A signed analog of the Birkhoff transform
Let \(P\) be an \(n\)-poset naturally labeled by \(\{1,\dots,n\}\) ordered by \(\leq_p\), i.e., \(p<_pq\) means \(p<q\) as integers. Then \(\pm P\) is the poset on \(\{\pm1,\dots,\pm n\}\) such that \(p<\pm q\) iff \(|p|<_p|q|\). A signed \(P\)-filter is a filter \(X\) of \(\pm P\) such that if \(p\) is a generator of \(X\), then \(-p\not\in X\). The signed Birkhoff transform of \(P\) is the poset \(B(P)\) consisting of the set of signed \(P\)-filters ordered by inclusion, with \(B(P)^\wedge\) obtained from \(B(P)\) by adding a maximal element \(\widehat{\mathitı}\). Posets isomorphic to \(B(P)\) or \(B(P)^\wedge\) are signed Birkhoff posets. In considering classes of posets as being ``nice'' one looks for qualities of an algebraic, geometric, or combinatorial nature such classes may have. In many important situations all three types may be connected by ``bridge theorems'' which illustrate the meaning of one type of result in terms of a different setting. Such classes include distributive lattices, Eulerian posets, Gorenstein posets for example. One is therefore always interested in bringing out another class whose construction can be described well enough to permit determination of many proporties analogously to known results to a sufficient extent that one obtains a satisfactory theory of adequate depth and detail. The class of signed Birkhoff posets has all these characteristics, e.g., \(B(P)^\wedge\) is Eulerian, whence the entire combinatorial panoply of such posets can be applied to the specific class at hand, including the Gorenstein\(^*\) property, and further results on generating functions, order-polynomials, \(f\)-flag vectors, the \(\overline c\overline d\)-index along with derived information on the homology of the order-complex and the strict geometric (algebraic-topological) implications thereof.
scientific article; zbMATH DE number 5012959
scientific article