ORTHOMODULAR (PARTIAL) ALGEBRAS AND THEIR REPRESENTATIONS
Peter Burmeister, Maciej Mączyński · Demonstratio Mathematica · 1994
8-13 June 1993.5 Note that by (A2) or (A3) it would only follow that the constant 0 is always defined in each non-empty orthomodular algebra, since the axioms (Al) through (A9) allow an empty model. 6 Actually, in order to conclude the usual form of associativity, one would also have expected the axiom (A5')However, this follows from (A4) and (A5) (we argue here semantically): Let a,b,c € A for some orthomodular partial algebra A, and let a φ (6 φ c) exist, then, by (A4), also (c φ 6) φ α exists, and therefore one has by (A5) and (A4) that c φ (6 φ α) exists, and that (α φ i) φ c = c φ (b φ a) = (c φ b) φ a = a φ (¡> φ c). 7Observe that by the commutativity of φ and by (Al) this is also equivalent to b' Β is a homomorphism between the orthomodular algebras A = (Λ;φ,',0) and Β = (ΰ;φ/,0), if /(0) = 0, if, for «ill it, ν Ç A, one has