Orthomodular lattices that are $Z_2$-rich

Abstract

We study the orthomodular lattices (OMLs) that have an abundance of $Z_2$-valued states. We call these OMLs $Z_2$-rich. The motivation for the investigation comes from a natural algebraic curiosity that reflects the state of the (orthomodular) art, the consideration also has a certain bearing on the foundation of quantum theories (OMLs are often identified with “quantum logics”) and mathematical logic ($Z_2$-states are fundamental in mathematical logic). Before we launch on the subject proper, we observe, for a potential application elsewhere, that there can be a more economic introduction of $Z_2$-richness - the $Z_2$-richness in the orthocomplemented setup is sufficient to imply orthomodularity. In the further part we review basic examples of OMLs that are $Z_2$-rich and that are not. Then we show, as a main result, that the $Z_2$-rich OMLs form a large and algebraicly “friendly” class—they form a variety. In the appendix we note that the OMLs that allow for a natural introduction of a symmetric difference provide a source of another type of examples of $Z_2$-rich OMLs. We also formulate open questions related to the matter studied.

Appendix: The orthomodular lattices with a symmetric difference are \(Z_2\)-rich

In this paragraph we briefly (without proofs) indicate the link of $Z_2$-rich OMLs with the OMLs that allow for a natural introduction of symmetric difference. The latter class has come into existence independently of $Z_2$-RICH and been investigated in [4, 13, 15,16,17]. The definition reads as follows.

Definition 4

Let $L = (X,\wedge ,\vee ,^\perp ,0,1,\mathbin {\triangle })$, where $(X,\wedge ,\vee ,^\perp ,0,1)$ is an orthomodular lattice and $\mathbin {\triangle }: X^2 \rightarrow X$ is a binary operation. Then L is said to be an orthocomplemented difference lattice (abbr., an ODL) if the following formulas hold in L:

$(\mathrm{D}_1)$ :

$x \mathbin {\triangle }(y\mathbin {\triangle }z) = (x \mathbin {\triangle }y)\mathbin {\triangle }z$,

$(\mathrm{D}_2)$ :

$x \mathbin {\triangle }1 = x^\perp$, $1 \mathbin {\triangle }x = x^\perp$,

$(\mathrm{D}_3)$ :

$x \mathbin {\triangle }y \le x \vee y$.

Theorem 2

Each ODL is $Z_2$-rich.

The proof of 2 uses a specific ODL reasoning and can be found in [16]. Let us note that the study of ODLs brought about brand new types of $Z_2$-rich OMLs. The reader might find them interesting. Also, the investigation advanced to formulating related open question. Let us end up this paper by formulating one of them: Is the variety $Z_2$-RICH generated by the class of all ODLs?