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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References
Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society Colloquium Publications (American Mathematical Society, RI) (1973)
Blount K., Tsinakis C.: The structure of residuated lattices. Internat J. Algebra Comput. 13, 437–461 (2003)
Blyth T.S., Janowitz M.F.: Residuation Theory. Pergamon Press, New York (1972)
Dvurečenskij A., Kühr J.: On the structure of linearly ordered pseudo-BCK-algebras. Arch. Math. Logic 48(8), 771–791 (2009)
Fuchs L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151. Studies in Logic and the Foundations of Mathematics, p. 532 (2007)
Georgescu, G., Iorgulescu, A.: Pseudo-BCK algebras: an extension of BCK algebras. In: Proceedings of DMTCS ’01: Combinatorics, Computability and Logic, London, pp. 97–114 (2001)
Gierz, G.K., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous Lattices and Domains, Encycl. of Math. and its Appl. 93. Cambridge University Press, Cambridge (2003)
Iseki K., Tanaka S.: An introduction to the theory of BCK-algebras. Math. Jpn. 23, 1–26 (1978)
Jenei S.: Equality algebras. Studia Logica 100, 1201–1209 (2012)
Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, Kluwer, Dordrecht, pp. 19–56 (2002)
Kabziński, J., Wroński, A.: On equivalential algebras. In: Proceedings of the 1975 International Symposium on Multiple-valued Logic. Indiana University, Bloomington, pp. 231–243 (1975)
Kühr J.: Pseudo BCK-algebras and residuated lattices. Control Gen. Algebra 16, 139–144 (2005)
Ursini A.: On subtractive varieties I. Algebra Universalis 31, 204–222 (1994)
Ward M., Dilworth R.P.: Residuated lattices. Trans. Am. Math. Soc. 45, 335–354 (1939)
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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