Abstract
In this paper, we introduce the concept of monadic pseudo-equality algebras and obtain some related properties. We study the relations between monadic pseudo-equality algebras, monadic bounded hoops and monadic pseudo-BCI-algebras. Then, we introduce and study monadic deductive systems and monadic congruences and discuss the relations between them. We use monadic deductive systems to define uniform topology on pseudo-equality algebras and obtain some topological results. Finally, the corresponding logic (monadic invariant pseudo-equality logic) is constructed and the soundness and completeness of this logic are proved based on monadic invariant pseudo-equality algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bezhanisvili G (1999) On geVarieties of monadic Heyting algebras I. Stud Log 61:367–402
Bosbach B (1966) Komplementäre Halbgruppen, Axiomatik und Arithmetik. Fundam Math 64:257–277
Cattaneo G (1998) Abstract approximation spaces for rough theories. In: Polkowski L, Skowron A (eds) Rough sets in knowledge discovery 1: methodology and applications. Physica-Verlag, Wurzburg, pp 59–98
Ciungu LC (2014) On pseudo equality algebras. Arch Math Log 53:561–570
Ciungu LC (2016) Commutative pseudo equality algebras. Soft Comput 21(16):4601–4616
Ciungu LC (2018) State pseudo equality algebras. J Multi-Valued Log Soft Comput 30(4–6):573–613
Di Nola A, Grigolia R (2004) On monadic MV-algebras. Ann Pure Appl Log 128:125–139
Dudek WA, Jun YB (2008) Pseudo-BCI algebras. East Asian Math J 24:187–190
Dvurečenskij A, Zahiri O (2016) Pseudo equality algebras-revision. Soft Comput 20:2091–2101
Esteva F, Godo L (2001) Monoidal t-norm-based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124:271–288
Figallo-Orellano A (2016) A topological duality for monadic MV-algebras. Soft Comput 21(23):7119–7123
Georgescu G, Iorgulescu A (2001) Pseudo-BCK algebras: an extension of BCK-algebras, DMTCS01: combinatorics, computability and logic. Springer, London, pp 97–114
Grigolia R (2006) Monadic BL-algebras. Georgain Math J 13:267–276
Halmos RP (1962) Algebraic logic. Chelsea Publ Co, New York
Iorgulescu A (2004) On pseudo-BCK algebras and porims. Sci Math Jpn 10:293–305
Iséki K (1966) An algebra related with a propositional calculus. Proc Jpn Acad 42:26–29
Iséki K (1980) On BCI-algebras. Math Semin Notes (Kobe Univ) 8(1):125–130
Jenei S (2012) Equality algebras. Stud Log 100:1201–1209
Jenei S, Kóródi L (2013) Pseudo equality algebras. Arch Math Log 52:469–481
Joshi KD (1997) Introduction to general topology. New Age Publisher, New Delhi
Novák V (2011) EQ-algebra-based fuzzy type theory and its extensions. Log J IGPL 19:512–542
Novák V, De Baets B (2009) EQ-algebras. Fuzzy Sets Syst 160:2956–2978
Novák V, Dyba M (2011) EQ-logics: non-commutative fuzzy logics based on fuzzy equality. Fuzzy Sets Syst 172:13–32
Rachunek J, Šalounová D (2008) Monadic bounded commutative residuated l-monoids. Order 25:157–175
Rachunek J, Šalounová D (2013) Monadic bounded residuated lattices. Order 30:195–210
Rutledge JD (1959) A preliminary of the infinitely many-valued predicate calculus. Ph.D. Thesis, Cornell University
Wang J, Xin X, He P (2018) Monadic bounded hoops. Soft Comput 22:1749–1762
Xin X, Fu Y, Lai Y, Wang J (2019) Monadic pseudo BCI-algebras and corresponding logics. Soft Comput 23:1499–1510
Acknowledgements
The author would like to express her thanks to referees for their comments and suggestions which improved the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that she has no conflict ofinterest.
Human and animal rights
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by A. Di Nola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ghorbani, S. Monadic pseudo-equality algebras. Soft Comput 23, 12937–12950 (2019). https://doi.org/10.1007/s00500-019-04243-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-019-04243-5