Group Closures of Injective Order-Preserving Transformations

Transcription

1 International Journal of Algebra, Vol. 7, 2013, no. 15, HIKARI Ltd, Group Closures of Injective Order-Preserving Transformations Paula Catarino 1 Departamento de Matemática Universidade de Trás-os-Montes e Alto Douro Vila Real, Portugal pcatarin@utad.pt Inessa Levi Department of Mathematics Columbus State University Columbus, GA 31907, USA levi inessa@columbusstate.edu Copyright c 2013 Paula Catarino and Inessa Levi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Given a group G of permutations of a finite n-element set X n and a transformation f of X n, the G-closure f : G of f is the semigroup generated by all the conjugates of f by permutations in G. A semigroup S of transformations of X n is G-normal if G S = G, where G S consists of all the permutations h of X n such that h 1 fh S for all f S. We may assume that X n is a chain and we let POI n be the semigroup of all the partial and total one-to-one order preserving transformations of X n. In the present paper we describe the group Γ = G POIn, characterize all the inverse Γ-closures of transformations in POI n, and characterize all the inverse Γ-closures that are also Γ-normal. Mathematics Subject Classification: 20M20, 20M18, 20M17 1 Nember of the Research Center CIDMA of the University of Aveiro, Portugal

2 704 Paula Catarino and Inessa Levi Keywords: semigroup, transformation, one-to-one transformation, orderpreserving transformation, group-closure, G-normal 1 Introduction Let X n = {1, 2,...,n} be a non-empty set, and let PT n be the semigroup of all the partial and total transformations of X n. For an f PT n and a subgroup G of the symmetric group S n of all permutations of X n, we let f : G denote the subsemigroup {h 1 fh : h G} of PT n generated by all the conjugates of f by the elements of G. The semigroup f : G is referred to as a group closure or G-closure of f (see [12]). For a semigroup S of transformations of X n let G S denote the group of all permutations h of X n such that h 1 fh S for all f S. Given a subgroup G of the symmetric group S n, a semigroup S of transformations of X n is said to be G-normal if G S = G (see [10]). For example, the semigroup PT n and its subsemigroup T n of all the total transformations are S n -normal semigroups, and the description of all S n -normal semigroups may be found in [11]. A number of results in the semigroup literature are related to the properties of G-closures as well as the relations between G-closures and G-normal semigroups. For example, it was shown in [12] that the alternating group A n can not serve as the group G S for any semigroup of total transformations of X n. Indeed, for any f T n the A n -closure f : A n is S n -normal semigroup, that is G f:an = S n. The A n -normal semigroups of partial transformations were described in [13] and [14]. The dihedral group D n may serve as the group G S for semigroups of total transformations of X n, and large classes of D n -normal semigroups of total transformations were described in [2]. It was shown recently in [1] that for n 10 the group closure f : G is idempotent-generated and regular for all f T n \S n if and only if G = A n or S n (with a special characterization of groups G in cases of n 9). A natural connection between automorphisms of a semigroup S and the group G S is discussed in Section 5.1. In the present paper we address the following inter-related problems that were raised in [12], [10] and [15]. Problem 1 What subgroups of the symmetric group S n groups G S for semigroups S of transformations of X n? may serve as the Problem 2 Given a subgroup G of S n, describe properties of G-closures. Problem 3 Given a subgroup G of S n, characterize the G-closures that are G-normal semigroups, that is characterize all transformations f PT n such that G f:g = G.

