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We study the existence and the number of k -neighborly reorientations of an oriented matroid. This leads to k -variants of McMullen's problem and Roudneff's conjecture, the case k = 1 being the original statements. Adding to results of Larman and García-Colín, we provide new bounds on k -McMullen's problem and prove the conjecture for several ranks and k by computer. Further, we show that k -Roudneff's conjecture for fixed rank and k reduces to a finite case analysis. As a consequence we prove the conjecture for odd rank r and k = r − 1 2 as well as for rank 6 and k = 2 with the aid of the computer. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Möbius function μ S associated to a (locally finite) poset arising from a semigroup S of Z m . We introduce and develop a new approach to study μ S by using the Hilbert series of S . The latter enables us to provide formulas for μ S when S belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given. [ABSTRACT FROM AUTHOR]