On semigroups of endomorphisms of a chain with restricted range

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

On Maximal Subsemigroups of Partial Baer-Levi Semigroups

Suppose that X is an infinite set with |X|≥q≥ℵ0 and I(X) is the symmetric inverse semigroup defined on X. In 1984, Levi and Wood determined a class of maximal subsemigroups MA (using certain subsets A of X) of the Baer-Levi semigroup BL(q)={α∈I(X): dom α=X and |X∖Xα|=q}. Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups of BL(q), but these are far more complicated to describe. It is known that BL(q) is a subsemigroup of the partial Baer-Levi semigroup PS(q)={α∈I(X):|X∖Xα|=q}. In this paper, we characterize all maximal subsemigroups of PS(q) when |X|>q, and we extend MA to obtain maximal subsemigroups of PS(q) when |X|=q

On Semigroups of Orientation-preserving Transformations with Restricted Range

Let Xn be a chain with n elements (n ∈ ℕ), and let n be the monoid of all orientation-preserving transformations of Xn. In this article, for any nonempty subset Y of Xn, we consider the subsemigroup n(Y) of n of all transformations with range contained in Y: We describe the largest regular subsemigroup of n(Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type n(Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid ℛn of all either orientation-preserving or orientation-reversing transformations of Xn.info:eu-repo/semantics/publishedVersio

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

The original article has been updated. Open Access funding enabled and organized by Projekt DEAL.Gefördert im Rahmen des Projekts DEA