mathematics Article Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section Chollawat Pookpienlert 1, Preeyanuch Honyam 2,* and Jintana Sanwong 2 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand; [email protected] 2 Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand; [email protected] * Correspondence: [email protected] Received: 4 June 2018; Accepted: 2 August 2018; Published: 4 August 2018 Abstract: Let TpX, Yq be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation r on X, let rˆ be the restriction of r on Y, R a cross-section of Y{rˆ and define TpX, Y, r, Rq to be the set of all total transformations a from X into Y such that a preserves both r (if pa, bq P r, then paa, baq P r) and R (if r P R, then ra P R). TpX, Y, r, Rq is then a subsemigroup of TpX, Yq. In this paper, we give descriptions of Green’s relations on TpX, Y, r, Rq, and these results extend the results on TpX, Yq and TpX, r, Rq when taking r to be the identity relation and Y “ X, respectively. Keywords: transformation semigroup; Green’s relations; equivalence relation; cross-section MSC: 20M20 1. Introduction Let X be a nonempty set and TpXq denote the semigroup containing all full transformations from X into itself with the composition. It is well-known that TpXq is a regular semigroup, as shown in Reference [1]. Various subsemigroups of TpXq have been investigated in different years. One of the subsemigroups of TpXq is related to an equivalence relation r on X and a cross-section R of the partition X{r (i.e., each r-class contains exactly one element of R), namely TpX, r, Rq, which was first considered by Araújo and Konieczny in 2003 [2], and is defined by TpX, r, Rq “ ta P TpXq : Ra Ď R and pa, bq P r ñ paa, baq P ru, where Za “ tza : z P Zu. They studied automorphism groups of centralizers of idempotents. Moreover, they also determined Green’s relations and described the regular elements of TpX, r, Rq in 2004 [3]. Let Y be a nonempty subset of the set X. Consider another subsemigroup of TpXq, which was first introduced by Symons [4] in 1975, called TpX, Yq, defined by TpX, Yq “ ta P TpXq : Xa Ď Yu, when Xa denotes the image of a. He described all the automorphisms of TpX, Yq and also determined when TpX1, Y1q is isomorphic to TpX2, Y2q. In 2009, Sanwong, Singha and Sullivan [5] described all the maximal and minimal congruences on TpX, Yq. Later, in Reference [6], Sanwong and Sommanee studied other algebraic properties of TpX, Yq. They gave necessary and sufficient conditions for TpX, Yq Mathematics 2018, 6, 134; doi:10.3390/math6080134 www.mdpi.com/journal/mathematics Mathematics 2018, 6, 134 2 of 12 to be regular and also determined Green’s relations on TpX, Yq. Furthermore, they obtained a class of maximal inverse subsemigroups of TpX, Yq and proved that the set FpX, Yq “ ta P TpX, Yq : Xa Ď Yau contains all regular elements in TpX, Yq, and is the largest regular subsemigroup of TpX, Yq. From now on, we study the subsemigroup TpX, Y, r, Rq of TpX, Yq defined by TpX, Y, r, Rq “ ta P TpX, Yq : Ra Ď R and pa, bq P r ñ paa, baq P ru, where r is an equivalence relation on X and R is a cross-section of the partition Y{rˆ in which rˆ “ r X pY ˆ Yq. If Y “ X, then TpX, Y, r, Rq “ TpX, r, Rq; and if r is the identity relation, then TpX, Y, r, Rq “ TpX, Yq, so we may regard TpX, Y, r, Rq as a generalization of TpX, r, Rq and TpX, Yq. Green’s relations play a role in semigroup theory, and the aim of this paper is to characterize Green’s relations on TpX, Y, r, Rq. As consequences, we obtain Green’s relations on TpX, r, Rq and TpX, Yq as corollaries. 