Polyadic-like algebras without the amalgamation property

TL;DR

This paper presents an example of a polyadic-like algebraic variety with infinitary substitutions that notably lacks the amalgamation property, challenging previous assumptions about such classes.

Contribution

It introduces a novel example of a polyadic-like algebraic class with infinitary substitutions that does not possess the amalgamation property.

Findings

Provides the first known example of such a class without amalgamation

Shows that infinitary substitutions do not guarantee amalgamation

Highlights limitations in the algebraic structure theory of polyadic-like algebras

Abstract

Usually when we have polyadic-like algebras, meaning that we have infinitary substitutions (that is substitutions moving infinitely many points) in the similarity type, then we get the superamalgamation property especially if this class of algebras happen to be a variety. This for example happens for (full) polyadic algebras, full Heyting algebras and reducts of those using only finitely many infinitary substitutions, like Sains Boolean and Heyting algebras. (The last is studied by Sayed Ahmed) . In cylindric-like algebras (like quasi-polyadic algebras) when we do not have infinitary substitutions, we do not get even the amalgamation property . In this paper, we give an example of a polyadic like variety (we have infinitary substitutions, in fact infinitely many of them) for which the amalgamation property fails.