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Lattice Ordered Polynomial Algebras

Authors:	Padmanabhan R Penner P

Institution:	(1) Dept. of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. E-mail

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.

Keywords:	Boolean algebra Cayley theorems hyperidentities lattice order polynomial algebra Urbanik algebra
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