Variants of finite full transformation semigroups

The variant of a semigroup S with respect to an element a S, denoted Sa, is the semigroup with underlying set S and operation ∗ defined by x∗y = xay for x,y S. In this paper, we study variants Xa of the full transformation semigroup X on a finite set X. We explore the structure of Xa as well as its subsemigroups Reg(Xa) (consisting of all regular elements) and RegXa (consisting of all products of idempotents), and the ideals of Reg(Xa). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size. Note: Some of the scientific symbols cannot be represented correctly in the abstract. Please read with caution and refer to the original publication