Digraphs and the semigroup of all functions on a finite set

Abstract

Let T n denote the full transformation semigroup on the finite set n = {1, 2, ..., n), that is the set of all mappings from h to n, with function composition as the semigroup operation. In this paper algorithms are introduced to solve equations such as ax m b = c and ax = xb (a, b, c e T n ), which employ a representation of members of T n as special directed graphs.Algorithms for equations in T n . The object of study here is T n , the full transformation semigroup on n = {1, 2, . . . , n}, which means the semigroup of all mappings from n to itself under composition. This semigroup is to the theory of algebraic semigroups what the symmetric group (the group of all permutations on a set) is to group theory. It is easy to prove a "Cayley Theorem" to the effect that any semigroup S can be faithfully represented as a subsemigroup of T s i, the full transformation semigroup on the set S 1 (the semigroup S 1 is the semigroup S, with an adjoined identity 1, if 5 does not already possess one (see [1])). The full transformation semigroup also enjoys the important property of regularity, meaning that each element a e T n has an inverse x (not necessarily unique) in the sense that a = axa and x = xax.In [3] the author introduced a method for the calculation of all square roots of a given a e T n , the full transformation semigroup on n = {1, 2,. . . , n), as an alternative to the necessary and sufficient conditions given in [6] for or to be a square. The technique, which relies on a representation of a as a special directed graph, is extended here to furnish algorithms for the solution of the equations:(1) and ax =xb (a,b,ceT n , n 3=1).(2) The following graph theoretic definitions and results come from [2]. A digraph is said to be weak if it is connected when viewed as a graph. A functional digraph is a weak digraph in which every point has outdegree one. An in-tree is a digraph with a sink (point of outdegree zero), which is a tree when regarded as a graph.

Second Centralizers of Partial Transformations

“…. , m}, there is t i ∈ AE ∪ {−∞} that satisfies condition (2). Moreover, it follows from Lemma 8 and Lemma 9 that for each i ∈ {1, .…”

“…λ m is a join of its cells. (For applications of the digraph representation of full transformations on X, see [2] and [1, 6.2].) If α is a permutation on X, then α is a join of its circuits.…”

Centralizers in the Full Transformation Semigroup

“…For example, for various S: the elements of centralizers have been described in [14], [28], [31], [34], [35], and [39]; Green's relations and regularity have been determined in [23], [24], and [25]; and some representation theorems have been obtained in [29], [30], and [37]. See also [1] for the semigroup generated by the idempotents of regular centralizers; and [2] for some centralizers related to maps preserving digraphs.…”

Regular, inverse, and completely regular centralizers of permutations

“…It is clear that C(a) is a subsemigroup of S. Centralizers in T n were studied by Higgins [1], Liskovec and Feȋnberg [6], [7], and Weaver [8]. The author studied centralizers in the semigroup P T n [3], [4], [5].…”