3 Group closures of injective order-preserving transformations 705 We focus on semigroups of order-preserving one-to-one transformations. A transformation f of X n is said to be an order-preserving [order-reversing] if x y implies xf yf [xf yf] for all x, y in the domain of f. The semigroup of all one-to-one order-preserving transformations of X n is denoted by POI n. Semigroup POI n is an inverse semigroup, its congruences, rank and its presentation were described in [4]. Semigroups of order-preserving and order-reversing transformations were studied extensively (see, for example, [3]- [8]). Let γ be the unique order-reversing permutation of X n defined by xγ = n+1 x for all x X n and let Γ = γ be the two-element group generated by γ. We will show that for n 3, G POIn = Γ, that is γ is the unique non-identity permutation of X n such that for any f POI n its conjugate γ 1 fγ is also in POI n. In view of this role of Γ in POI n we will focus on the Γ-closures f :Γ of f POI n. We characterize all the Γ-closures that are inverse semigroups, and all the Γ-closures that are both, inverse semigroups and Γ- normal subsemigroups of POI n, contributing to solutions of Problems 1, 2 and 3. 2 Decomposition of one-to-one order preserving transformations We start by describing a framework for studying one-to-one order-preserving transformations through a decomposition into elementary components. As a permutation of X n may be written uniquely as a product of disjoint cycles (up to the order in which the cycles appear), a partial one-to-one transformation of X n may be written basically uniquely as a join of disjoint cycles and chains (see Theorem 3.2 in Lipscomb [16]). Specifically, for a subset A = {a 1,a 2,...,a k } of X n with k 2 let (a 1,a 2,...,a k ] be a one-to-one transformation with domain A\{a k } that maps each a i onto a i+1 for i =1, 2,...,k 1. The transformation (a 1,a 2,...,a k ]isak-chain. For each a X n,(a] denotes the empty transformation; (a] is referred to as a 1-chain. The k-chains with k 2 are referred to as proper chains. A partial one-to-one transformation is a k-cycle if it is a cycle on its domain. A path is either a chain or a cycle; we will use Greek letters to denote chains and cycles, the elementary components of one-to one partial transformations. The only exception is γ, the order-reversing permutation of X n. For f POI n, let im(f) and dom(f) denote the image and the domain of f respectively. For a path α let set(α) = im(α) dom(α). Two paths α and β are said to be disjoint if the sets set(α) and set(β) are

4 706 Paula Catarino and Inessa Levi disjoint. The join g of two disjoint paths α and β is a partial one-to-one transformation with domain dom(g) = dom(α) dom(β) that coincides with α on dom(α) and with β on dom(β). Every non-empty partial one-to-one transformation f is a join of disjoint paths of length at least two, and this join representation of f is unique up to the order in which the paths are written. Following the practise established in [16] we will not use a symbol for joins of chains and cycles, and we will omit writing explicitly 1-chains and 1-cycles unless it is necessary for clarity. When f is a partial one-to-one order-preserving transformation, its paths have a restricted form. Proposition 2.1 Let f be a non-empty transformation in POI n. Then f is a join of some (possibly none) disjoint proper chains (a 1,a 2,...,a k ] such that either a 1 <a 2 < <a k or a 1 >a 2 > >a k, and, some (possibly none) disjoint 1-cycles and 1-chains. Proof. Suppose f contains a path α with set(α) ={a 1,a 2,...,a k } such that a i α = a i+1 for i =1, 2,...,k 1, and a k α = a 1 if α is a cycle. Assume without loss of generality that a 1 <a 2. Then a 2 = a 1 α<a 2 α = a 3,...,a k 1 = a k 2 α<a k 1 α = a k, so that a 1 <a 2 <a 3 < <a k.ifαis a proper chain, the result follows. If α is a cycle with k 2 we also have that a k = a k 1 α< a k α = a 1,soa k <a 1, a contradiction to our prior conclusion. In view of the above result we refer to a chain (a 1,a 2,...,a k ]asincreasing if a 1 <a 2 < <a k ;itisdecreasing if a 1 >a 2 > >a k. Given a proper chain α =(a 1,a 2,...,a k ] we let min(α) = min(set(α)) and max(α) = max(set(α)). For a, b X n with a<b, let [a, b] denote as usual the set of all x X n such that a x b. Thus the set [min(α), max(α)] contains the set(α), and this containment may be proper. Observe that if α =(a 1,a 2,...,a k ]is a proper chain in POI n then min(im(α)) min(dom(α)). Indeed, if α is increasing, min(im(α)) = a 2 and min(dom(α)) = a 1 <a 2. If α is decreasing, then min(im(α)) = a k while min(dom(α))= a k 1 >a k. Let fix(f) ={x X n : xf = x}. Lemma 2.2 Let f POI n have a proper chain α in its join representation, and assume that fix(f). Then the sets fix(f) and [min(α), max(α)] are disjoint. Proof. Let α =(a 1,a 2,...,a k ]. Take b fix(f), and assume that min(α) < b<max(α). If α is increasing there exist a i,a i+1 set(α) such that a i <b< a i+1. Then since f is a one-to-one order-preserving transformation we have that a i+1 = a i f<bf= b, a contradiction. Similarly, if α is decreasing, there

