Ideals and congruences in pseudo-BCH algebras
-
Andrzej Walendziak
Abstract
We introduce the concept of a translation ideal in pseudo-BCH algebras and investigate its properties. We prove that for any pseudo-BCH algebra 𝔛 there is a one-to-one order-preserving correspondence between the closed translation ideals of 𝔛 and the relative congruences on 𝔛.
(Communicated by Jan Kühr)
Acknowledgement
The author is indebted to the referee for his/her very careful reading and suggestions.
References
[1] Bhatti, S. A.—Zhang, X. H.: Strong ideals, associative ideals and p-ideals in BCI-algebras, Punjab Univ. J. Math. 27 (1994), 113–120.Search in Google Scholar
[2] Chaudhry, M. A.: On BCH-algebras, Math. Japonica 36 (1991), 665–676.Search in Google Scholar
[3] Dudek, W. A.—Jun, Y. B.: Pseudo-BCI-algebras, East Asian Math. J. 24 (2008), 187–190.Search in Google Scholar
[4] Dudek, W. A.—Thomys, J.: On decompositions of BCH-algebras, Math. Japonica 35 (1990), 1131–1138.Search in Google Scholar
[5] Dymek, G.: On two classes of pseudo-BCI-algebras, Discuss. Math. Gen. Algebra Appl. 31 (2011), 217–230.10.7151/dmgaa.1184Search in Google Scholar
[6] Dymek, G.: p-semisimple pseudo-BCI-algebras, J. Mult.-Valued Logic Soft Comput. 19 (2012), 461–474.Search in Google Scholar
[7] Dymek, G.: On the category of pseudo-BCI-algebras, Demonstratio Math. 46 (2013), 631–644.10.1515/dema-2013-0479Search in Google Scholar
[8] Dymek, G.: On pseudo-BCI-algebras, Ann. Univ. Mariae Curie-Skodowska Sect. A 69 (2015), 59–71.10.17951/a.2015.69.1.59Search in Google Scholar
[9] Georgescu, G.—Iorgulescu, A.: Pseudo-MV algebras: a noncommutative extension of MV algebras. In: The Proc. of the Fourth International Symp. on Economic Informatics, Bucharest, Romania, May 1999, pp. 961–968.Search in Google Scholar
[10] Georgescu, G.—Iorgulescu, A.: Pseudo-BL algebras: a noncommutative extension of BL algebras. In: Abstracts of the Fifth International Conference FSTA 2000, Slovakia, February 2000, pp. 90–92.Search in Google Scholar
[11] Georgescu, G.—Iorgulescu, A.: Pseudo-BCK algebras: an extension of BCK algebras. In: Proc. of DMTCS’01: Combinatorics, Computability and Logic, Springer, London, 2001, pp. 97–114.10.1007/978-1-4471-0717-0_9Search in Google Scholar
[12] Hu, Q. P.—Li, X.: On BCH-algebras, Math. Seminar Notes 11 (1983), 313–320.Search in Google Scholar
[13] Iorgulescu, A.: New generalizations of BCI, BCK and Hilbert algebras - Part I, J. Mult.-Valued Logic Soft Comput. 27 (2016), 353–406.Search in Google Scholar
[14] Iorgulescu, A.: New generalizations of BCI, BCK and Hilbert algebras - Part II, J. Mult.-Valued Logic Soft Comput. 27 (2016), 407–456.Search in Google Scholar
[15] Imai, Y.—Iséki, K.: On axiom systems of propositional calculi XIV, Proc. Japan Acad. Ser. A Math. Sci. 42 (1966), 19–22.10.3792/pja/1195522169Search in Google Scholar
[16] Iséki, K.: An algebra related with a propositional culculus, Proc. Japan Acad. Ser. A Math. Sci. 42 (1966), 26–29.10.3792/pja/1195521766Search in Google Scholar
[17] Jun, Y. B.—Kim, H. S.—Neggers, J.: On pseudo-BCI ideals of pseudo-BCI algebras, Mat. Vesnik 58 (2006), 39–46.Search in Google Scholar
[18] Kim, Y. H.—So, K. S.: On minimality in pseudo-BCI-algebras, Commun. Korean Math. Soc. 27 (2012), 7–13.10.4134/CKMS.2012.27.1.007Search in Google Scholar
[19] Lee, K. J.—Park, C. H.: Some ideals of pseudo-BCI-algebras, J. Appl. Math. Inform. 27 (2009), 217–231.Search in Google Scholar
[20] Lu, Y.—Zhang, X.: Q-type pseudo-BCI algebras and 2-type pseudo-q filters, Journal of Ningbo University 24 (2011), 56–60.Search in Google Scholar
[21] Roh, E. H.—Kim, S. Y.—Jun, Y. B.: On a problem in BCH-algebras, Sci. Math. Jpn. 52 (2000), 279–283.Search in Google Scholar
[22] Walendziak, A.: Pseudo-BCH-algebras, Discuss. Math. Gen. Algebra Appl. 35 (2015), 1–15.10.7151/dmgaa.1233Search in Google Scholar
[23] Walendziak, A.: On ideals of pseudo-BCH-algebras, Ann. Univ. Mariae Curie-Skodowska Sect. A 70 (2016), 81–91.10.17951/a.2016.70.1.81Search in Google Scholar
[24] Walendziak, A.: Strong ideals and horizontal ideals in pseudo-BCH-algebras, Ann. Univ. Paedagog. Crac. Stud. Math. 15 (2016), 15–25.10.1515/aupcsm-2016-0002Search in Google Scholar
[25] Wroński, A.: BCK-algebras do not form a variety, Math. Japonica 28 (1983), 211–213.Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations
Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations
