A Representation of Lattice Effect Algebras by Means of Right Near Semirings with Involution

Abstract

Since every lattice effect algebra decomposes into blocks which are MV-algebras and since every MV-algebra can be represented by a certain semiring with an antitone involution as shown by Belluce, Di Nola and Ferraioli, the natural question arises if a lattice effect algebra can also be represented by means of a semiring-like structure. This question is answered in the present paper by establishing a one-to-one correspondence between lattice effect algebras and certain right near semirings with an antitone involution.Keywords Effect algebra · Lattice effect algebra · Right near semiring · Antitone involution · Effect near semiring Effect algebras were introduced by Foulis and Bennett [8] in order to axiomatize unsharp logics of quantum mechanics. Although the definition of an effect algebra looks elementary, these algebras have several very surprising properties. Concerning these properties the reader is referred to the monograph [7] by Dvurečenskij and Pulmannová. In particular, every effect algebra induces a natural partial order relation and thus can be considered as a bounded poset. If this poset is a lattice, the effect algebra is called a lattice effect algebra.

The Groupoid-Based Logic for Lattice Effect Algebras

“…The previous Corollaries I. 5 and I.6 motivate us to set up an appropriate logic also for lattice effect algebras. Of course, we will formulate the axioms and rules in the language of effect groupoids as derived in the previous part.…”

“…the so-called effect near semiring, see e.g. [5] for details. However, we can derive another algebra which has only one binary operation, i.e., a groupoid enriched by unary and nullary operations.…”

Derivations in Lukasiewicz semirings

“…The advantage of this approach is that we can apply also methods from the theory of semirings. This was done by the authors in [6].…”