PC-lattices: A Class of Bounded BCK-algebras

Sadegh Khosravi Shoar; Rajab Ali Borzooei; R. Moradian; Atefe Radfar

Bulletin of the Section of Logic (2018)

  • Volume: 47, Issue: 1
  • ISSN: 0138-0680

Abstract

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In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition (S) and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition (S) is a distributive BCK-algebra.  

How to cite

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Sadegh Khosravi Shoar, et al. "PC-lattices: A Class of Bounded BCK-algebras." Bulletin of the Section of Logic 47.1 (2018): null. <http://eudml.org/doc/295528>.

@article{SadeghKhosraviShoar2018,
abstract = {In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition (S) and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition (S) is a distributive BCK-algebra.  },
author = {Sadegh Khosravi Shoar, Rajab Ali Borzooei, R. Moradian, Atefe Radfar},
journal = {Bulletin of the Section of Logic},
keywords = {PC-lattice; BCK-lattice; Involutory BCK-algebras; Bounded commutative BCK-algebras},
language = {eng},
number = {1},
pages = {null},
title = {PC-lattices: A Class of Bounded BCK-algebras},
url = {http://eudml.org/doc/295528},
volume = {47},
year = {2018},
}

TY - JOUR
AU - Sadegh Khosravi Shoar
AU - Rajab Ali Borzooei
AU - R. Moradian
AU - Atefe Radfar
TI - PC-lattices: A Class of Bounded BCK-algebras
JO - Bulletin of the Section of Logic
PY - 2018
VL - 47
IS - 1
SP - null
AB - In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition (S) and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition (S) is a distributive BCK-algebra.  
LA - eng
KW - PC-lattice; BCK-lattice; Involutory BCK-algebras; Bounded commutative BCK-algebras
UR - http://eudml.org/doc/295528
ER -

References

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  1. [1] C. Bărbăcioru, Positive implicative BCK-algebras, Mathematica Japonica 36 (1967), pp. 11–59. 
  2. [2] R. A. Borzooei, S. Khosravi Shoar, Implication Algebras are Equivalent to the Dual Implicative BCK-algebras, Scientiae Mathematicae Japonicae 633 (2006), pp. 429–431. 
  3. [3] R. A. Borzooei, S. Khosravi Shoar, R. Ameri, Some new filters in MTL-algebras, Fuzzy Sets and Systems 187(1) (2012), pp. 92–102. 
  4. [4] B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990, 2002. 
  5. [5] G. Grätzer, General Lattice Theory, Academic Press, 1978. 
  6. [6] Y. Huang, BCI-algebras, Science Press, 2006. 
  7. [7] Y. Huang, On involutory BCK-algebras, Soochow Journal of Mathematics 32(1) (2006), pp. 51–57. 
  8. [8] Y. Imai, K. Iséki, On axioms systems of propositional calculi XIV, Proceedings of the Japan Academy 42 (1966), pp. 19–22. 
  9. [9] K. Iséki, BCK-algebras with condition (S), Mathematica Japonica 24 (1979), pp. 107–119. 
  10. [10] K. lséki, On positive implicative BCK-algebras with condition (S), Mathematica Japonica 24 (1979), pp. 107–119. 
  11. [11] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica 23 (1978), pp. 1–26. 
  12. [12] J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa Co, Seoul, Korea, 1994. 
  13. [13] S. Tanaka, A new class of algebras, Mathematics Seminar Notes 3 (1975), pp. 37–43. 
  14. [14] S. Tanaka, On ^-commutative algebras, Mathematics Seminar Notes 3 (1975), pp. 59–64. 

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