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

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.


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


Zbl 1014.08001