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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
##### Keywords:
categories of orthomodular posets; reflections; embeddings
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##### References:
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