Azeef Muhammed P A | Searcharxiv

arXiv2017

Normal category of partitions of a set

Let $T_X$ be the semigroup of all non-invertible transformations on an arbitrary set $X$. It is known that $T_X$ is a regular semigroup. The principal right(left) ideals of a regular semigroup $S$ with partial left(right) translations as morphisms form a normal category $\mathcal{R}( S )$($\mathcal{L}( S )$). Here we consider the category $Π(X)$ of partitions of a set $X$ and show that it admits a normal category structure and that $Π(X)$ is isomorphic to the category $\mathcal{R}( T_X )$. We also consider the normal dual $N^\ast \mathscr{P}(X)$ of the power-set category $\mathscr{P}(X)$ associated with $X$ and show that $N^\ast \mathscr{P}(X)$ is isomorphic to the partition category - $Π(X)$ of the set $X$.

math.GR↗

arXiv2015

Cross-connections of completely simple semigroups

A completely simple semigroup S is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that S is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup. In the study of structure theory of regular semigroups, K.S.S. Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left(right) ideals using cross-connections. A normal category C is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories C and D is a local isomorphism frm D to N*C where N*C is the normal dual of the category C. In this paper, we identify the normal categories associated with a completely simple semigroup S and show that the semigroup of normal cones TL(S) is isomorphic to a semi-direct product of semigroups. We characterize the cross-connections in this case and show that each sandwich matrix P correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.

math.GR↗