On transformation semigroups which are ℬ𝒬‐semigroups
Sansanee Nenthein, Yupaporn Kemprasit · International Journal of Mathematics and Mathematical Sciences · 2006
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 of T(X), where ∅≠Y⊆X and . If W is a subspace of V, the subsemigroup of LF(V) will be defined analogously. In this paper, it is shown that is a ℬ𝒬‐semigroup if and only if Y = X, |Y| = 1, or |X| ≤ 3, and is a ℬ𝒬‐semigroup if and only if (i) W = V, (ii) W = {0}, or (iii) F = ℤ2, dimFV = 2, and dimFW = 1 .