We characterize the left-handed noncommutative frames that arise from sheaves on topological spaces. Further, we show that a general left-handed noncommutative frame $A$ arises from a sheaf on the dissolution locale associated to the commutative shad
ow of $A$. Both constructions are made precise in terms of dual equivalences of categories, similar to the duality result for strongly distributive skew lattices in arXiv:1206.5848.
In this note, we correct an error in arXiv:1702.04949 by adding an additional assumption of join completeness. We demonstrate with examples why this assumption is necessary, and discuss how join completeness relates to other properties of a skew lattice.
Antilattices $(S;lor, land)$ for which the Greens equivalences $mathcal L_{(lor)}$, $mathcal R_{(lor)}$, $mathcal L_{(land)}$ and $mathcal R_{(land)}$ are all congruences of the entire antilattice are studied and enumerated.
In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in gen
eral, and thus different from solutions obtained from braces and other algebraic structures. Our main result concerns a description of a set-theoretic solution of the Yang-Baxter equation, obtained from an arbitrary skew lattice. We also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets.
We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classica
l Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and so-called override. Our duality shows that there is a canonical choice for this embedding. Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.
We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with
intersections is dual to the category of surjective etale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section.
