A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups
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A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups Lei Sun1 Received: 6 July 2018 / Revised: 26 October 2019 / Accepted: 5 November 2019 © Iranian Mathematical Society 2019
Abstract Mitsch defined the natural partial order ≤ on a semigroup S as follows: a ≤ b if and only if a = xb = by, a = xa for some x, y ∈ S 1 . Let T X be the full transformation semigroup on a finite set X = {1, 2, . . . , n}. Let ρ be an equivalence relation on X and be a total order on the partition set X /ρ of X induced by ρ. Denote by x the ρ-class containing x ∈ X . In this paper, we endow the partition order-decreasing transformation subsemigroup of T X defined by T (ρ, ) = { f ∈ T X : ∀ x ∈ X , f (x) x} with the natural partial order and give a characterization for this order. Then we determine the compatibility of their elements and find all the minimal and maximal elements. Keywords Transformation semigroup · Natural partial order · Compatibility · The minimal (maximal) elements Mathematics Subject Classification 20M20
1 Introduction Throughout the paper, let ≤ be a partial order on a semigroup S and be a total order on the partition set X /ρ of a set X induced by the equivalence ρ on X . Also, for two sets A and B, A ⊆ B means that A is a subset of B. In the study of algebraic semigroups, an order relation on a semigroup may be defined via the multiplication of the semigroup. An important such order relation is
Communicated by Hamid Mousavi.
B 1
Lei Sun [email protected] School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China
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Bulletin of the Iranian Mathematical Society
the natural partial order. There are many studies in the literature about this order. For any semigroup S, the set E S of all the idempotents is partially ordered by e ≤ f if and only if e = e f = f e (e, f ∈ E S ).
(1.1)
Since idempotents give a good deal of information on the semigroup, this partial order can be used successfully in the study of semigroup S as a whole. There have been numerous attempts to extend this order from E S to all of S defining it by means of the multiplications of S and postulating that its restriction to E S coincides with the partial order (1.1). The reason for the interest in such natural orders lies in the obvious fact that such an order can provide additional information on a given semigroup since it reflects its multiplications in a particular way. An important class of semigroups for which such a natural partial order was found is that of inverse semigroups. It was defined by Wagner [18] in 1952 as follows: a ≤ b if and only if a = eb for some e ∈ E S .
(1.2)
The natural partial order is in fact compatible with multiplication of S, in the sense that a ≤ b implies ac ≤ bc and ca ≤ cb for all c ∈ S. It proved very useful in the theory of inverse semigroups. This order was generalized to the much larger class of regular semigroups by Hartwig [4] and Nambooripad [8], respectively. The most commonly used definition for regular semigroups is the followin
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