5 Group closures of injective order-preserving transformations 707 exist a i,a i+1 such that a i >b>a i+1, and we have that a i+1 = a i f>bf= b, again a contradiction. Note that if α and β are distinct proper chains in the decomposition of f then the intervals [min(α), max(α)] and [min(β), max(β)] are not necessarily disjoint. For example the transformation f =(1, 3, 5](2, 4, 6] POI 6 is a join of two disjoint proper chains α =(1, 3, 5] and β =(2, 4, 6]. The sets [min(α), max(α)] = {1, 2, 3, 4, 5} and [min(β), max(β)] = {2, 3, 4, 5, 6} have a non-empty intersection. However, the following restriction on α and β holds. Lemma 2.3 Let f POI n have proper chains α and β in its join representation. If the intervals [min(α), max(α)] and [min(β), max(β)] are not disjoint then α and β are either both increasing or both decreasing. Proof. Assume without loss of generality that α = (a 1,a 2,...,a k ] is increasing while β =(b 1,b 2,...,b m ] is decreasing. Let x [min(α), max(α)] [min(β), max(β)], so there exist s, t such that a t x a t+1 and b s+1 x b s, hence a t x b s with t<kand s<m. Therefore a t+1 = a t f b s f = b s+1. Since α and β are disjoint, a t+1 <b s+1 x, a contradiction since x a t+1 by our choice of t. The next result may be found in [16]. Lemma 2.4 Suppose f is a partial one-to-one transformation, and let h S n. Then 1. fix(h 1 fh)=(fix(f))h, and 2. if (a 1,a 2,...,a k ] is a chain in the join representation of f then (a 1 h, a 2 h,...,a k h] is a chain in the join representation of h 1 fh. The following result will be useful. Its proof is straightforward. Lemma 2.5 Let f be a transformation of X n and let h S n. Then im(h 1 fh)=(im(f))h and dom(h 1 fh)=(dom(f))h. Observe that a transformation f POI n is uniquely determined by its domain and its image. Indeed, since f is one-to-one, its domain and image have the same number of elements. If dom(f) ={a 1,a 2,...,a k } with a 1 < a 2 < <a k and im(f) ={b 1,b 2,...,b k } with b 1 <b 2 < <b k, then since f is order-preserving, a i f = b i, for i =1, 2,...,k. Therefore we have the following. Lemma 2.6 Transformations f and g POI n are equal if and only if dom(f) =dom(g) and im(f) =im(g). For a subset A of X n let i A denote the identity transformation with domain A. Clearly i A is an idempotent in POI n. Moreover if e is an idempotent in POI n then e = i A for some A X n.

6 708 Paula Catarino and Inessa Levi 3 POI n is a Γ-normal semigroup Recall that γ denotes the unique non-identity order-reversing permutation of X n. Since γ 1 = γ, we have that Γ = γ = {γ,i Xn }. For any order-preserving transformation f, its conjugate γfγ is also an order-preserving transformation, hence Γ is a subgroup of G POIn. We show that in fact an equality holds. Theorem 3.1 Let g =(n, n 1,...,1] POI n be an n-chain. 1. If S is a subsemigroup of POI n containing g then G S Γ. 2. G POIn =Γ. Proof. Since the second result follows directly from the first and the discussion above the theorem, we just need to prove the first result. To this end, note that for any h G S we have that h 1 gh S POI n. By Lemma 2.4 (nh, (n 1)h,...,1h] =h 1 gh POI n, therefore by Proposition 2.1 we have that either nh < (n 1)h < < 1h or nh > (n 1)h > > 1h. Since the set {nh, (n 1)h,...,1h} consists of n distinct elements, we have that either nh =1, (n 1)h =2,...,1h = n and h = γ; ornh = n, (n 1)h = n 1,...,1h = 1 and h = i Xn, therefore G S Γ as required. 4 Inverse semigroups and Γ-closures In this section we characterize all f POI n such that the Γ-closure f :Γ is an inverse semigroup, and we provide a descriptions of these f in terms of their join representation. Proposition 4.1 Let f POI n and f. Then f and γfγ are mutually inverse if and only if im(f) =(dom(f))γ. Proof. Assume that f and γf γ are mutually inverse. Since f = f(γf γ)f, for one-to-one transformations f and γfγ we have that im(f) = dom(γfγ) = (dom(f))γ, by Lemma 2.5. Conversely, if im(f) = (dom(f))γ = dom(γfγ), then dom(f) = (im(f))γ = im(γfγ), so the image and domain of fγfγf coincide with that of f, and so fγfγf = f by Lemma 2.6. Similarly, γfγfγfγ = γfγ, sof 1 = γfγ. In view of the above result we let Ϝ = {f POI n : either im(f) = (dom(f))γ, or f = i A for some A X n }, so Ϝ consists of all the one-to-one order-preserving transformations f with f 1 = γfγ or f 1 = f. Given a transformation f in the inverse semigroup

