A note on orthopseudorings and Boolean quasirings. (English) Zbl 1040.06003

Chajda, I.; Eigenthaler, G.

Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 207, 83-94 (1998).

Summary: The concept of an orthopseudoring was introduced by the first author in Czech. Math. J. 46, 405–411 (1996; Zbl 0879.06003), where it was shown that there is a one-to-one corresondence between orthopseudorings and ortholattices, generalizing the well-known correspondence between Boolean rings and Boolean algebras.
Independently, D. Dorninger, H. Länger and M. Maczyński [Demonstr. Math. 30, 215–232 (1997; Zbl 0879.06005)] defined other ring-like structures, so-called Boolean quasirings, and again a one-to-one correspondence between these structures and ortholattices was established.
In this paper, we first study connections between orthopseudorings and Boolean quasirings, which are both based on generalizations of symmetric differences [see, e.g., G. Dorfer, A. Dvurečenskij and H. Länger, Math. Slovaca 46, 435–444 (1996; Zbl 0890.06006)]. Then we introduce the concept of an ideal and classify those ideals which are congruence kernels. We show that, contrary to rings, an ideal can be the congruence kernel of more than one congruence.

MSC:

06C15	Complemented lattices, orthocomplemented lattices and posets
06E20	Ring-theoretic properties of Boolean algebras
06B10	Lattice ideals, congruence relations

Keywords:

orthopseudoring; Boolean quasirings; generalizations of symmetric differences; ideal; congruence kernels

Citations:

Zbl 0879.06003; Zbl 0879.06005; Zbl 0890.06006