Lattice Ordered Polynomial Algebras

Abstract

In this paper we define a lattice order on a set F of binary functions. We then provide necessary and sufficient conditions for the resulting algebra \(\mathfrak{L}\)F to be a distributive lattice or a Boolean algebra. We also prove a ‘Cayley theorem’ for distributive lattices by showing that for every distributive lattice \(\mathfrak{L}\), there is an algebra \(\mathfrak{L}\)F of binary functions, such that \(\mathfrak{L}\) is isomorphic to\(\mathfrak{L}\)F and we show that \(\mathfrak{L}\)F is a distributive lattice iff the operations ∨ and ∧ are idempotent and cummutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of ∨ and ∧. We also examine the equational properties of an Algebra \(\mathfrak{U}\) for which \(\mathfrak{L}_\mathfrak{U}\), now defined on the set of binary \(\mathfrak{U}\)-polynomials is a lattice or Boolean algebra.