7 Group closures of injective order-preserving transformations 709 POI n, the least inverse subsemigroup of POI n containing f is precisely the subsemigroup of POI n generated by f and its inverse in POI n (see for example [9], Proposition 3). Therefore if f Ϝ, then the Γ-closure f : Γ, generated by f and its inverse γfγ, is an inverse semigroup. Conversely, if f / Ϝ, then f and γfγ are not mutually inverse and f is not an idempotent. Moreover, in this case f and γfγ are the only elements of f :Γ whose image contains im(f) elements, indeed, since im(f) (dom(f))γ, we have that (im(f))γ dom(f), and im(f γfγ), im(γf γf) contain fewer elements than im(f). So f has no inverse in f :Γ, and f :Γ is not an inverse semigroup. We summarize this in the result below. Theorem 4.2 Let f POI n. Then f :Γ is an inverse semigroup if and only if f Ϝ. An inverse semigroup of the form f :Γ will be referred to as an inverse Γ- closure. To describe the join representation of non-idempotent transformations in Ϝ, for f POI n we let M k (f) ={set(α) :α is a k-chain in the join representation of f} for each integer k 2. Note that M k (f) = if f has no k-chains in its join representation. We let M k (f) be the subset of M k(f) consisting of all the sets A i = set(α i ) such that α i is an increasing chain in f, and let M k (f) be the subset of M k (f) consisting of all the sets A j = set(α j ) such that α j is a decreasing chain in f. The next result is immediate. Lemma 4.3 Let f,g POI n. Then g = f 1 if and only if fix(f) =fix(g), M k (f) = M k (g), and M k (f) = M k (g). Lemma 4.4 Let f POI n. Then 1. M k (γfγ) =(M k (f))γ for k 2; 2. M k (γfγ) =(M k (f))γ for k 2. Proof. By Lemma 2.4 for any proper chain α =(a 1,a 2,...,a k ]inf, γαγ = (a 1 γ,a 2 γ,...,a k γ] is a chain in γfγ, and, since γ is order-reversing, γαγ is increasing if and only if α is decreasing. Since set(γαγ) = (set(α))γ, assertions (1) and (2) of this lemma follow. The lemmas above are used below to describe the join representation of the transformations that give rise to inverse Γ-closures. Proposition 4.5 Let f be a non-idempotent transformation in POI n. Then f Ϝ if and only if

