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A note on the symmetric difference in lattices. (English) Zbl 1083.06012

Let \((L,\cdot,+;0,1)\) be a lattice with the usual order \(\leq\) defined by \(a\leq b\) iff \(a\cdot b = a\) for \(a,b\) in \(L\). A lattice with negation is a lattice \(L\) with a unary operation \(': L\to L\) satisfying: \(0' = 1\); \(1' = 0\); if \(a\leq b\), then \(b'\leq a'\) for \(a, b\) in \(L\); and \(a'' = a\) for \(a\) in \(L\). In a lattice with negation, the authors introduce a definition of symmetric difference as the binary operation \(\Delta: L\times L\to L\) defined by \(\Delta(a,b)=(a+b)(a\cdot b)'\) for \(a, b\) in \(L\). The authors give many general properties of \(\Delta\) and characterize ortholattices (lattices with negation which satisfy \(a\cdot a' = 0\) and \(a+b = ( a'\cdot b')'\) for \(a, b\) in \(L\)), orthomodular lattices (ortholattices with relative complement), de Morgan algebras (distributive lattices with negation which satisfy \(a + b =(a'\cdot b')'\) for \(a\), \(b\) in \(L\)), Kleene algebras (de Morgan algebras which satisfy \(a\cdot a' \leq b + b'\) for \(a, b\) in \(L\)) and Boolean algebras (distributive ortholattices) in terms of properties of \(A\). For example, the authors prove the following:
Theorem. An orthomodular lattice is a Boolean algebra if and only if \(A\) is an associative operation.

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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