Turkish Journal of Mathematics Turk J Math (2019) 43: 2218 – 2225 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1902-23 Relative ranks of some partial transformation semigroups Ebru YİĞİT1, Gonca AYIK2;,∗ Hayrullah AYIK2 1Department of Informatics, Faculty of Engineering, Yalova University, Yalova, Turkey 2Department of Mathematics, Faculty of Arts and Sciences, Çukurova University, Adana, Turkey Received: 05.02.2019 • Accepted/Published Online: 16.07.2019 • Final Version: 28.09.2019 Abstract: Let Pn , Tn , In , and Sn be the partial transformation semigroup, the (full) transformation semigroup, the symmetric inverse semigroup, and the symmetric group on Xn = f1; : : : ; ng, respectively. For 1 ≤ r ≤ n − 1, let PKn;r be the subsemigroup consisting α 2 Pn such that jimαj ≤ r and let SP Kn;r = PKn;r n Tn . In this paper, we first examine the subsemigroup In;r = In [ PKn;r and we find the necessary and sufficient conditions for any nonempty subset of PKn;r to be a (minimal) relative generating set of the subsemigroup In;r modulo In . Then we examine the subsemigroups PIn;r = SIn [ PKn;r and SIn;r = SIn [ SP Kn;r for 1 ≤ r ≤ n − 1 where SIn = In n Sn and compute their relative rank. Key words: (Partial) transformation semigroup, symmetric inverse semigroup, symmetric group, (minimal) generating set, relative rank 1. Introduction The partial transformation semigroup PX , the (full) transformation semigroup TX and the symmetric inverse semigroup IX on a set X have been extensively studied over the last sixty years, both in the finite and in the infinite cases. Among recent contributions are[1–6, 13, 16]. Here we are concerned solely with the case where f g X = Xn = 1; : : : ; n , and we denote the semigroups PXn , TXn , and IXn , by Pn , Tn , and In , respectively. Moreover, we denote the subsemigroup In n Sn by SIn where Sn is the symmetric group on Xn . It is well known that In is an inverse semigroup and every finite inverse semigroup S is embeddable in In , the analog of Cayley’s theorem for finite groups. Hence, as emphasized in[1], the importance of In to inverse semigroup theory is similar to that of the symmetric group Sn to group theory. Moreover, Gomes and Howie remarked in [11] that very little has been written on the symmetric inverse semigroups. Despite the appearance of the books of Lipscomb [18], and Ganyushkin and Mazorchuk [8], as well as a handful of papers (for example, [10]), the study of In is still in its infancy compared to that of Tn . 2 An element α of Pn is called an idempotent if α = α. We denote the set of all idempotents in any subset U of any semigroup by E(U). Let S be a semigroup and let A be a nonempty subset of S . Then the subsemigroup generated by A, that is the smallest subsemigroup of S containing A, is denoted by hAi. If a semigroup S has a finite subset A such that S = hAi, then S is called a finitely generated semigroup. The rank of a finitely generated semigroup S is defined by rank(S) = minf jAj : hAi = S g. For a fixed subset G of ∗Correspondence: [email protected] 2010 AMS Mathematics Subject Classification: 20M20 2218 This work is licensed under a Creative Commons Attribution 4.0 International License. YİĞİT et al./Turk J Math a semigroup S , if there exists a subset A of S such that hA [ Gi = S , then A is called a relative generating set of S modulo G. Then the relative rank of a finitely generated semigroup S modulo G is defined by rerank(S : G) = minfjAj : hA [ Gi = Sg: For 1 ≤ r ≤ n, let Kn;r = fα 2 Tn : jim(α)j ≤ rg; Tn;r = Sn [ Kn;r; PKn;r = fα 2 Pn : jim(α)j ≤ rg; PTn;r = Sn [ PKn;r; SP Kn;r = PKn;r n Tn = PKn;r n Kn;r;An;r = An [ Kn;r; PAn;r = An [ PKn;r; In;r = In [ PKn;r; SIn;r = SIn [ SP Kn;r and PIn;r = SIn [ PKn;r; where An denotes the alternating group on Xn . Howie and McFadden proved in [15] that the rank of Kn;r is S(n; r), the Stirling number of the second kind, for 2 ≤ r ≤ n − 1. Recall that the Stirling number S(n; r) of the second kind is defined by S(n; 1) = S(n; n) = 1 and S(n; r) = S(n − 1; r − 1) + r · S(n − 1; r) for 2 ≤ r ≤ n−1. Moreover, Garba proved in [9] that the rank of the subsemigroup PKn;r of Pn is S(n+1; r+1) for 2 ≤ r ≤ n − 1. + For n; r 2 Z with r ≤ n, let Pr(n) be the set of all integer solutions of the equation x1 + x2 + ··· + xr = n with x1 ≥ x2 ≥ · · · ≥ xr ≥ 1; and let pr(n) = jPr(n)j. If an r-tuple (n1; n2; : : : ; nr) is a solution of the equation given above, then it is called a partition of n with r terms (see [12]). Ayık et al. developed a notation for certain primitive elements of Tn , called path-cycle, and described an algorithm to decompose an arbitrary transformation α in Tn into a product of path-cycles in [2]. In addition, they used these techniques to obtain some informations about generators of Tn , and proved that, rerank(Tn;r : Sn) = pr(n) for 1 ≤ r ≤ n − 1 (see also [17, Theorem 8]). In [19], we obtained the necessary and sufficient conditions for any nonempty subset U of Kn;r (PKn;r) to be a (minimal) relative generating set of Tn;r (PTn;r) modulo Sn for 1 ≤ r ≤ n − 1. Then we concluded the same result in [2, 17] that rerank(Tn;r : Sn) = pr(n) and, we obtained the new result nX−r rerank(PTn;r : Sn) = pr(n − s) s=0 for 1 ≤ r ≤ n − 1. Moreover, we showed that nX−r rerank(An;r : An) = pr(n) and rerank(PAn;r : An) = pr(n − s) s=0 for each 1 ≤ r ≤ n − 1. In this paper, we first find the necessary and sufficient conditions for any nonempty subset U of PKn;r to be a (minimal) relative generating set of In;r (respectively PIn;r ) modulo In (respectively SIn ) 2219 YİĞİT et al./Turk J Math for 1 ≤ r ≤ n − 1. Moreover, we find the necessary and sufficient conditions for any subset U of SP Kn;r to be a (minimal) relative generating set of SIn;r modulo SIn . Then we conclude that rerank(In;r : In) = pr(n); rerank(SIn;r : SIn) = pr(n − 1) and rerank(PIn;r : SIn) = S(n; r) for 1 ≤ r ≤ n − 1. 2. Preliminaries The height and the kernel of any partial transformation α 2 Pn are defined by h(α) = jim(α)j and ker(α) = f(x; y) 2 Xn × Xn : either x; y 2 dom(α) and xα = yα or x; y2 = dom(α)g; respectively. For any α; β 2 Pn (also α; β 2 Tn ), recall that dom(αβ) ⊆ dom(α), im(αβ) ⊆ im(β), ker(α) ⊆ ker(αβ), and that (α; β) 2 D , h(α) = h(β) (α; β) 2 H , im(α) = im(β); ker(α) = ker(β) and dom(α) = dom(β) where the equivalences D and H denote Green’s relations (see, for examples [14] and [8, Theorem 4.5.1]). For ≤ ≤ D T 1 r n, we denote that Green’s -class, consists of all elements in Tn (respectively Pn ) of height r, by Dr P (respectively Dr ). If the implied semigroup is clear from the context, we will use the simpler notation Dr . D T It is shown in [7] that Green’s -class Dr is generated by its idempotents, and so it follows from [15, h T i ≤ ≤ − D Lemma 4] that Kn;r = E(Dr ) for 2 r n 1. It is also shown in [9] that a subset A of Green’s -class P P ⊆ h i ≤ ≤ − Dr is a generating set of PKn;r if and only if E(Dr ) A for 2 r n 1. Therefore, to show a subset P P ⊆ h i A of Dr is a generating set of PKn;r , it is enough to prove Dr A . For a given nonempty set X and a positive integer r where 1 ≤ r ≤ jXj, let A1;:::;Ar be a collection of nonempty disjoint subsets of X . Then ξ = fA ;:::;A g is called a partition of X (with r terms) if S 1 r r f g j j ≥ · · · ≥ j j X = i=1 Ai . A partition A1;:::;Ar of X is called an ordered partition if A1 Ar , and is P denoted by (A1;:::;Ar). For 1 ≤ r ≤ n, it is clear that α 2 D if and only if there exists a unique partition S r f g r+1 × n A1;:::;Ar of dom(α) such that ker(α) = i=1 (Ai Ai) where Ar+1 = Xn dom(α) = cdom(α); or equivalently, there exists a unique subset fa1; : : : ; arg of Xn with cardinality r, such that im(α) = fa1; : : : ; arg. Without loss of generality suppose that Aiα = ai for each 1 ≤ i ≤ r, and so α can be written in the following tabular form: ( )( ( ) ) ··· ··· A1 Ar Ar+1 A1 Ar 2 T α = and α = if α Dr : a1 ··· ar − a1 ··· ar 2 T 2 For α Dr , written in the above tabular form, there clearly exists a permutation σ Sr such that (jA1σj;:::; jArσj) is a partition of n with r terms.