Vol. 58, No. 2, 1975

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Semimodularity in the completion of a poset

Barbara Jeffcott and William Thomas Spears

Vol. 58 (1975), No. 2, 467–474

M. D. 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.

Mathematical Subject Classification
Primary: 06A30
Received: 16 April 1974
Published: 1 June 1975
Barbara Jeffcott
William Thomas Spears