On elementary, odd, semimagic and other classes of antilattices

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 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 Latin antilattices which arise from the construction of semimagic squares from pairs of orthogonal Latin squares.