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Considering the inference rules in generalized logics, J.C. Abbott arrives to the notion of orthoimplication algebra (see Abbott (1970) and Abbott (Stud. Logica. 2:173–177, XXXV)). We show that when one enriches the Abbott orthoimplication algebra with a falsity symbol and a natural $$\mathbb {XOR}$$ XOR -type operation, one obtains an orthomodular difference lattice as an enriched quantum logic (see Matoušek (Algebra Univers. 60:185–215, 2009)). Moreover, we find that these two structures endowed with the natural morphisms are categorically equivalent. We also show how one can introduce the notion of a state in the Abbott $$\mathbb {XOR}$$ XOR algebras strenghtening thus the relevance of these algebras to quantum theories.

No abstract

Let us denote by ℒ ℱ $[\mathcal{L}\mathcal{F}$ the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L ∈ ℒ ℱ $[L\in \mathcal{L}\mathcal{F}$ provided each finite subset of L generates in L a finite subOML). In this note, we first show how one can obtain new locally finite OMLs from the initial ones and enlarge thus the class ℒ ℱ $[\mathcal{L}\mathcal{F}$ . We find ℒ ℱ $[\mathcal{L}\mathcal{F}$ considerably large though, obviously, not all OMLs belong to ℒ ℱ $[\mathcal{L}\mathcal{F}$ . Then we study states on the OMLs of ℒ ℱ $[\mathcal{L}\mathcal{F}$ . We show that local finiteness may to a certain extent make up for distributivity. For instance, we show that if L ∈ ℒ ℱ $[L\in \mathcal{L}\mathcal{F}$ and if for any finite subOML K there is a state s: K → [0, 1] on K, then there is a state on the entire L. We also consider further algebraic and state properties of ℒ ℱ $[\mathcal{L}\mathcal{F}$ relevant to the quantum logic theory.