Searcharxiv⌕ Search

arXiv subjects

Karin Cvetko-Vah

Publications and source records attributed to Karin Cvetko-Vah.

14 recordsLinked to original sources
arXiv2021

On elementary, odd, semimagic and other classes of antilattices

An \emph{antilattice} is an algebraic structure based on the same set of axioms as a lattice except that the two commutativity axioms for $\land$ and $\lor$ are replaced by anticommutative counterparts. In this paper we study certain classes of antilattices, including elementary (no nontrivial subantilattices), odd (no subantilattices of order $2$), simple (no nontrivial congruences) and irreducible (not expressible as a direct product). In the finite case, odd antilattices are the same as Leech's \emph{Latin} antilattices which arise from the construction of semimagic squares from pairs of orthogonal Latin squares.

math.RA↗
arXiv2020

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

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 general, 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.

math.QA↗
arXiv2019

Duality for noncommutative frames

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 shadow 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.

math.RA↗
arXiv2019

Noncommutative Frames Revisited

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.

math.RA↗
arXiv2019

Regular Antilattices

Antilattices $(S;\lor, \land)$ for which the Green's 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.

math.RA↗
arXiv2019

Flat Coset Decompositions of Skew Lattices

Skew lattices are non-commutative generalizations of lattices, and the cosets represent the building blocks that skew lattices are built of. As by Leech's Second Decomposition Theorem any skew lattice embeds into a direct product of a left-handed skew lattice by a right-handed one, it is natural to consider the so called flat coset decompositions, i.e. decompositions of a skew lattice into right and left cosets, thus finding the smallest atoms that compose the structure.

math.RA↗
arXiv2017

Noncommutative frames

We explore algebraic properties of noncommutative frames. The concept of noncommutative frames is due to Le Bruyn, who introduced it in connection with noncommutative covers of the Connes-Consani arithmetic site.

math.RA↗
arXiv2017

What is a noncommutative topos?

In 1702.04949 noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a topological space. In this paper we extend this notion to noncommutative Grothendieck topologies and their associated noncommutative toposes of sheaves of sets.

math.RA↗
arXiv2016

On Skew Heyting Algebras

In the present paper we generalize the notion of a Heyting algebra to the non-commutative setting and hence introduce what we believe to be the proper notion of the implication in skew lattices. We list several examples of skew Heyting algebras, including Heyting algebras, dual skew Boolean algebras, conormal skew chains and algebras of partial maps with poset domains.

math.RA↗
arXiv2013

A non-commutative Priestley duality

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 classical 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.

math.RA↗
arXiv2011

Stone Duality for Skew Boolean Algebras with Intersections

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.

math.RA↗
arXiv2007

Semitransitive and transitive subsemigroups of the inverse symmetric semigroups

We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups.

math.GR↗
arXiv2007

Towards a skew lattice approach to quantum (computational) logic

In quantum mechanics, each observable is assigned a collection of projections. Two observables are compatible (can be measured simultaneously) if and only if any two projections that are assigned to them commute. This led to the study of noncommuting sets of idempotents which further led to the skew lattice theory that was founded and explored by Jonathan Leech. In the present paper we shall see how noncommuting projections can be regarded as generators of a skew lattice.

math.QA↗