Regular elements of some order-preserving transformation semigroups

Abstract Let X be a chain and OT (X) the full order-preserving transformation semigroup on X. In this paper, we give a necessary and sufficient condition for an element of OT (X) to be regular. For ∅ = Y ⊆ X, we may count the order-preserving transformation semigroup OT (X, Y ) = {α ∈ OT (X) | ran α ⊆ Y } as a generalization of OT (X). In addition, we show that an element α ∈ OT (X, Y ) is regular in OT (X, Y ) if and only if ran α = Y α and α is regular in OT (X)

Cross-connections of linear transformation semigroups

Cross-connection theory developed by Nambooripad is the construction of a regular semigroup from its principal left (right) ideals using categories. We use the cross-connection theory to study the structure of the semigroup Sing(V) of singular linear transformations on an arbitrary vector space V over a field K. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the variants of the linear transformation semigroup. © 2018, Springer Science+Business Media, LLC, part of Springer Nature

On transformation semigroups which are ℬ-semigroups

A semigroup whose bi-ideals and quasi-ideals coincide is called a ℬ-semigroup. The full transformation semigroup on a set X and the semigroup of all linear transformations of a vector space V over a field F into itself are denoted, respectively, by T(X) and LF(V). It is known that every regular semigroup is a ℬ-semigroup. Then both T(X) and LF(V) are ℬ-semigroups. In 1966, Magill introduced and studied the subsemigroup T¯(X,Y) of T(X), where ∅≠Y⊆X and T¯(X,Y)={α∈T(X,Y)|Yα⊆Y}. If W is a subspace of V, the subsemigroup L¯F(V,W) of LF(V) will be defined analogously. In this paper, it is shown that T¯(X,Y) is a ℬ-semigroup if and only if Y=X, |Y|=1, or |X|≤3, and L¯F(V,W) is a ℬ-semigroup if and only if (i) W=V, (ii) W={0}, or (iii) F=ℤ2, dimFV=2, and dimFW=1

Green's relations, regularity and abundancy for semigroups of quasi-onto transformations

Let V be an infinite-dimensional vector space and for every infinite cardinal n such that n≤dimV, let AE(V,n) denote the semigroup of all linear transformations of V whose defect is less than n. In 2009, Mendes-Gonçalves and Sullivan studied the ideal structure of AE(V,n). Here, we consider a similarly-defined semigroup AE(X,q) of transformations defined on an infinite set X. Quite surprisingly, the results obtained for sets differ substantially from the results obtained in the linear setting.This research was financed by FEDER Funds through “Programa Operacional Factores de Competitividade – COMPETE” and by Portuguese Funds through FCT - “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, within the project PEst-C/MAT/UI0013/2011