an:06347441
Zbl 1314.03003
Botur, Michal; Chajda, Ivan; Hala??, Radom??r; K??hr, Jan; Paseka, Jan
Algebraic methods in quantum logic
EN
Olomouc: Palack?? University, Faculty of Science (ISBN 978-80-244-4166-5/pbk). viii, 195~p. (2014).
2014
b
03-02 03G12 03G25 06D35 06F25 08A55 81P10
MV-algebra; tense MV-algebra; lattice effect algebra; commutative basic algebra; non-associative BL-algebra; state-morphism
The monograph (based on recent results of the authors) is devoted to the study of various algebraic structures that model non-classical logics, in particular MV-algebras, lattice effect algebras, commutative basic algebras, and non-associative BL-algebras (generalizations of H??jek's BL-logics). MV-algebras are studied in Chapter~2. A direct proof of Di Nola's representation theorem (and its extensions) is presented using Farkas' lemma and the finite embedding theorem. A new proof of the completeness of the ??ukasiewicz axioms is obtained as a by-product. It is proved that a tense semisimple MV-algebra is induced by a time frame. A tense MV-algebra is an MV-algebra with a couple of unary operators expressing universal time quantifiers. Lattice effect algebras are studied in Chapter~3. Finitely generated varieties of distributive lattice effect algebras are axiomatized and the free \(n\)-generator algebras in these varieties are described. Tense operators are constructed.
Chapter~4 is devoted to non-associative logics. A subdirectly irreducible commutative basic algebra that is not an MV-algebra is constructed for an arbitrary infinite cardinality. Basic properties of states on commutative basic algebras are presented. It is shown that the class of non-associative BL-algebras forms a variety generated by non-associative t-norms. The last chapter deals with state-operators. Characterizations of subdirectly irreducible state BL-algebras and subdirectly irreducible state-morphism BL-algebras are presented. A general theory of state-morphism algebras is given. Generators of varieties of state-morphism algebras are described, in particular for BL-algebras, MTL-algebras, non-associative BL-algebras and pseudo MV-algebras.
Josef Tkadlec (Praha)