Coset laws for categorical skew lattices

Open problems from NCS 2018

“…Problem 9.3. The index theorems presented for some varieties of skew lattices in [3,17] and [19] show a combinatorial perspective on these algebras as a direct consequence of their coset structure. Can we always obtain such results?…”

On the coset structure of distributive skew lattices

“…Transversals allows us to define the index of B in A, first presented in [4] and denoted by The nature of the coset structure of a skew lattice permits such instances of combinatorial implications that are arise frequently in the literature. These combinatorial properties enabled us to derive coset laws to characterize varieties of symmetric skew lattices and cancellative skew lattices, in [15] and [4] respectively, or in the first author's research in [5,6,18], and his PhD thesis in [19]. Similar characterizations are also available for normal (and conormal) skew lattices, as a direct consequence of Proposition 2.9.…”

“…[18]). A skew lattice S is normal iff for each comparable pair of Dclasses A > B in S and all x, x ∈ B, A ∧ x ∧ A = A ∧ x ∧ A. Dually, S is conormal iff for all comparable pairs of D-classes A > B in S and all x, x ∈ A, B∨x∨B = B∨x ∨B.Proposition 2.9 above essentially states that S is a normal skew lattice if and only if, for each comparable pair of D-classes A > B in S and for each x ∈ A, a unique y ∈ B exists such that y ≤ x.…”

“…[18]). Given a skew chain A > B > C in a skew lattice S with both A and C finite, then|B| ≥ ω[A, B] • ω[B, C] ω[A, C] .Furthermore, A > B > C is strictly categorical if and only if|B| = ω[A, B] • ω[B, C] ω[A, C] (5.2)so that in particular B is also finite.…”

My journey into noncommutative lattices and their theory

“…Here is a nice result: a strictly categorical skew lattice S is distributive iff its maximal lattice image S/D is distributive. A nice counting theorem quoted in this paper came from João's 2012 Algebra Universalis publication "Coset laws for categorical skew lattices" [53]: While on the topic of skew lattice architecture, further research in this area has been carried out by Karin and/or João. The relevant published papers, all appearing since 2010, are often recognized by such phrases as "coset structure" or "coset laws" appearing in the title.…”