Weakly extendable partial clones on an infinite set

Abstract

We consider local partial clones defined on an uncountable set E having the form Polp(${\mathfrak{R}}$), where ${\mathfrak{R}}$ is a set of relations on E. We investigate the notion of weak extendability of partial clones of the type Polp(${\mathfrak{R}}$) (in the case of E countable, this coincides with the notion of extendability previously introduced by the author in 1987) which allows us to expand to uncountable sets results on the characterization of Galois-closed sets of relations as well as model-theoretical properties of a relational structure ${\mathfrak{R}}$. We establish criteria for positive primitive elimination sets (sets of positive primitive formulas over ${\mathfrak{R}}$ through which any positive primitive definable relation over ${\mathfrak{R}}$ can be expressed without existential quantifiers) for finite ${\mathfrak{R}}$ as well as for ${\mathfrak{R}}$ having only finite number of positive primitive definable relations of any arity. Emphasizing the difference between countable and uncountable sets, we show that, unlike in the countable case, the characterization of Galois-closed sets InvPol(${\mathfrak{R}}$) (that is, all relations which are invariant under all operations from the clone Pol(${\mathfrak{R}}$) defined on an uncountable set) cannot be obtained via the application of finite positive primitive formulas together with infinite intersections and unions of updirected sets of relations from ${\mathfrak{R}}$.