Fixed-point theory in the varieties D_n

Sabine Frittella (1) , Luigi Santocanale (2)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille
2 LIF - Laboratoire d'informatique Fondamentale de Marseille - UMR 6166

The varieties of lattices D_n, n >=0, were introduced in [Nation90] and studied later in [Semenova05]. 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 φ^2(\bot) = φ(\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 Dn, for any lattice term φ(x) and for x in {\bot,\top}. Moreover, we prove that the equations φ^{n + 1}(x) = φ^{n}(x), x = \bot,\top, might not hold in the variety D_n nor in the variety D_n \cap (D_n)^{op}, where (D_n)^{op} is the variety containing the lattices L^{op}, for L in D^{n}.

Domaines: Logique en informatique [cs.LO]; Mathématiques générales [math.GM]

Sabine Frittella, Luigi Santocanale. Fixed-point theory in the varieties D_n. RAMICS, Apr 2014, Marienstatt im Westerwald, Germany. pp.446--462, ⟨10.1007/978-3-319-06251-8_27⟩. ⟨hal-01260836⟩