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.