Cross-connections and variants of the full transformation semigroup

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 $\mathscr{T}_X^\theta$ of the full transformation semigroup $(\mathscr{T}_X,\cdot)$ for an arbitrary $\theta \in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^\theta= (\mathscr{T}_X,\ast)$ with the binary operation $\alpha \ast \beta = \alpha\cdot\theta\cdot\beta$ where $\alpha, \beta \in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part $Reg(\mathscr{T}_X^\theta)$ of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of $Reg(\mathscr{T}_X^\theta)$ 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 $Reg(\mathscr{T}_X^\theta)$ as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup

Qubit Entanglement Driven by Remote Optical Fields

We examine the entanglement between two qubits, supposed to be remotely located and driven by independent quantized optical fields. No interaction is allowed between the qubits, but their degree of entanglement changes as a function of time. We report a collapse and revival of entanglement that is similar to the collapse and revival of single-atom properties in cavity QED.Comment: v3, major changes, published in Optics Letter