Subalgebras of FA-presentable algebras

TL;DR

This paper investigates the properties of FA-presentable algebras, showing that while the class of finitely generated FA-presentable algebras is not always closed under subalgebras, certain subclasses with unary operations are closed and decidable.

Contribution

It proves that finitely generated subalgebras of unary FA-presentable algebras are FA-presentable and that the membership problem for these subalgebras is decidable.

Findings

Finitely generated FA-presentable algebras are not closed under subalgebras.

Subalgebras of unary FA-presentable algebras are FA-presentable.

Membership problem for unary FA-presentable subalgebras is decidable.

Abstract

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. However, it is proven that a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras, and that the membership problem for such subalgebras is decidable.