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Categories of orthomodular posets. (English) Zbl 0583.06005
The author considers categories of orthomodular posets \(O_ 1\), \(O_ 2\), \(O_ 3\) which differ only by the choice of morphisms \(f: P\to Q\) for orthomodular posets P, Q. These maps satisfy (i) \(a\leq b\) implies f(a)\(\leq f(b)\), (ii) \(f(a')=f(a)'\) and (iii) \(a\in C(P)\) implies f(a)\(\in C(Q).\)
For \(O_ 1\) it is required that (iv) \(f(a\vee b)=f(a)\vee f(b)\) for \(a\in C(P)\) and \(a\leq b'\) holds, for \(O_ 2\) that (v) \(f(a\vee b)=f(a)\vee f(b)\) for \(a\leq b'\) holds and for \(O_ 3\) that (vi) \(f(a\vee b)=f(a)\vee f(b)\) holds whenever \(a\vee b\) exists.
It is \(O_ 1\supset O_ 2\supset O_ 3\) and Boolean algebras are reflective and coreflective in all three categories. Several conditions for reflections to be embeddings are given.
Reviewer: G.Kalmbach

MSC:
06A06 Partial orders, general
06C15 Complemented lattices, orthocomplemented lattices and posets
06B25 Free lattices, projective lattices, word problems
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