Representing quantum structures as near semirings

TL;DR

This paper introduces near semirings with involution to represent quantum structures like basic algebras and orthomodular lattices, providing new algebraic frameworks and decomposition theorems.

Contribution

It generalizes semiring theory to include near semirings with involution, enabling algebraic representation of quantum structures such as basic algebras and orthomodular lattices.

Findings

Representation of basic algebras as near semirings

Representation of orthomodular lattices via orthomodular near semirings

Establishment of the variety of involutive integral near semirings as a Church variety

Abstract

In this paper we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called \L ukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a \L ukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of \L ukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the paper, we discuss several universal algebraic properties of \L ukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras.