A short note on categorical equivalences of proper weak pseudo EMV-algebras

Document Type : Original Article

Author

1 Mathematical Institute, Slovak Academy of Sciences, Stef´anikova 49, SK-814 73 Bratislava, Slovakia ˇ

2 Palack´y University Olomouc, Faculty of Sciences, tˇr. 17. listopadu 12, CZ-771 46 Olomouc, Czech Republi

Abstract

We study the class of weak pseudo EMV-algebras without top element that are a non-commutative generalization of MV-algebras, pseudo MV-algebras and generalized Boolean algebras. We present their categorical equivalences to a special category of pseudo MV-algebras with a fixed maximal and normal ideal as well as to a special category of unital l-groups with a fixed maximal and normal l-ideal.

Keywords

References

[1] P. Bahls, J. Cole, N. Galatos, P. Jipsen, C. Tsinakis, Cancellative residuated lattices, Algebra Universalis,
50 (2003), 83–106.
[2] B. Bosbach, Concerning semiclans, Archiv der Mathematik, 37 (1981), 316–324.
[3] C.C. Chang, Algebraic analysis of many-valued logics, Transactions of the American Mathematical
Society, 88 (1958), 467–490.
[4] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Springer
Science and Business Media, Dordrecht, 2000.
[5] P. Conrad, M.R. Darnel, Generalized Boolean algebras in lattice-ordered groups, Order, 14 (1998),
295–319.
[6] M.R. Darnel, Theory of lattice-ordered groups, Marcek Dekker, Inc., New York, Basel, Hong Kong,
1995.
[7] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL-algebras: Part I, Multi-Valued Logic, 8 (2002),
673–714.
[8] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL-algebras: Part II, Multi-Valued Logic, 8 (2002),
715–750.
[9] A. Dvurečenskij, States on pseudo MV-algebras, Studia Logica, 68 (2001), 301–327.
[10] A. Dvurečenskij, Pseudo MV-algebras are intervals in ℓ-groups, Journal of the Australian Mathematical
Society, 72 (2002), 427–445.
[11] A. Dvurečenskij, O. Zahiri, On EMV-algebras, Fuzzy Sets and Systems, 373 (2019), 116–148.
[12] A. Dvurečenskij, O. Zahiri, The Loomis–Sikorski theorem for EMV-algebras, Journal of the Australian
Mathematical Society, 106 (2019), 200–234.
[13] A. Dvurečenskij, O. Zahiri, Pseudo EMV-algebras. I. Basic properties, Journal of Applied Logics–
IFCoLog Journal of Logics and their Applications, 6 (2019), 1285–1327.
[14] A. Dvurečenskij, O. Zahiri, Pseudo EMV-algebras. II. Representation and states, Journal of Applied
Logics–IFCoLog Journal of Logics and their Applications, 6 (2019), 1329–1372.
[15] A. Dvurečenskij, O. Zahiri, A variety containing EMV-algebras and Pierce sheaves of EMV-algebras,
Fuzzy Sets and Systems, 418 (2021), 101–125, Doi:10.1016/j.fss.2020.09.011.
[16] A. Dvurečenskij, O. Zahiri, Weak pseudo EMV-algebras. I. Basic properties, Journal of Applied Logic –
IfCoLog Journal of Logics and their Applications, 8 (2021), 2365–2399.
[17] A. Dvurečenskij, O. Zahiri, Weak pseudo EMV-algebras. II. Representation and subvarieties, Journal
of Applied Logic – IfCoLog Journal of Logics and their Applications, 8 (2021), 2401–2433.
[18] G. Georgescu, A. Iorgulescu, Pseudo MV-algebras, Multiple-Valued Logics, 6 (2001), 193–215.
[19] G. Georgescu, L. Leuştean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic and Soft
Computing, 11 (2005), 153–184.
[20] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, New York, Heidelberg, Berlin,
1971.
[21] D. Mundici, Interpretation of AF C∗-algebras in Łukasiewicz sentential calculus, Journal of Functional
Analysis, 65 (1986), 15–63.
[22] J. Rachůnek, A non-commutative generalization of MV-algebras, Czechoslovak Mathematical Journal,
52 (2002), 255–273.
[23] W. Rump, Y. Yang, A note on Bosbach’s cone algebras, Studia Logica, 98 (2011), 375–386.
Journal of Algebraic Hyperstructures and Logical Algebras
Volume 3, Issue 1
Special Issue
February 2022
Pages 35-44

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