Introduction

1. Introduction

The concept of cross-connection between partially ordered sets was originally introduced by Grillet [1974a,b,c] in order to classify the class of fundamental regular semigroups. Grillet deﬁnes a cross-connection between two regular partial ordered sets $I$ and $Λ$ as a pair of order-preserving mappings

Γ : I \to Λ^{*} and Δ : Λ \to I^{*}

(1)

satisfying certain axioms, where $I^{*}$ denote the dual of $I$ . See Grillet [1974b] 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 $Λ (S)$ of all principal left ideals of $S$ under inclusion and the partially ordered set $I (S)$ of all principal right ideals. In Nambooripad [1994], this was generalized to cross-connections between normal categories. Again, given a regular semigroup $S$ , category $𝕃 0 (S)$ of all principal left ideals is normal and similarly, the category $ℝ 0 (S)$ of all principal right ideals is also normal (see Propositions 4.3 and 4.4 for deﬁnitions of these categorioes). We can see that every regular semigroup $S$ induces a cross-connection between categories $𝕃 0 (S)$ and $ℝ 0 (S)$ .

We observe that the conﬁguration that occur in Grillet’s deﬁnition as well as its generalization in Nambooripad [1994] 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 ( $𝒮$ -category for short). A cross-connection between to $𝒮$ -categorys $𝒞$ and $𝒟$ consists of two set-valued bifunctors

Γ : 𝒞 \times 𝒟 \to S e t and Δ : 𝒞 \times 𝒟 \to S e t

and a natural isomorphism

χ : Γ \to Δ

between them. We can show that every cross-connection determines a semigroup. Also for any semigroup $S$ there is a cross-connection between $𝕃 0 (S)$ and $ℝ 0 (S)$ and there is a natural representation of $S$ by the cross-connection semigrpoup determined by the cross-connection between $𝕃 0 (S)$ and $ℝ 0 (S)$ .

In the following discussion, we will follow [Nambooripad, 1994] 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.

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