The category of L-algebras

Subject Areas : Transactions on Fuzzy Sets and Systems

Lavinia Corina  Ciungu ¹ (Department of Mathematics, St Francis College, Brooklyn Heights, USA.)

Received: 2022-05-28 Accepted : 2022-05-29 Published : 2022-11-07

Keywords: L-algebra‎, ‎Cyclic L-algebra‎, ‎MV-algebra‎, ‎Equalizer‎, ‎Product‎, Co-product‎,

Abstract :

In this paper, we define and study the category of L-algebras, proving that this category has equalizers, coequalizers, kernel pairs and products. We investigate the existence of injective objects in this category and show that an object in the subcategory of cyclic L-algebras is injective if and only if it is a complete and divisible cyclic L-algebra.

References:

[1] S. S. Ahn, On injective BCK-algebras, Honam Math. J., 29 (2007), 289-297.
[2] N. Akhlaghinia, R. A. Borzooei, M. A. Kologani and X. L. Xin, On the category of EQ-algebras, Bull. Sect. Logic, https://doi.org/10.18778/0138-0680.2021.01, (2021).

‎[3] B‎. ‎Bucsneag‎, ‎Categories of algebraic logic‎, ‎‎Editura Academiei Romane‎, ‎Bucharest, (2006)‎.

[4] B. Busneag and M. Ghita, Some properties of epimorphisms of Hilbert algebras, Centr. Eur. J. Math., 8 (2010), 41-52.
[5] C. C. Chang, Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467-490.
[6] M. A. Chaudhry and H. Fakhar-Ud-Din, Some categorical aspects of BCH-algebras, IJMMS, 27 (2003), 1739-1750.
[7] L. C. Ciungu, Results in L-algebras, Algebra Univ., 82(1) (2021), Paper no. 7.
[8] L. C. Ciungu, Quantifiers on L-algebras, Math. Slovaca, accepted for publication.
[9] A. Dvurecenskij and S. Pulmannova, New trends in Quantum Structures, Kluwer Academic Publishers, Dordrecht, Ister Science, Bratislava, (2000).
[10] A. Dvurecenskij and O. Zahiri, On epicomplete MV-algebras, FLAP, 5 (2018), 165-184.
[11] G. Dymek, On the category of pseudo BCI-algebras, Demonstr. Math., 4 (2013), 631-644.
[12] G. Dymek, An injective pseudo-BCI algebra is trivial, Discuss. Math. Gen. Algebra Appl., 39 (2019) 221-229.
[13] M. Ghita, Some categorical properties of Hilbert algebras, An. Univ. Craiova Ser. Mat. Inform., 36 (2009), 95-104.
[14] X. J. Hua, State L-algebras and derivations of L-algebras, Soft Comput., 25 (2021), 4201-4212.
[15] A. Iorgulescu, Classes of BCK-algebras-Part III, Preprint Series of the Institute of Mathematics of the Romanian Academy, 3 (2004), 1-42.

[16] B. Mitchell, Theory of Categories, Pure and Applied Mathematics, vol. XVII, Academic Press, New York, (1965).
[17] N. Popescu and A. Radu, Theory of categories and sheaves, Editura Stiintifica, Bucharest, (in Romanian), (1971).
[18] L. Popescu and N. Popescu, Theory of categories, Editura Academiei Romane, Bucharest and Sijthoff & Noordhoff International Publishers, (1979).
[19] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equations, Adv. Math., 193 (2005), 40-55.
[20] W. Rump, L-algebras, self-similarity, and ℓ-groups, J. Algebra, 320 (2008), 2328-2348.
[21] W. Rump and Y. Yang, Intervals in ℓ-groups as L-algebras, Algebra Univ., 67 (2012), 121-130.
[22] W. Rump, Right ℓ-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation, J. Algebra, 439 (2015), 470-510.
[23] W. Rump, The structure group of an L-algebra is torsion-free, J. Group Theory, 20 (2017), 309-324.
[24] W. Rump, The L-algebra of Hurwitz primes, J. Number Theory, 190 (2018), 394-413.
[25] W. Rump, Von Neumann algebras, L-algebras, Baer*-monoids, and Garside groups, Forum Math., 30 (2018), 973-995.
[26] W. Rump, The structure group of a generalized orthomodular lattice, Stud. Logica, 106 (2018), 85-100.
[27] E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag, (1999).
[28] Y. Wu, J. Wang and Y. Yang, Lattice-ordered effect algebras and L-algebras, Fuzzy Sets Syst., 369 (2019), 103-113.
[29] Y. Wu and Y. Yang, Orthomodular lattices as L-algebras, Soft Comput., 24 (2020), 14391-14400.
[30] Y. Yang and W. Rump, Pseudo-MV algebras as L-algebras, J. Mult.-Valued Logic Soft Comput., 19 (2012), 621-632.
[31] H. Yutani, Colimits in the category of BCK-algebras, Math. Japonica, 30 (1985), 527-534.
[32] S. M. A. Zaidi and S. Khan, The Category of p-semisimple BCI-algebra, Kyungpook Math. J., 43 (2003), 279-291.