Products of Classes of Finite Structures

TL;DR

This paper investigates how various structural properties of finite classes are preserved under different product operations, revealing that such products generally do not produce new classes of theories.

Contribution

It provides a detailed analysis of property preservation under products of finite structures and shows that these products typically do not generate new theories.

Findings

Preservation of indivisibility under certain products

No new classes of theories are produced by these products

Interactions between properties and product types are characterized

Abstract

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this paper do not yield new classes of theories.