On transformation semigroups which are ℬ𝒬-semigroups

Abstract

A semigroup whose bi-ideals and quasi-ideals coincide is called a ℬ𝒬-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 ℬ𝒬-semigroup. Then both T(X) and LF(V) are ℬ𝒬-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 ℬ𝒬-semigroup if and only if Y=X, |Y|=1, or |X|≤3, and L¯F(V,W) is a ℬ𝒬-semigroup if and only if (i) W=V, (ii) W={0}, or (iii) F=ℤ2, dimFV=2, and dimFW=1 .