A model theoretic construction for layered semifields

TL;DR

This paper develops a model theoretic framework for uniform layered semifields, proving their theory is complete, model complete, and admits quantifier elimination, with applications to max-plus algebras.

Contribution

It introduces a new model theoretic construction for layered semifields and proves their theory is complete and model complete, extending understanding of their algebraic properties.

Findings

Theory of uniform L-layered divisibly closed semifields is complete.

The theory admits quantifier elimination and is model complete.

Model completeness ensures polynomial equality extends to all uniform L-layered extensions.

Abstract

In this paper we introduce a model theoretic construction for the theories of uniform layered domains and semifields introduced in the paper of Izhakian, Knebusch and Rowen. We prove that, for a given layering semiring L, the theory of uniform L-layered divisibly closed semifields is complete. In the process of doing so, we prove that this theory has quantifier elimination and consequently is model complete. Model completeness of uniform L-layered divisibly closed has some important consequences regarding the uniform L-layered semifields theory. One example involves equating polynomials. Namely, model completeness insures us that if two polynomials are equal over a divisibly closed uniform L-layered semifield, then they are equal over any divisibly closed uniform L-layered extension of that semifield, and thus over any uniform L-layered domain extending the semifield (as it is contained in its divisible closure of its semifield of fractions). At the end of this paper we apply our results to the theory of max-plus algebras as a special case of uniform L -layered domains.