Lattices of relative colour-families and antivarieties

Abstract

We consider general properties of lattices of relative colour-families and antivarieties. Several results generalise the corresponding assertions about colour-families of undirected loopless graphs, see [1]. Conditions are indicated under which relative colour-families form a lattice. We prove that such a lattice is distributive. In the class of lattices of antivarieties of relation structures of finite signature, we distinguish the most complicated (universal) objects. Meet decompositions in lattices of colour-families are considered. A criterion is found for existence of irredundant meet decompositions. A connection is found between meet decompositions and bases for anti-identities.

Antivarieties of unars

“…The relation → induces a partial order on the quotient set U f / ≡. The resulting poset is denoted by Core(U) (see [7]). …”

“…In [2,6,7], such formulas were called anti-identities. For any class K of algebraic systems, a K-antivariety is a class definable in K by some (possibly empty) set of anti-identities.…”

“…A characterization theorem for antivarieties, an analog of Birkhoff's HSP theorem, was proved in [6]. Lattices of antivarieties of algebraic systems were examined in [7].…”