Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups

TL;DR

This paper establishes that determining whether finite ordered complemented semigroups or lattice-ordered semigroups can be represented as binary relation algebras is undecidable, with extensions to infinite cases under certain conditions.

Contribution

It proves the undecidability of the representability problem for these algebraic structures, advancing understanding of their computational complexity.

Findings

Representability problems are undecidable for finite structures.

Undecidability extends to infinite representations with universal complementation.

Results impact the theoretical limits of algebraic logic and relation algebra representations.

Abstract

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with respect to a universal relation, this result can be extended to infinite representations of ordered complemented semigroups.