8 710 Paula Catarino and Inessa Levi 1. (fix(f))γ = fix(f), 2. (M k (f))γ = M k (f) for all k 2, and 3. (M k (f))γ = M k (f) for all k 2. Proof. If f Ϝ, then f 1 = γfγ, so by Lemma 4.3 fix(f) = fix(f 1 )= fix(γfγ) = (fix(f))γ, and assertion (1) follows. Also by Lemma 4.4 we have that (M k (f))γ = M k (γfγ) =M k (f 1 )=M k (f) by Lemma 4.3, so assertion (2) above follows. Assertion (3) follows in a similar manner. Conversely, if assertions (1)-(3) above hold, then Lemmas 4.3 and 4.4 imply that f and γfγ are mutually inverse, so f Ϝ. 5 Group G f:γ and Centralizers Take f to be a transformation in POI n, and let C Sn (f) ={h S n : fh = hf} be the centralizer of f in the symmetric group S n. For brevity we write C(f) for C Sn (f). Recall that if e and p are idempotents POI n then e = i A, p = i B for some subsets A and B of X n and ep = i A B = pe. We utilize Proposition 4.1 to obtain the following description of G f:γ. Theorem 5.1 G f:γ = (C(f) γ(c(f))γ) (γ(c(f)) (C(f))γ) for f POI n. Proof. Take h G f:γ, and note that im(f) = im(h 1 fh) by Lemma 2.5. We show that either h 1 fh = f and h 1 γfγh = γfγ, or (1) h 1 fh = γfγ and h 1 γfγh = f. (2) If f is an idempotent, f = i A for some A X n, then γfγ = i Aγ, and fγfγ = i A Aγ = γfγf. Thusf and γfγ are the only elements in f :Γ whose image contains im(f) elements, and either Equation (1) or (2) holds. If f is not an idempotent, h 1 fh is not an idempotent also. We show that f and γfγ are the only non-idempotent elements in f :Γ whose image contains im(f) elements. Since f is not an idempotent, it contains at least one proper chain α =(a 1,,a k ] with k 2 in its join representation. Thus a k im(f) \ im(f 2 ) and so im(f 2 ) < im(f). Iff Ϝ then im(f) (dom(f))γ, so im(fγfγ) < im(f). Hence we also have im(γfγf) = im(γ(fγfγ)γ) = im(fγfγ) < im(f). Iff Ϝ then by Theorem 4.2 we have that fγfγ and γfγf are idempotents. Since h 1 fh and h 1 γfγh are also non-idempotents in f : Γ whose image contains im(f) elements, we have that either Equation (1)

9 Group closures of injective order-preserving transformations 711 or (2) holds. Now if Equation (1) holds then h C(f) γ(c(f))γ and if Equation (2) holds then h γ(c(f)) (C(f))γ. Since f and γfγ generate f :Γ it is easy to verify that (C(f) γ(c(f))γ) (γ(c(f)) (C(f))γ) G f:γ. If f = i A is an idempotent then C(f) ={h S n : Ah = A}, γ(c(f))γ = {h S n : Aγh = Aγ}, γ(c(f)) = {h S n : Aγh = A} and (C(f))γ = {h S n : Ah = Aγ}. Therefore C(f) γ(c(f))γ γ(c(f)) (C(f))γ, and G ia :Γ = {h S n : either Ah = A, Aγh = Aγ, or Aγh = A, Ah = Aγ}. (3) It is easy to check that G ia :Γ =Γifn 3 and A 2. If n 4 and if there exist distinct a, b in either A Aγ or A \ Aγ or X n \ (A Aγ), and the permutation h =(a, b) G ia :Γ \ Γ, since γ has at most one fixed point. If n 4 and and we assume that no such a, b exist, then n = 4 and A =2; in this case Aγ and A are either equal or disjoint, in either case violating the previously assumed restrictions on sizes of A Aγ and A \ Aγ. We summarize these observations below. Proposition 5.2 For an idempotent f = i A we have that G ia :Γ =Γif and only if n 3 and A 2. Theorem 5.1 provides a motivation for a description of C(f) for f POI n. While a general description of centralizers of one-to-one partial transformations in the semigroup of all one-to-one transformations can be found in Lipscomb [16] (Theorem 10.1), the special case of order-preserving transformations and their centralizers in S n is not immediately derivable from this theorem, and we present as a self-contained result in Proposition 5.3 below. For a fixed k =2, 3,...,n 1, and i =1, 2 we let A i = {a i1,a i2,...,a ik } be k-element subsets of X n. Assume that the elements of A i are arranged in the increasing order: a i1 <a i2 < <a ik. Let π A1 A 2 and ρ A1 A 2 be respectively the order-preserving and the order-reversing bijections from A 1 onto A 2,so that for j =1, 2,...,k, a 1j π A1 A 2 = a 2j and a 1j ρ A1 A 2 = a 2,k+1 j Proposition 5.3 Let f POI n.ifh C(f) then there exists a permutation h of fix(f), and for each k 2 with M k (f) there exists a permutation h k of M k (f) such that for x A M k (f) { xπa,ahk if either A, Ah xh = k M k (f) or A, Ah k M k (f), otherwise. xρ A,Ahk (4) Conversely, take any permutation h of fix(f), any family of permutations h k of M k (f), for each k 2 with M k (f), and any permutation h of X n \

