Cross-connections and variants of the full transformation semigroup

Abstract

Cross-connection theory propounded by K. S. S. Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant ${\mathcal{T}}_X^{\theta }$ of the full transformation semigroup $({\mathcal{T}}_X,\cdot )$ for an arbitrary $\theta \in {\mathcal{T}}_X$ is the semigroup ${\mathcal{T}}_X^{\theta }=({\mathcal{T}}_X,\ast )$ with the binary operation $\alpha \ast \beta =\alpha \cdot \theta \cdot \beta$ where $\alpha ,\beta \in {\mathcal{T}}_X$. In this article, we describe the ideal structure of the regular part $Reg({\mathcal{T}}_X^{\theta })$ of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of $Reg({\mathcal{T}}_X^{\theta })$ and describe how they are \emph{cross-connected} by a functor induced by the sandwich transformation $\theta$. This lead us to a structure theorem for the semigroup and give the representation of $Reg({\mathcal{T}}_X^{\theta })$ as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.

References in corpus (3)

• Variants of finite full transformation semigroups
• Cross-connections of completely simple semigroups
• Cross-connections of the singular transformation semigroup

Cited by in corpus (1)

• Cross-connections of linear transformation semigroup