Fixed-Point Theory in the Varieties $\mathcal{D}_{n}$

Abstract

The varieties of lattices $\mathcal{D}_n$, n ≥ 0, were introduced in [Nat90] and studied later in [Sem05]. These varieties might be considered as generalizations of the variety of distributive lattices which, as a matter of fact, coincides with $\mathcal{D}_{0}$. It is well known that least and greatest fixed-points of terms are definable on distributive lattices; this is an immediate consequence of the fact that the equation $\phi^{2}(\bot) = \phi(\bot)$ holds on distributive lattices, for any lattice term φ(x). In this paper we propose a generalization of this fact by showing that the identity φ ^n + 2(x) = φ ^n + 1(x) holds in $\mathcal{D}n$, for any lattice term φ(x) and for $x \in \{\top,\bot\}$. Moreover, we prove that the equations φ ^n + 1(x) = φ ⁿ(x), $x = \bot,\top$, do not hold in the variety $\mathcal{D}_{n}$ nor in the variety $\mathcal{D}_{n} \cap \mathcal{D}_{n}^{op}$, where $\mathcal{D}_{n}^{op}$ is the variety containing the lattices L ^op, for $L \in \mathcal{D}_{n}$.