10 712 Paula Catarino and Inessa Levi (dom(f) im(f)). Define a transformation h of X n by letting xh = x h if x fix(f), xh be calculated as in (4) above if x A M k (f) for some k 2, and xh = x h if x X n \ (dom(f) im(f)). Then h C(f). Proof. Take h C(f) S n. Lemma 2.4(1) implies that fix(f)h =fix(f), and so h induces a permutation h of fix(f) defined by x h = xh for all x fix(f). Lemma 2.4(2) implies that h maps the set of k-chains in f onto the set of k-chains in f, so that h induces a permutation h k of M k (f). Moreover, Proposition 2.1 assures that every chain α in f is either increasing or decreasing. Take A = {a 1,a 2,...,a k } = set(α) and let Ah k = {b 1,b 2,...,b k } = set(β), a 1 <a 2 < <a k and b 1 <b 2 < <b k. If α, β M k (f) then α = (a 1,a 2,...,a k ], β =(b 1,b 2,...,b k ] and a j h = b j for each j =1, 2,...,k, therefore a j h = a j π A,Ahk. Similarly, if α, β M k (f) then α =(a k,a k 1,...,a 1 ], β =(b k,b k 1,...,b 1 ] and a j h = b j for each j =1, 2,...,k again. If α M k (f) while β M k (f), we have that α =(a 1,a 2,...,a k ], β =(b k,b k 1,...,b 1 ], and a 1 h = b k,a 2 h = b k 1,...,a k h = b 1, so that a j h = a j ρ A,Ahk,j =1, 2,...,k. The remaining case when α M k (f) while β M k (f) is similar. For the converse, take arbitrary permutations h of fix(f), h k of M k (f), for each k 2, and h of X n \ (dom(f) im(f)). Define h as stated in the converse part of the statement of the proposition. Since the transformation f generates a partition of dom(f) im(f) into subsets fix(f), if non-empty, and {A M k (f) :k 2}, we have that h S n. We show that xhf = xfh for any x dom(f). If x fix(f), we have that xh = x h fix(f), and so (xh)f = xh =(xf)h. If x dom(f) \ fix(f) then x A M k (f) for some k 2, where A = {a 1,a 2,...,a k } with a 1 <a 2 < <a k. Suppose A M k (f), then A = set(α) for an increasing chain α = (a 1,a 2,...,a k ] and x = a i for some i =1, 2,...,a k 1. Let B = {b 1,b 2,...,b k } with b 1 <b 2 < <b k such that B = Ah k. If B M k (f) then B = set(β) for an increasing chain β =(b 1,b 2,...,b k ] and so by Equation (4) we have that xh = a i h = a i π A,Ahk. Therefore xhf =(a i π A,Ahk )f = b i f = b i+1 = a i+1 π A,Ahk = a i+1 h =(a i f)h = xfh. If B M k (f) then B = set(β) for a decreasing chain β =(b k,b k 1,...,b 1 ], xh = a i h = a i ρ A,Ahk and so by Equation (4) again xhf =(a i ρ A,Ahk )f = b k+1 i f = b k i = a i+1 ρ A,Ahk = a i+1 h =(a i f)h = xfh. If A M k (f), similar computations show that xhf = xfh, as required. In the remainder of this section we will focus on a description of G f:γ for a non-idempotent f Ϝ; the result below simplifies this task.

