Commutative pseudo BE-algebras


Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419, Usa


The aim of this paper is to introduce the notion of commutative pseudo BE-algebras and investigate their properties.
We generalize some results proved by A. Walendziak for the case of commutative BE-algebras.
We prove that the class of commutative pseudo BE-algebras is equivalent to the class of commutative pseudo BCK-algebras. Based on this result, all results holding for commutative pseudo BCK-algebras also hold for commutative pseudo BE-algebras. For example, any finite commutative pseudo BE-algebra is a BE-algebra, and any commutative pseudo BE-algebra is a join-semilattice. Moreover, if a commutative pseudo BE-algebra is a meet-semilattice, then it is a distributive lattice. We define the pointed pseudo-BE algebras, and introduce and study the relative negations on pointed pseudo BE-algebras. Based on the relative negations we construct two closure operators on a pseudo BE-algebra.
We also define relative involutive pseudo BE-algebras, we investigate their properties and prove equivalent conditions for a relative involutive pseudo BE-algebra.
We define the relative Glivenko property for a relative good pseudo BE-algebra and show that any relative
involutive pseudo BE-algebra has the relative Glivenko property.


[1] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algebras, Scientiae Mathematicae
Japonicae, 68(2) (2008), 279-285.

[2] S. S. Ahn, Y. H. Kim and J. M. Ko, Filters in commutative BE-algebras, Communications
of the Korean Mathematical Society, 27(2) (2012), 233-242.
[3] A. Borumand Saeid, Smarandache BE-algebras, Education Publisher, Columbus, Ohio, USA,
[4] R. A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, On pseudo BE-
algebras, Discussiones Mathematicae General Algebra and Applicationes, 33(1) (2013), 95-
[5] R. A. Borzooei, A. Borumand Saeid and R. Ameri, States on BE-algebras, Kochi Journal of
Mathematics, 9(1) (2014), 27-42.
[6] R. A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, Distributive pseudo
BE{algebras, Fasciculi Mathematici, 54(1) (2015), 21-39.
[7] R. Cignoli and A. Torrens, Glivenko like theorems in natural expansions of BCK-logic, Mathematical
Logic Quaterly, 50(2) (2004), 111-125.
[8] R. Cignoli and A. Torrens, Free Algebras in Varieties of Glivenko MTL-algebras Satisfying
the Equation 2(x2) = (2x)2, Studia Logica, 83(1-3) (2006), 157-181.
[9] Z. Ciloglu and Y. Ceven, Commutative and bounded BE-algebras, Hindawi Publishing Corporation,
2013(1) (2013), Article ID 473714.
[10] L. C. Ciungu and A. Dvurecenskij, Measures, states and de Finetti maps on pseudo-BCK
algebras, Fuzzy Sets and Systems, 161(22) (2010), 2870-2896.
[11] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part I, Archive for
Mathematical Logic, 52(3-4) (2013), 335-376.
[12] L. C. Ciungu and J. Kuhr, New probabilistic model for pseudo-BCK algebras and pseudo-
hoops, Journal of Multiple-Valued Logic and Soft Computing, 20(3-4) (2013), 373-400.
[13] L. C. Ciungu, Non-commutative multiple-valued logic algebras, Springer, Cham, Heidelberg,
New York, Dordrecht, London, 2014.
[14] L. C. Ciungu, Relative negations in non-commutative fuzzy structures, Soft Computing,
18(1) (2014), 15-33.
[15] A. Dvurecenskij and O. Zahiri, Pseudo equality algebras: revision, Soft Computing, doi:
10.1007/s00500-015-1888-x, (2015).
[16] G. Georgescu and A. Iorgulescu, Pseudo MV-algebras, Multiple-Valued Logic, 6(1-2) (2001),
[17] G. Georgescu and A. Iorgulescu, Pseudo-BCK algebras: An extension of BCK-algebras, Proceedings
of DMTCS'01: Combinatorics, Computability and Logic, Springer, London, (2001),
[18] Y. Imai and K. Iseki, On axiom systems of propositional calculi XIV, Proceedings of the
Japan Academy, 42(1) (1966), 19-22.
[19] A. Iorgulescu, Classes of pseudo-BCK algebras - Part I, Journal of Multiple-Valued Logic
and Soft Computing, 12(1-2) (2006), 71-130.
[20] A. Iorgulescu, Algebras of logic as BCK-algebras, ASE Ed., Bucharest, 2008.
[21] H. S. Kim and Y. H. Kim, On BE-algebras, Scientiae Mathematicae Japonicae, 66(1) (2007),
[22] K. H. Kim and Y. H. Yon, Dual BCK-algebra and MV-algebra, Scientiae Mathematicae
Japonicae, 66(2) (2007), 247-254.
[23] J. Kuhr, Pseudo-BCK semilattices, Demonstratio Mathematica, 40(3) (2007), 495-516.
[24] J. Kuhr, Pseudo-BCK algebras and related structures, Habilitation thesis, Palacky University
in Olomouc, 2007.
[25] J. Kuhr, Commutative pseudo BCK-algebras, Southeast Asian Bulletin of Mathematics,
33(3) (2009), 451-475.
[26] K. J. Lee, Pseudo-valuations on BE-algebras, Applied Mathematical Sciences, 7(125) (2013),
[27] B. L. Meng, CI-algebras, Scientiae Mathematicae Japonicae, 71(1) (2010), 11-17.
[28] B. L. Meng, On lters in BE-algebras, Scientiae Mathematicae Japonicae, 71(2) (2010),

[29] J. Rachunek, A non-commutative generalization of MV-algebras, Czechoslovak Mathematical
Journal, 52(127) (2002), 255-273.
[30] A. Rezaei and A. Borumand Saeid, On fuzzy subalgebras of BE-algebras, Afrika Matematika,
22(2) (2011), 115-127.
[31] A. Rezaei and A. Borumand Saeid, Some results in BE-algebras, Annals of Oradea University
- Mathematics Fascicola, XIX(1) (2012), 33-44.
[32] A. Rezaei and A. Borumand Saeid, Commutative ideals in BE-algebras, Kyungpook Mathematical
Journal, 52(4) (2012), 483-494.
[33] A. Rezaei, A. Borumand Saeid and R. A. Borzooei, Relation between Hilbert algebras and
BE-algebras, Applications and Applied Mathematics, 8(2) (2013), 573-584.
[34] A. Rezaei, A. Borumand Saeid, A. Radfar and R. A. Borzooei, Congruence relations on
pseudo BE-algebras, Annnals of the University of Craiova, Mathematics and Computer Sciences
Series, 41(2) (2014), 166-176.
[35] A. Rezaei, L. C. Ciungu and A. Borumand Saeid, States on pseudo BE-algebras, submitted.
[36] A. Walendziak, On commutative BE-algebras, Scientiae Mathematicae Japonicae, 69(2)
(2009), 281-284.
[37] A. Walendziak, On normal lters and congruence relations in BE-algebras, Commentationes
Mathematicae, 52(2) (2012), 199-205.
[38] H. Zhou and B. Zhao, Generalized Bosbach and Riecan states based on relative negations in
residuated lattices, Fuzzy Sets and Systems, 187(1) (2012), 33-57.