Every cutset meets every fibre in certain poset products

Abstract

AbstractA cutset of a partially ordered set is a subset which meets every maximal chain, and a fibre of a partially ordered set is a subset which meets every maximal antichain. A poset is called skeletal if every cutset meets every fibre. K1,n stands for the linear sum of a singleton and an n-element antichain. Duffus et al. (1990) showed that any Boolean lattice K1,1 × ··· × K1,1 is skeletal. Gibson and Maltby (1993) showed that K1,m × K1,n is skeletal and asked if every K1,n1 × ··· ×x K1,nk is skeletal. We prove that K1,1 × ··· × K1,1 × K1,m × K1,n and K1,1 × K1,m × K1,n are skeletal