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On double basic algebras and pseudo-effect algebras. (English) Zbl 1235.06012
Double basic algebras have been introduced in [I. Chajda, “Double basic algebras”, Order 26, No. 2, 149–162 (2009; Zbl 1173.06002)] as a counterpart of bounded lattices with order-antiautomorphisms on principal filters. Now their independent axiomatization is given. Double basic algebras represent a non-commutative extension of basic algebras which generalize orthomodular lattices in a similar way as MV-algebras generalize Boolean algebras. Lattice pseudo-effect algebras are characterized as a subvariety of double basic algebras and pairs of compatible elements of lattice pseudo-effect algebras are described in terms of double basic algebras.
MSC:
06D35 MV-algebras
03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
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