On superrings of polynomials and algebraically closed multifields

TL;DR

This paper explores the model theory of multialgebras, focusing on algebraically closed structures and polynomial superrings, extending the understanding of hyperring and multifield algebraic systems.

Contribution

It introduces the class of algebraically closed multialgebras and provides a variant proof of quantifier elimination, based on new results on superrings of polynomials.

Findings

Established model-theoretic properties of algebraically closed multialgebras.

Proved quantifier elimination for the class of these multialgebras.

Extended the theory of superrings of polynomials in this context.

Abstract

The concept of multialgebraic structure -- an "algebraic like" structure but endowed with multiple valued operations -- has been studied since the 1930's; in particular, the concept of hyperrings was introduced by Krasner in the 1950's. Some general algebraic study has been made on multialgebras: see for instance \cite{golzio2018brief} and \cite{pelea2006multialgebras}. More recently the notion of multiring have obtained more attention: a multiring is a lax hyperring, satisfying an weak distributive law, but hyperfields and multifields coincide. Multirings has been studied for applications in abstract quadratic forms theory (\cite{marshall2006real}, \cite{worytkiewiczwitt2020witt}) and tropical geometry (\cite{jun2015algebraic}); a more detailed account of variants of concept of polynomials over hyperrings is even more recent (\cite{jun2015algebraic}, \cite{ameri2019superring}). In the present work we start a model-theoretic oriented analysis of multialgebras introducing the class of algebraically closed and providing variant proof of quantifier elimination flavor, based on new results on superring of polynomials (\cite{ameri2019superring}).