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.
