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