Groupoids on a skew lattice of objects

TL;DR

This paper explores algebraic structures combining skew lattices with groupoids of isomorphisms, extending classical models and providing a new framework for understanding these mathematical entities.

Contribution

It introduces a novel algebraic structure that extends skew lattices with locally invertible elements and establishes a correspondence with certain groupoids.

Findings

Defined a new algebraic structure with an orthodox semigroup signature

Established conditions for actions of objects on morphisms

Reconstructed groupoids from the algebraic structures

Abstract

Motivated by some alternatives to the classical logical model of boolean algebra, this paper deals with algebraic structures which extend skew lattices by locally invertible elements. Following the meme of the Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of isomorphisms) in which the set of objects carries the structure of a skew lattice. The objects act on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Conditions are placed on the actions, from which pseudoproducts may be defined. This gives an algebra of signature (2,2,1), in which each binary operation has the structure of an orthodox semigroup. In the reverse direction, a groupoid of the kind described may be reconstructed from the algebra.