Semimodularity in the completion of a poset
Barbara Jeffcott, William Spears · Pacific Journal of Mathematics · 1975
MacLaren examined semimodularity in the completion by cuts of a lattice L, and showed that if L is semimodular, atomic, and orthocomplemented then L is semimodular [Pacific J. Math.14 (1964)].We study here semimodularity in an orthomodular poset P and its completion by cuts P. In particular, we show that if P is semimodular and orthomodular and contains no infinite chains, then P is semimodular if and only if P is isomorphic to P. Hence, contrary to the result of MacLaren for lattices, semimodularity is never preserved in the completion by cuts of an orthomodular poset with no infinite chains which is not a lattice.More generally, we show that if P is orthomodular, atomic, and orthocomplete, then the covering condition in P is carried over to P if and only if P is isomorphic to P. As a result, MacLaren's theorem cannot be generalized to posets.