Cross-connections and variants of the full transformation semigroup

Cross-connection theory propounded by Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant T_X ^θ of the full transformation semigroup (TX, ·) for an arbitrary θ ∈ TX is the semigroup T_X ^θ = (TX, ∗) with the binary operation α ∗ β = α · θ · β where α, β ∈ TX . In this article, we describe the ideal structure of the regular part Reg(T_X ^θ ) of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of Reg(T_X ^θ ) and describe how they are cross-connected by a functor induced by the sandwich transformation θ. This leads us to a structure theorem for the semigroup and gives the representation of Reg(T_X ^θ ) as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.