Ideals and D-systems in orthoimplication algebras. (English) Zbl 1078.03050

Halaš, Radomír

J. Mult.-Val. Log. Soft Comput. 11, No. 3-4, 309-316 (2005).

An orthoimplication algebra is an axiomatization of the connective implication in the so-called logic of quantum mechanics which was introduced by J. C. Abbott in 1976. The author introduces a concept of deductive system in orthoimplication algebra and proves that it coincides with an ideal (for exact definitions, see e.g. the monograph [I. Chajda, G. Eigenthaler and H. Länger, Congruence classes in universal algebra. Lemgo: Heldermann Verlag (2003; Zbl 1014.08001)]). He derives the so-called ideal terms and proves that subsets closed under these terms are just the congruence kernels.

Reviewer: Ivan Chajda (Přerov)

MSC:

03G10	Logical aspects of lattices and related structures
03B22	Abstract deductive systems
06C15	Complemented lattices, orthocomplemented lattices and posets

Keywords:

orthoimplication algebra; implication algebra; D-system; ideal; deductive system

Citations:

Zbl 1014.08001