On elementary, odd, semimagic and other classes of antilattices

Abstract

An antilattice is an algebraic structure based on the same set of axioms as a lattice except that the two commutativity axioms for ∧ and ∨ are replaced by anticommutative counterparts. In this paper, we study certain classes of antilattices, including elementary (no nontrivial subantilattices), odd (no subantilattices of order [Formula: see text]), simple (no nontrivial congruences) and irreducible (not expressible as a direct product). In the finite case, odd antilattices are the same as Leech’s Latin antilattices which arise from the construction of semimagic squares from pairs of orthogonal Latin squares.