Finite models for positive combinatorial and exponential algebra

TL;DR

This paper demonstrates the existence of finite models for the equational theory of certain algebraic structures using hypergraphs, and explores their logical properties and representations.

Contribution

It introduces finite models for the equational theory of semirings with extended operations, showing the non-axiomatizability and decidability aspects.

Findings

Finite models exist for the equational theory of nonnegative integer semirings.

Decidability of equational logical entailment is established.

Representation of ordinal $oldsymbol{ extepsilon_0}$ via factorial functions.

Abstract

We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of nonnegative integers whose equational theory has no finite axiomatisation, and show this also holds if factorial, fixed base exponentiation and operations for binomial coefficients are adjoined. We also derive the decidability of the equational logical entailment operator $\vdash$ for antecedents true on $\mathbb{N}$ by way of a form of the finite model property. Two appendices contain additional basic development of combinatorial operations. Amongst the observations are an eventual dominance well-ordering of combinatorial functions and consequent representation of the ordinal $\epsilon_0$ in terms of factorial functions; the equivalence of the equational logic of combinatorial algebra over the natural numbers and over the positive reals; and a candidate list of elementary axioms.