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.
CITATION STYLE
Muhammed, P. A. A., & Volkov, M. V. (2019). Inductive groupoids and cross-connections of regular semigroups. Acta Mathematica Hungarica, 157(1), 80–120. https://doi.org/10.1007/s10474-018-0888-6
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