Inductive groupoids and cross-connections of regular semigroups

Abstract

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann-Schein-Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

A Tale of Two Categories: Inductive groupoids and Cross-connections

“…The two approaches of Nambooripad seemed unrelated if not orthogonal to each other. However, the present authors [7] established an equivalence between inductive groupoids and cross-connections that 'bypasses' regular semigroups in the sense that the equivalence between IG and Cr exposited in [7] was not a mere composition of the aforementioned categorical equivalences found in [22,23]. Still, the equivalence from [7] remained in the realm of regular semigroups: what we did is that we explored the inter-relationship between the idempotent structure and ideal structure in an arbitrary regular semigroup to establish how one can be retrieved from the other.…”

Structure theory of regular semigroups

“…The subject of Section 4.2 is an alternative description of the structure of regular semigroups due to Nambooripad which generalizes Grillet's approach (Section 3.2) for all regular semigroups. Additionally, we discuss a recent work by Muhammed and Volkov [44,45] on the relationship between Nambooripad's two approaches. We close this section by outlining a generalization of Nambooripad's description of regular semigroups via ordered groupoids over biordered sets for a class of semigroups with distinguished sets of idempotents.…”

“…Motivated by Theorems 4.1 and 4.2, Muhammed and Volkov have investigated the interrelations between cross-connections and regular inductive groupoids. In [44] they construct the regular inductive groupoid of a regular semigroup directly from the cross-connection representation of the semigroup, and vice versa, by analyzing the rather complicated relationship between the idempotent structure and ideal structure of an arbitrary regular semigroup. It should be emphasized that the equivalence of categories established is not the composition of the equivalences given in the proofs of Theorems 4.1 and 4.2.…”

Cross-connection structure of concordant semigroups

“…It must be mentioned here that this 'jump' in cross-connections from regular partially ordered sets to normal categories is indeed a 'giant leap' and it looks quite aloof from the ideas and techniques developed in [20]. But recently the first two authors of this article have established [2,3] an expected, but a completely non-trivial direct equivalence between the category of cross-connections and category of regular inductive groupoids 2 .…”

“…First, Nambooripad [20,Theorem 7.6] characterised those biordered sets which are pseudo-semilattices; in particular, he showed that a biordered set E is a pseudosemilattice if and only if for all e, f ∈ E, the sandwich set S(e, f ) contains exactly one element. Conversely, in [21,Theorem 2], he characterised all pseudosemilattices which form biordered sets and showed that the class of pseudo-semilattices forms a variety of algebras of type (2).…”