Orthomodular lattices with rich state spaces (Q1866795)

A new construction technique for orthomodular lattices is introduced. For an OML \(L\) and \(a,b \in L\) their distance \(d_L(a,b)\) in \(L\) is the minimal \(n\) for which there exists a sequence \((B_1 ,\ldots ,B_n)\) of blocks (i.e. maximal Boolean subalgebras of \(L\)) such that \(a \in B_1\), \(b \in B_n\) and \(B_i\cap B_{i+1}\supset \{0,1\}\) for \(i=1,\dots ,n-1\). If no such sequence exists then \(d_L(a,b)=\infty \) and \(d_L(a,a)=0\) for all \(a \in L\). A \(K\)-valued state \((K\subseteq I=[0,1]\subseteq {\mathbb R})\) on \(L\) is a mapping \(s:L\rightarrow K\) with \(s(1)=1\), and \(s(a\vee b)=s(a)+s(b)\) whenever \(a\bot b\). \(I\)-valued states are called states. A principal construction tool of the paper is the so-called regulator: a regulator of order \(n\) is a finite OML \(L\) with a sequence of atoms \((x_1 ,\ldots ,x_n;y_1,\ldots ,y_n)\) (entries) satisfying the following three conditions: (R1) The mutual distance of each pair of entries is at least 5; (R2) Each state \(s\) on \(L\) satisfies the characteristic relation of a regulator: \[ \sum _{i \leq n} {s(x_i)}=\sum _{i \leq n} {s(y_i)};\tag{E} \] (R3) Let \(a,b\) be non-orthogonal atoms of \(L\), and let \(s:\{a,b,x_1, \ldots ,x_n,y_1,\ldots ,y_n\}\rightarrow \{0,1\}\) be an acceptable function such that \(s(a)=s(b)=1\) and (E) is satisfied. Then \(s\) can be extended to a \(\{0,1\}\)-valued state on \(L\). In contrast to the preceding constructions, this one admits rich spaces of states, i.e. for each pair of incomparable elements \(a,c\) there is a state \(s\) such that \(s(a)=1>s(c)\). This allowed a progress in many questions that were open for a long time; among others it is proved that there is a continuum of varieties of orthomodular lattices with rich state spaces and hence the authors solve a problem formulated by Mayet in 1985. As a by-product of this research, the uniqueness problem for bounded observables (posed by Gudder in 1966) is solved. As a tool also a construction of identification of atoms in an OML is introduced.