SFN:Math-test

$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}} \newcommand\sbcat{\mathit{Category}} \newcommand{\lcat}[1][S]{\mathbb{L}(#1)} \newcommand{\rcat}[1][S]{\mathbb{R}(#1)} \newcommand{\cc}{\mathcal{C}} \newcommand{\cd}{\mathcal{D}} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\set}{\mathsf{Set}} \newcommand{\sbcat}{\mathcal{S}-\text{Category}} \newcommand{\preord}{\mathcal{P}} \newcommand{\scat}{\mathsf{Cat_{o}}} \newcommand{\vrt}[1]{\boldsymbol{\mathfrak{v}}{#1}} \newcommand{\Grp}{\mathsf{Grp}} \newcommand{\abgrp}{\mathsf{Ab}} \def\inc#1#2{{\jmath}_{{#1}}^{{#2}}} \newcommand{\cod}{\operatorname{cod}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\e}{\epsilon} \newcommand{\im}{\operatorname{im}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\vmap}[1][{}]{{#1}_{\mathfrak{A}}} \def\relax{} $

Introduction

The concept of cross-connection between partially ordered sets was originally introduced by ??? in order to classify the class of fundamental regular semigroups. Grillet defines a cross-connection between two regular partial ordered sets $I$ and $\Lambda $ as a pair of order-preserving mappings \begin{equation} \label{eq:1} \Gamma:I\to\Lambda^{*}\quad\text{and}\quad\Delta:\Lambda\to I^{*} \end{equation} satisfying certain axioms, where $I^{*}$ denote the dual of $I$. See ? for definition of regular partially ordered sets, duals and axioms for cross-connections. Every regular semigroup $S$ induces a cross-connection between the partially ordered set $\Lambda S$ of all principal left ideals of $S$ under inclusion and the partially ordered set $IS$ of all principal right ideals. In ?, this was generalized to cross-connections between normal categories. Again, given a regular semigroup $S$, category $\lcat $ of all principal left ideals is normal and similarly, the category $\rcat $ of all principal right ideals is also normal (see Propositions ?? and ?? for definitions of these categorioes). We can see that every regular semigroup $S$ induces a cross-connection between categories $\lcat $ and $\rcat $.

We observe that the configuration that occur in Grillet’s definition as well as its generalization in ? occures in many areas in mathematics. Our aim here is to describe cross-connections of a more general class of categories. We first identify an appropriate class of categories for which cross-section can be defined. These are categories with subobjects in which objects are sets, morhisms are mappings and inclusions are inclusions of sets. We refer to these categories as set-based categories ($\relax \mathcal {S}$-category for short). A cross-connection between to $\relax \mathcal {S}$-categorys $\cc $ and $\cd $ consists of two set-valued bifunctors \[ \Ga :\cc \times \cd \to \set \quad \text {and}\quad \De :\cc \times \cd \to \set \] and a natural isomorphism \[ \chi :\Ga \to \De \] between them. We can show that every cross-connection determines a semigroup. Also for any semigroup $S$ there is a cross-connection between $\lcat $ and $\rcat $ and there is a natural representation of $S$ by the cross-connection semigrpoup determined by the cross-connection between $\lcat $ and $\rcat $.

In the following discussion, we will follow (?) for ideas regarding categories, subobject relations, etc. Moreover, to save repetition, we shall assume that categories under consideration are small, unless otherwise provided, so that they may be treated as partial algebras.