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.
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