arXiv subjects
Publications and source records attributed to Jonathan Leech.
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 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.
We study the structure and properties of free skew Boolean algebras. For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements and calculate the number of such elements. These results are used to study minimal generating sets of finite skew Boolean algebras. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.
Distributive skew lattices satisfying $x\wedge (y\vee z)\wedge x = (x\wedge y\wedge x) \vee (x\wedge z\wedge x)$ and its dual are studied, along with the larger class of linearly distributive skew lattices, whose totally preordered subalgebras are distributive. Linear distributivity is characterized in terms of the behavior of the natural partial order between comparable $\DD$-classes. This leads to a second characterization in terms of strictly categorical skew lattices. Criteria are given for both types of skew lattices to be distributive.
Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.