On the coset laws for skew lattices

Flat coset decompositions of skew lattices

“…This approach enables us to classify certain varieties of skew lattices. The results of Section 4 were motivated by the earlier studies of [17], [5], [21], [7], [22] and [23]. We demonstrate the impact of the study of the flat coset structure in the case of skew lattices of matrices, following the work of [8], [9] and [4].…”

“…The following technical result was proven in [7]. We shall make frequent use of this result in the remainder of the paper.…”

“…Dual results hold, having a similar statement (cf. [7]). All cosets and image sets are rectangular sub skew lattices.…”

“…In the remainder of the paper we will give a further insight to the flat coset decomposition of cancellative skew lattices for which the lattice image is distributive, and therefore the ones permitting the coset laws established in [7]. Remark 4.3.1.…”

“…Already in the 1989 foundation paper [15] certain aspects of the coset structure of a skew lattice were analyzed, however it was fully explored in [17] where Leech studied what he referred to as the global geometry of skew lattices. The coset structure was later used in [6] and [7] to characterize certain sub-varieties of skew lattices, and in [14] for the purpose of studying the distributivity of skew lattices, an approach proposed in [23]. By Leech's First Decomposition Theorem a skew lattice is a lattice of its Green's Dclasses.…”