The Hahn embedding theorem for a class of residuated semigroups

TL;DR

This paper extends Hahn's embedding theorem to a specific class of residuated semigroups, providing a new representation using linearly ordered abelian groups and a novel partial sublex product construction.

Contribution

It introduces the partial sublex product construction and proves a representation theorem for certain residuated chains, generalizing Hahn's theorem to this algebraic setting.

Findings

Established a representation theorem for odd involutive commutative residuated chains.

Introduced the partial sublex product construction for residuated lattices.

Extended Hahn's embedding theorem to a new class of residuated semigroups.

Abstract

Hahn's embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn's theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial sublex product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial sublex product construction is presented.