11 Group closures of injective order-preserving transformations 713 Proposition 5.4 Take f POI n with f f If C(f) =γ(c(f))γ then G f:γ =ΓC(f). In this case G f:γ =Γif and only if C(f) Γ. 2. If f Ϝ then C(f) =γ(c(f))γ. Proof. If C(f) =γ(c(f))γ, then γ(c(f))=(c(f))γ, so by Theorem 5.1 we have that G f:γ = (C(f) γ(c(f))γ) (γ(c(f)) (C(f))γ) = C(f) γ(c(f)) =ΓC(f), and (1) follows. To show that (2) holds, it suffices to show that γ(c(f))γ C(f) since C(f) is finite. Note that C(f) =C(f 1 ) and for a non-idempotent f Ϝ we have that f 1 = γfγ, soc(f) =C(γfγ). Therefore for h C(f) we have that hγfγ = γfγh, and γhγf = fγhγ, that is h γ(c(f))γ. We apply this result to characterize all f Ϝ with f f 2 for which G f:γ =Γ. Lemma 5.5 Take f POI n with f f 2. If there exists an A M k (f) for some k 2 such that either A, Aγ M k (f) or A, Aγ M k (f) then Γ is a proper subgroup of G f:γ. Proof. Assume without loss of generality that A, Aγ M k (f) and define h S n such that xπ A,Aγ if x A xh = xπ Aγ,A if x Aγ x if x X n \ (A Aγ). By Proposition 5.3 h C(f). Let A = {a 1,a 2,...,a k } with a 1 < a 2 < <a k. Since each non-trivial cycle in h has the form (a i,a k i+1 γ) for i = 1, 2,...,k, we also have that γhγ = h C(f), that is h C(f) γ(c(f))γ G f:γ by Theorem 5.1. However, h γ, for example a 1 h = a k γ a 1 γ. The above results aid in characterization of f Ϝ such that G f:γ =Γ. Note that if n = 2, we have that S 2 = Γ and G f:γ =Γ. Ifn = 3, the nonidempotents f in Ϝ are (1, 3](2], (3, 1](2], (1, 2, 3] and (3, 2, 1], and C(f) ={i X3 } for each of these transformations. In this case in view of Proposition 5.2, Γ- closure f :Γ is Γ-normal for any f Ϝ, f i X3. However for larger values of n there exist non-identity transformations of X n for which Γ-closures are not Γ-normal, for example, if n =4,f =(1, 3](2, 4] Ϝ, C(f) ={(1, 2)(3, 4),i X4 }, and G f:γ is the Klein 4-group. The result below contributes to the body of research on Problems 1, 2, and 3 in the Introduction.

12 714 Paula Catarino and Inessa Levi Theorem 5.6 Let n 4. Iff Ϝ, and f f 2 then G f:γ =ΓC(f). For f Ϝ the semigroup f :Γ is Γ-normal if and only if f f 2, M k (f) 1 for all k 2, and one of the following holds: 1. im(f) dom(f) =X n and fix(f) =, or 2. n is odd, and either (a) im(f) dom(f) =X n \{(n +1)/2} and fix(f) =, or (b) im(f) dom(f) =X n and fix(f) ={(n +1)/2}. Proof. We have that G f:γ =ΓC(f) by Proposition 5.4 for non-idempotent f. Suppose that G f:γ = Γ. Then by Proposition 5.2 we have that f f 2, and we may assume that M k (f) 2 for some k 2. By Lemma 5.5 we can deduce that M k (f) ={A, B} such that A M k (f) and B M k (f). Moreover, since f Ϝ we have by Proposition 4.5 that {A} = M k (f) = (M k (f))γ = {Aγ}. Similarly we can deduce that Bγ = B. By Lemma 2.3 the intervals [min(a), max(a)] and [min(b), max(b)] are disjoint. Assume without loss of generality that max(a) < min(b). Since γ is order-reversing, we have that min(a) = max(a)γ > min(b)γ = max(b), a contradiction, since min(a) < max(a) < min(b) < max(b) by our previous assumption. Hence M k (f) 1 for all k 2. Now assume that there exist distinct a, b either in fix(f) orinx n \(im(f) dom(f)). Take h =(a, b) to be a permutation of X n. Then by Proposition 5.3 we have that h C(f). Moreover, since fix(f)γ =fix(f) and (X n \ (im(f) dom(f)))γ = X n \ (im(f) dom(f)) for an f Ϝ, we have that γhγ C(f) as well. Thus h G f:γ by Theorem 5.1. An assumption that h = γ leads to a contradictory conclusion that n 3. Hence we conclude that fix(f) 1 and X n \ (im(f) dom(f)) 1. If a is the unique element of fix(f) or X n \(im(f) dom(f)) then aγ = a (since f Ϝ), so n is odd and a =(n+1)/2. Conversely, assume that f f 2, M k (f) 1 for all k 2, and conditions (1)-(2) of the Theorem hold. Let h C(f), then by Proposition 5.3, xh = x for all x {A : A M k (f)} for k 2, thus we can deduce that h = i Xn, and so G f:γ =ΓC(f) =Γ. 5.1 The group G S and the inner automorphisms of S We conclude by looking at an important connection between the automorphisms of S and the group G S. An automorphism ϕ = ϕ h of a semigroup S of transformations of X n is said to be inner if there exists a permutation h of X n such that ϕ h (g) =hgh 1 for all g S. The set Inn(S) of all the inner automorphisms of S forms a subgroup of the group Aut(S) of all the automorphisms of S. The group Inn(S) is a homomorphic image of G S. Observe

