  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.
