On transformation semigroups which are ℬ-semigroups

A semigroup whose bi-ideals and quasi-ideals coincide is called a -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 -semigroup. Then both T(X) and LF(V) are -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) | Yα ⊆ Y}. If W is a subspace of V , the subsemigroup LF(V ,W) of LF(V) will be defined analogously. In this paper, it is shown that T(X ,Y) is a -semigroup if and only if Y = X , |Y | = 1, or |X| ≤ 3, and LF(V ,W) is a -semigroup if and only if (i) W = V , (ii) W = {0}, or (iii) F = Z2, dimF V = 2, and dimFW = 1.