13 Group closures of injective order-preserving transformations 715 that permutations h and q of X n give rise to equal automorphisms ϕ h and ϕ q of S if and only if h 1 q C(g) for all g S, so that if we denote the group {C(g) :g S} by C(S) we have, as in [12], that Inn(S) = G S /C(S). Now if S = f :Γ for f Ϝ, then C(γfγ) =γ(c(f))γ = C(f) by Proposition 5.4. Since f and γfγ generate S, we have that C(f) C(g) for any g S, so C(S) = {C(g) :g S} = C(f). Thus in view of Proposition 5.4 again, we have for f Ϝ that Inn( f :Γ ) = Γ. References [1] J.Araújo, J.D. Mitchell, Csaba Schneider, Groups that together with any transformation generate regular semigroups or idempotent generated semigroups, Journal of Algebra, 343 (2011), [2] P. Catarino, I. Levi, D n -normal semigroups of transformaions, Semigroup Forum, 76 (2008), [3] V. H. Fernandes, Semigroups of order preserving mappings on a finite chain: a new class of divisors, Semigroup Forum, 54 (1997), no. 2, [4] V. H. Fernandes, The monoid of all injective order-preserving partial transformations on a finite chain, Semigroup Forum, 62 (2001), [5] V.H. Fernandes, G.M.S. Gomes, M.M. Jesus, Congruences on monoids of order-preserving or order-reversing transformations of a finite chain, Glasgow Math. J., 47 (2005), [6] G. U. Garba, On the idempotent ranks of certain semigroups of orderpreserving transformations, Portugal. Math., 51 (1994), no. 2, [7] G.M.S. Gomes, J.M. Howie, On the ranks of certain semigroups of orderpreserving transformations, Semigroup Forum, 45 (1992), no. 3, [8] P.H. Higgins, Combinatorial results for semigroups of order-preserving mappings, Math. Proc. Cambridge Philos. Soc., 113 (1993), no. 2, [9] M.V. Lawson, Inverse Semigroups, The Theory of Partial Symmetries, World Scientific, [10] I. Levi, S. Seif, Finite normax semigroups, Semigroup Forum, 57 (1998),

14 716 Paula Catarino and Inessa Levi [11] I. Levi, R. B. McFadden, S n -normal semigroups, Proc. Edinburgh Math. Soc., 37 (1994), [12] I. Levi, On the inner automorphisms of finite transformation semigroups, Proc. Edinburgh Math. Soc., 39 (1996), [13] I. Levi, D.B. McAlister, R.B. McFadden, Groups associated with finite transformation semigroups, Semigroup Forum, 61 (2000), [14] I. Levi, D.B. McAlister, R.B.McFadden, A n -normal Semigroups, Semigroup Forum, 62 (2001), [15] I. Levi, Group closures of one-to-one transformations, Bull. Austral. Math. Soc., 64 (2001), [16] Lipscomb, S., Summetric Inverse Semigroups, Mathematical Surveys and Monographs, Vol 46, Received: September 15, 2013