Abstract
A new structure, called pseudo equality algebras, will be introduced. It has a constant and three connectives: a meet operation and two equivalences. A closure operator will be introduced in the class of pseudo equality algebras; we call the closed algebras equivalential. We show that equivalential pseudo equality algebras are term equivalent with pseudo BCK-meet-semilattices. As a by-product we obtain a general result, which is analogous to a result of Kabziński and Wroński: we provide an equational characterization for the equivalence operations of pseudo BCK-meet-semilattices. Our result treats a much more general algebraic structure, namely, pseudo BCK-meet-semilattice instead of Heyting algebras, on the other hand, we also need to use the meet operation. Finally, we prove that the variety of pseudo equality algebras is a subtractive, 1-regular, arithmetical variety.
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The research of the first author was supported by the OTKA-76811 grant, the EC MC grant 267589, the project MSM 6198898701 of the MŠMT ČR., and the SROP-4.2.1.B-10/2/KONV-2010-0002 grant.
The research of the second author was supported by the SROP-4.2.1.B-10/2/KONV-2010-0002 grant.
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Jenei, S., Kóródi, L. Pseudo equality algebras. Arch. Math. Logic 52, 469–481 (2013). https://doi.org/10.1007/s00153-013-0325-z
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DOI: https://doi.org/10.1007/s00153-013-0325-z
Keywords
- Residuated lattices
- BCK-meet-semilattices
- Heyting algebras
- Equivalential algebras
- Equivalential fragment
- Term equivalence
- Equational characterization