On the size of Boolean combinations of subgroups of finite Abelian groups (Q5935841)

Let \(L\) be a modular lattice of finite length and \(P(L)\) its set of prime quotients. An edge valuation \(\nu\) is a map from \(P(L)\) into the natural numbers such that \(\nu(a/b)=\nu(c/d)\) whenever \(c=a\vee d\) and \(b=a\wedge d\). A representation of \((L,\nu)\) is a lattice morphism \(\phi\) of \(L\) into the subgroup lattice of some finite Abelian group such that \(\phi(0_L)=0\) and such that the cardinality of the quotient subgroup \(\phi(a)/\phi(b)\) is \(\nu(a/b)\) for every prime quotient \(a/b\). Let \(N_L\) be the set of all \(\nu\) such that \((L,\nu)\) has a representation. A Boolean expression over a set \(S\) is a term \(\beta=\beta(a_1,\dots,a_n)\) built from elements of \(S\), the binary lattice operations, the unary operation \(\urcorner\), and the constants \(0,1\). Given a map \(\phi\colon S\to{\mathcal P}(G)\) we get a subset \(\phi(\beta)\) of \(G\) by interpretation in the power set algebra \({\mathcal P}(G)\). The following result is proved: Theorem 1. For every Boolean expression \(\beta\) over a finite length modular lattice \(L\) there is an order preserving function \(f_{\beta,L}\) from \(N_L\) (with pointwise order) to the natural numbers such that \(f_{\beta,L}(\nu)=|\phi(\beta)|\) for every \(\nu\in N_L\) and representation \(\phi\) of \((L,\nu)\). Nothing is said about the existence of representations (actually, not much is known).