On elementary, odd, semimagic and other classes of antilattices

TL;DR

This paper investigates various classes of antilattices, algebraic structures similar to lattices but with anticommutative properties, focusing on their properties, classifications, and connections to Latin squares.

Contribution

It introduces and analyzes classes of antilattices such as elementary, odd, simple, and irreducible, and links odd antilattices to Latin antilattices from semimagic square constructions.

Findings

Elementary antilattices have no nontrivial subantilattices.

Odd antilattices coincide with Latin antilattices in finite cases.

Characterization of simple and irreducible antilattices provided.

Abstract

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.