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 \(L \in \mathcal{D}_{n}\).