The strong amalgamation property into union

TL;DR

This paper introduces the Strong Amalgamation Property into union (SAPU), a strengthened form of SAP that allows merging structures over their union, enabling preservation theorems and broad applicability in algebraic and relational theories.

Contribution

The paper defines SAPU, demonstrates its preservation under theory merging, and shows its applicability to various relational and algebraic structures with complex properties.

Findings

SAPU holds for theories with multiple binary relations satisfying various properties.

SAPU is applicable to algebraic structures like Maltsev varieties, bounded directoids, and order algebras.

Preservation theorems enable merging theories while maintaining SAPU.

Abstract

We consider the situation in which some class of structures has the Strong Amalgamation Property (SAP) with the further requirement that the amalgamating structure can be taken over the set theoretical union of (the images of) the domains of the structures to be amalgamated. We call this property SAPU. The main advantage of SAPU over SAP is that there are many preservation theorems showing that we can merge different theories with SAPU still obtaining a theory with SAPU, hence with SAP. In particular, we get SAPU for various theories with many binary relations, each relation satisfying any set of properties chosen among transitivity, reflexivity, symmetry, antireflexivity, antisymmetry. We may also add unary operations, possibly satisfying some coarseness, isotonicity and closure conditions. SAPU is not limited to relational theories: the varieties defining the most usual Maltsev conditions in universal algebra have SAPU. Other examples include bounded directoids, order algebras and various generalization.