Preservation of superamalgamation by expansions

TL;DR

This paper explores how the superamalgamation property in ordered structures can imply strong amalgamation in expanded classes, with implications for algebraic logic and model theory.

Contribution

It demonstrates that superamalgamation leads to strong amalgamation in classes with added operations under certain conditions, extending its relevance.

Findings

Superamalgamation implies strong amalgamation in classes with added operations.

The theory of join semilattices with a closure operation has a model completion.

Universal consequences of certain algebraic theories are decidable.

Abstract

The superamalgamation property is a strong form of the amalgamation property which applies to ordered structures; it has found many applications in algebraic logic. We show that superamalgamation has some interest also from the pure model-theoretical point of view. Under a completion assumption, we prove that the superamalgamation property for some class of ordered structures implies strong amalgamation for classes with added operations, including isotone, idempotent, extensive, antitone and closure operations. Thus, for example, partially ordered sets, semilattices, lattices, Boolean algebras and Heyting algebras with an isotone extensive operation (or an operation as above) have the strong amalgamation property. The theory of join semilattices with a closure operation has model completion. The set of universal consequences of the theory of Boolean algebras (or posets, semilattices, distributive lattices) with a closure or isotone, etc., operation is decidable.