Obstacles to extending Mirsky's theorem (Q698450)

A poset \((P,\leq)\) is called locally finite if for any \(a,b\in P\), \(a\leq b\), the interval \([a,b]= \{x\in P:a\leq x\leq b\}\) is finite. In this paper the following theorem is proved. Theorem: For each infinite cardinal \(\kappa\) there is a locally finite poset \(P\) such that (1) \(P\) has no infinite chain and (2) \(P\) cannot be covered by \(\kappa\) antichains. The local finiteness reminds the reviewer of his study of many years ago, that is, a very special case of a semilattice \(S\) satisfying the condition: for any \(a,b\in S\), \(a\leq b\), \([a,b]\) is a finite chain [J. Gakugei Tokushima Univ. Natur. Sci. Math. 7, 9-17 (1956; Zbl 0073.01803)]. However, it contains an infinite chain and there is no direct relation with this paper.