Realizable posets (Q1176978)

Let \((P,\leq)\) be a partially ordered set and \(F\) a field. The paper deals with \(k\)-dimensional representations of \(P\), i.e. sequences \((V,A(a)\mid a\in P)\), where \(V\) is a \(k\)-dimensional vector space over \(F\) and \(A(a)\) are subspaces of \(V\) with \(A(a)\subseteq A(b)\) if \(a\leq b\). \(P\) is called realizable, if there is a \(k\)-dimensional representation of \(P\) with i) \(k<| P|\), ii) \(A(a)\subseteq A(b)\) if and only if \(a\leq b\), iii) \(A(a)\neq 0\) or \(A(a)\neq V\) for all \(a\in P\). The orderings that are not realizable are characterized. They are of width at most two and (besides further conditions) they are ``cross-connected''. This problem occurs in realizing particular subrings of a central \(\mathbb{Q}\) division algebra \(D\) as endomorphism rings of \(Z_ p\)-submodules of \(D\).