Regular identities in lattices

Abstract:

An algebraic system $\mathfrak {A} = \langle A; + , \circ \rangle$ is called a quasilattice if the two binary operations + and $\circ$ are semilattice operations such that the natural partial order relation determined by + enjoys the substitution property with respect to $\circ$ and vice versa. An identity “$f = g$” in an algebra is called regular if the set of variables occurring in the polynomial $f$ is the same as that in $g$. It is called $n$-ary if the number of variables involved in it is at the most $n$. In this paper we show that the class of all quasilattices is definable by means of ternary regular lattice identities and that these identities span the set of all regular lattice identities and that the arity of these defining equations is the best possible. From these results it is deduced that the class of all quasilattices is the smallest equational class containing both the class of all lattices and the class of all semilattices in the lattice of all equational classes of algebras of type $\langle 2,2\rangle$ and that the lattice of all equational classes of quasilattices is distributive.