Strong properties in partially ordered sets II

Abstract

An [α, β)-normal poset with (α, β)-logarithmic concave Whitney numbers is a normal poset with logarithmic concave Whitney numbers, with the additional condition that, without mentioning trivial cases, in the definitional inequalities for normality and logarithmic concavity equality can only hold in the exceptional intervals [α, β) or (α, β), respectively. A theorem is proved, where some conditions for the posets P and Q are given such that P × Q is an [α, β)-normal poset with (α, β)-logarithmic concave Whitney numbers. Some corollaries are deduced from which the strong h-family property of many posets can be obtained.