On semigroups of endomorphisms of a chain with restricted range

Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\) . In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\) . Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks.     

Maximal and minimal congruences on some semigroups

In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T（X） consisting of all total transformations of an infinite set X under composition. Here, we determine all maximal congruences on the semigroup Zn under multiplication modulo n. And, when Y lohtain in X, we do the same for the semigroup T（X, Y） consisting of all elements of T（X） whose range is contained in Y. We also characterise the minimal congruences on T（X. Y）.     

A note on dual prehomomorphisms from a group into the Margolis–Meakin expansion of a group

Semigroup Forum - We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to...