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On the coset category of a skew lattice

Pita-Costa J.

Demonstratio Mathematica

2014

Vol. 47, nr 3

539--554

Skew lattices are noncommutative generalizations of lattices. The coset structure decomposition is an original approach to the study of these algebras describing the relation between its rectangular classes. In this paper, we will look at the category determined by these rectangular algebras and the morphisms between them, showing that not all skew lattices can determine such a category. Furthermore, we will present a class of examples of skew lattices in rings that are not strictly categorical, and present sufficient conditions for skew lattices of matrices in rings to constitute ^-distributive skew lattices.

Erratum to : On the coset structure of a skew lattice

Pita Costa J.

Demonstratio Mathematica

2013

Vol. 46, nr 2

245--246

On the coset structure of a skew lattice

Pita-Costa J.

Demonstratio Mathematica

2011

Vol. 44, nr 4

673-692

The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of sets where the Green's relation D is a congruence describing an adjunction to the category of lattices. In this paper we will discuss the relevance of this approach, revisit some known decompositions and relate the order structure of a skew lattice with its coset structure that describes the internal coset decomposition of the respective skew lattice.