Fixed-point theory in the varieties Dn

Abstract

The varieties of lattices Dn, 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 D0. 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 φ2(⊥) = φ(⊥) 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 φn + 2(x) = φn + 1(x) holds in Dn, for any lattice term φ(χ) and for χ ∈ {⊤, ⊥}. Moreover, we prove that the equations φn + 1(χ) = φn (χ), χ = ⊥, ⊤, do not hold in the variety nor in the variety Dn ∩ Dnop, where Dnop is the variety containing the lattices Lop, for L ∈ D n. © 2014 Springer International Publishing.