2. Preliminaries and Notations For any semigroup S, let S1 be a semigroup obtained from S by adjoining an identity if S has no identity and letting S1 “ S if it already contains an identity. Green’s relations of S are equivalence relations on the set S which were first defined by Green. According to such definitions, we define the L-relation as follows. For any a, b P S, aLb if and only if S1a “ S1b, or equivalently, aLb if and only if a “ xb and b “ ya for some x, y P S1. Furthermore, we dually define the R-relation as follows. aRb if and only if aS1 “ bS1, or equivalently, aRb if and only if a “ bx and b “ ay for some x, y P S1. Moreover, we define the J -relation as follows. aJ b if and only if S1aS1 “ S1bS1, or equivalently, aJ b if and only if a “ xby and b “ uav for some x, y, u, v P S1. Finally, we define H “ L X R and D “ L ˝ R, where ˝ is the composition of relations. Since the relations L and R commute, it follows that L ˝ R “ R ˝ L. In this paper, we write functions on the right; in particular, this means that for a composition ab, a is applied first. Furthermore, the cardinality of a set A is denoted by |A|. For each a P TpXq, we denote by kerpaq the kernel of a, the set of ordered pairs in X ˆ X having the same image under a, that is, kerpaq “ tpa, bq P X ˆ X : aa “ bau. Moreover, the symbol ppaq denotes the partition of X induced by the map a, namely ppaq “ txa´1 : x P Xau. We observe that kerpaq is an equivalence relation on X in which the partition X{ kerpaq and ppaq coincide. Moreover, for all a, b P TpXq, we have kerpaq “ kerpbq if and only if ppaq “ ppbq. Mathematics 2018, 6, 134 3 of 12 In addition, if r is an equivalence relation on the set X and a, b P X, we sometimes write a r b instead of pa, bq P r, and define ar to be the equivalence class that contains a, that is, ar “ tb P X : b r au. For the subsemigroup TpX, Y, r, Rq of TpXq where r is an equivalence relation on X, Y is a nonempty subset of X and R is a cross-section of Y{rˆ in which rˆ “ r X pY ˆ Yq, we see that if a P X and ar X Y ‰H, then there exists a unique r P R such that a r r, and we denote this element by ra. Furthermore, we observe that FpX, Yq X TpX, Y, r, Rq contains all constant maps whose images belong to R. This implies that FpX, Yq X TpX, Y, r, Rq is a subsemigroup of TpX, Y, r, Rq, which will be denoted by F. An element a in a semigroup S is said to be regular if there exists x P S such that a “ axa; and S is a regular semigroup if every element of S is regular. In general, TpX, Y, r, Rq is not a regular semigroup, so we cannot apply Hall’s Theorem to find the L-relation and the R-relation on TpX, Y, r, Rq. Now, we give an example of a non-regular element in TpX, Y, r, Rq. Let X “ t1, 2, 3, 4, 5u, Y “ t3, 4, 5u, X{r “ tt1, 2u, t3, 4, 5uu, Y{rˆ “ tt3, 4, 5uu and R “ t3u. Define a P TpX, Y, r, Rq by 1 2 3 4 5 a “ . ˜4 3 3 5 5¸ Suppose that a is regular. Then a “ aba for some b P TpX, Y, r, Rq. We see that 4 “ 1a “ 1pabaq “ p4bqa, which implies that 1 “ 4b P Y, a contradiction. Throughout this paper, the set X we study can be a finite or an infinite set. For convenience, we will denote TpX, Y, r, Rq by T. 3. Green’s Relations on TpX, Y, r, Rq Unlike TpX, r, Rq, in general T has no identity, as shown in the following example. Example 1. Let X “ t1, 2, 3, 4, 5, 6u, Y “ t1, 3u,X{r “ tt1, 2u, t3, 4u, t5, 6uu, Y{rˆ “ tt1u, t3uu and R “ t1, 3u. Suppose that # is an identity element in T. Consider a P T defined by 1 2 3 4 5 6 a “ . ˜1 1 1 1 3 3¸ We see that p5#qa “ 5p#aq “ 5a “ 3, which implies that 5# P t5, 6u. This leads to a contradiction, since both 5 and 6 are not in Y. Therefore, we use the semigroup T with identity adjoined, given by T1, in studying its Green’s relations. From now on, the notation La (Ra, Ha, Da) denote the set of all elements of T which are L-related (R-related, H-related, D-related) to a, where a P T. Let A and B be families of sets. If for each set A P A there is a set B P B such that A Ď B, we say that A refines B, denoted by A ãÑ B.