Semigroups, Springer-Verlag, [4] G.U. Garba, On the idempotent ranks of certain semigroups of order-preserving transformations, Portugal. Math. 51 (1994) [5] G.M.S. Gomes, J.M. Howie, On the ranks of certain semigroups of orderpreserving transformations, Semigroup Forum 45 (1992) [6] P.M. Higgins, The product of the idempotents and an H-class of the finite full transformation semigroup, Semigroup Forum 84 (2012) [7] J.M. Howie, Products of idempotents in certain semigroups of transformations, Proc. Edinburgh Math. Soc. 17 (1971) [8] J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, [9] J.M. Howie, R.B. McFadden, Idempotent rank in finite full transformation semigroups, Proc. Royal Soc. Edinburgh 114A (1990) [10] S. Sornsanam, R.P. Sullivan, Regularity conditions on order-preserving transformation semigroups, Southeast Asian Bull. Math. 28 (2004) [11] R.J. Wilson, J.J. Watkins, Graphs: An Introductory Approach, Wiley, [12] H. Yang, X. Yang, Automorphisms of partition order-decreasing transformation monoids, Semigroup Forum 85 (2012) [13] P. Zhao, B. Xu, M. Yang, A note on maximal properties of some subsemigroups of finite order-preserving transformation semigroups, Comm. Algebra 40 (2012)