Infinitely many reducts of homogeneous structures

TL;DR

This paper demonstrates that certain infinite homogeneous structures, like vector spaces over finite fields and Boolean algebras, have infinitely many first order definable reducts, using constructions related to Reed--Muller codes.

Contribution

It proves the existence of infinitely many reducts for specific infinite homogeneous structures, expanding understanding of their definability properties.

Findings

Countably infinite dimensional pointed vector space over finite fields has infinitely many reducts.

Countable homogeneous Boolean algebra has infinitely many reducts.

Construction over the 2-element field relates to Reed--Muller codes.

Abstract

It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts. Our construction over the 2-element field is related to the Reed--Muller codes.