Partial Order in the Normal Category Arising from Normal Bands

• Main Authors: Preenu, C S, Rajan, A R, Zeenath, K S
• Format: Journal Article
• Language: English
• Published: 03.05.2021
• Subjects: Mathematics - Category Theory; Mathematics - Group Theory
• DOI: 10.48550/arxiv.2105.01135

Advances in Mathematics: Scientific Journal, Volume 9, 2020 The notion of normal category was introduced by KSS Nambooripad in connection with the study of the structure of regular semigroups using cross connections\cite{nambooripad1994theory}. It is an abstraction of the category of principal left ideals of a regular semigroup. A normal band is a semigroup $B$ satisfying $a^2=a$ and $abca=acba$ for all $a,b,c \in B$. Since the normal bands are regular semigroups, the category $\mathcal{L}(B)$ of principal left ideals of a normal band $B$ is a normal category. One of the special properties of this category is that the morphism sets admit a partial order compatible with the composition of morphisms. In this article we derive several properties of this partial order and obtain a new characterization of this partial order.