Representing quantum structures as near semirings

作者：Bonzio Stefano; Chajda Ivan; Ledda Antonio

来源：Logic Journal of the IGPL, 2016, 24(5): 719-742.

DOI：10.1093/jigpal/jzw031

• Near semiring
• involution
• Lukasiewicz near semiring
• semiring
• basic algebra
• orthomodular lattice
• MV-algebra
• Church variety
• central element

In this article, 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 Lukasiewicz 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 Lukasiewicz 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 Lukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Lukasiewicz 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.

• 出版日期2016-10