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Groupoids on a skew lattice of objects. (English) Zbl 1481.20195

Summary: 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.

MSC:

20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
20M19 Orthodox semigroups
06B75 Generalizations of lattices
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References:

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