Many-valued quantum algebras. (English) Zbl 1219.06013

Chajda, Ivan; Halaš, Radomír; Kühr, Jan

Algebra Univers. 60, No. 1, 63-90 (2009).

Summary: We deal with algebras \(\mathbf A = (A,\oplus,\neg,0)\) of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) \(\mathbf A\) bears a natural lattice structure, (ii) the elements \(a\) for which \(\neg a\) is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety.

MSC:

06D35	MV-algebras
03G12	Quantum logic
06C15	Complemented lattices, orthocomplemented lattices and posets

Keywords:

De Morgan algebra; MV-algebra; orthomodular lattice; lattice effect algebra; sectional antitone involution; basic algebra; MVQ-algebra