Coordinatization of orthocomplemented and orthomodular posets

Abstract

Generalizations of Baer $^{\ast }$-semigroups called partial Baer $^{\ast }$-semigroups and OM-partial Baer $^{\ast }$-semigroups are introduced. It is shown that the set of closed projections of a (OM) partial Baer $^{\ast }$-semigroup form an (orthomodular) orthocomplemented poset. Conversely (orthomodular) orthocomplemented posets are coordinatized by (OM) partial Baer $^{\ast }$-semigroups. It is shown that these coordinatizing semigroups are minimal.