Algebras inspired by logics

W.A.Dudek

In this paper we give a short survey of author's results on algebras connected with non-classical logic. At first we describe groupoids inspired by Leśniewski's equivalential calculus. Next we present a new computer method of computation of Iséki's BCK-algebras of small orders, which may be used also to computation of Komori's BCC-algebras. In detail we describe the connection of initial segments and some types of ideals with congruences in BCC-algebras. Finally, we present a method of fuzzification of ideals in BCC-algebras.

1. Y. Arai, On axiom systems of propositional calculi. XVII, Proc. Japan Acad. 42 (1966), 351–354.

2. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

3. W. M. Bunder, BCK and related algebras and their corresponding logics, The Journal of non-classical logic, 7 (1983), 15–24.

4. W. A. Dudek, On BCC-algebras, Logique et Analyse, 129–130 (1990), 103–111.

5. W. A. Dudek, The number of subalgebras of finite BCC-algebras, Bul. Inst. Math. Acad. Sinica 20 (1992), 129–135.

6. W. A. Dudek, On proper BCC-algebras, Bul. Inst. Math. Acad. Sinica 20 (1992), 137–150.

7. W. A. Dudek, On constructions of BCC-algebras, Selected Papers on BCK- and BCI-algebras 1 (1992), 93–96.

8. W. A. Dudek, On the axioms system for BCI-algebras, Matematychni Studii 3 (1994), 5–9.

9. W. A. Dudek, Algebras inspired by logics, Abstract of Talks of the International Conference on Mathematical Logic, Novosibirsk 1994, 45–46.

10. W. A. Dudek, Algebras connected with the equivalential calculus, Math. Montisnigri 4 (1995), 13–18.

11. W. A. Dudek, Algebras motivated by the equivalential calculus, Rivista Mat. Pura et Appl. 17 (1996), 107–112.

12. W. A. Dudek, Remarks on the axioms system for BCI-algebras, Prace Naukowe WSP w Czestochowie, ser. Matematyka, 2 (1996), 46–61.

13. W. A. Dudek, A computer method of computation of BCK and BCI-algebras of small orders, Proceedings of the XI Conference on Applied Mathematics, PRIM'96, Novi Sad 1997, 41–64.

14. W. A. Dudek, On embedding Hilbert algebras in BCK-algebras, Math. Moravica 3 (1999), 25–28.

15. W. A. Dudek, Algebras inspired by the equivalential calculus, Italian J. Pure and Appl. Math. 8 (1999), (in print)

16. W. A. Dudek and Y. B. Jun, Fuzzy BCC-ideals in BCC-algebras, Math. Montisnigri 10 (1999), 21–30.

17. W. A. Dudek, Y. B. Jun and S. M. Hong, On fuzzy topological BCC-algebras, Discussiones Math., General Algebra and Appl. 20 (2000), ( in print).

18. W. A. Dudek, Y. B. Jun and Z. Stojaković, On fuzzy ideals in BCC-algebras, Fuzzy Sets and Systems ( in print).

19. W. A. Dudek, R. Rousseau, Set-theoretic relations and BCH-algebras with trivial structure, Zbornik Radova Prirod.-Mat. Fak. Univ. u Novom Sadu 25.1 (1995), 75–82.

20. W. A. Dudek, J. Thomys, On decompositions of BCH-algebras, Math. Japonica 35 (1990), 1131–1138.

21. W. A. Dudek and X. H. Zhang, On atoms in BCC-algebras, Discussiones Math., Algebra and Stoch. Methods 15 (1995), $81-85.$

22. W. A. Dudek, X. H. Zhang, Initial segments in BCC-algebras, International Algebraic Conference, Slovyans'k 1997, 29–30.

23. W. A. Dudek, X. H. Zhang, On ideals and congruences in BCC-algebras, Czech. Math. J. 48 (123) (1998), 21–29.

24. I. Fleischer, Subdirect product of totally ordered BCK-algebras, J. Algebra 111 (1987), 384–387.

25. H. G. Forder, J. A. Kalman, Implication in equational logic, Math. Gazette 46 (1962), 122–126.

26. K. Iséki, On axiom systems of propositional calculi. XV, Proc. Japan Acad. 42 (1966), 217–220.

27. K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica 23 (1978), 1–26.

28. K. Iséki, S. Tanaka, Ideal theory of BCK-algebras, Math. Japonica 21(1976), 351–366.

29. Z. Jordan, The development of mathematical logic and logical positivism in Poland between the two wars, being Polish Science and Learning No 6 (Oxford University Press, 1945).

30. Y. Komori, The variety generated by BCC-algebras is finitely based, Reports Fac. Sci. Shizuoka Univ. 17 (1983), 13–16.

31. S. Leśniewski, Grundrüge eines neuen Systems der Grundlagen der Mathematik, Fund. Math. 14 (1929), 1–81.

32. H.Rasiowa, An algebraic approach to non-classical logics, North-Holland, Amsterdam 1974.

33. B. Sobociński, An investigation of protothetic, Éditions de l'Institut d'Études Polonaises en Belgique (Brussels, 1949).

34. S. Tanaka, On axiom systems of propositional calculi. XVIII, Proc. Japan Acad. 42 (1966), 355–357.

35. T. Traczyk, On the variety of bounded commutative BCK-algebras, Math. Japonica 24 (1979), 283–294.

36. A. Wroński, BCK-algebras do not form a variety, Math. Japonica 28 (1983), 211–213.

37. Y. Yu, J. N. Mordeson and S. C. Cheng, Elements of L-algebra, Lecture Notes in Fuzzy Math. and Computer Sci., Creighton Univ., Omaha, Nebraska (USA) 1994.

38. L. A. Zadeh, Fuzzy sets, Inform. control. 8 (1965), 338–353.

	Algebras inspired by logics
Author	W.A.Dudek dudek@im.pwr.wroc.pl Institute of ,Mathematics WrocUniversity of Technology, ,Wybrzeże ,Wyspiańskiego 27, ,50-370 Wroc, Poland
Abstract	In this paper we give a short survey of author's results on algebras connected with non-classical logic. At first we describe groupoids inspired by Leśniewski's equivalential calculus. Next we present a new computer method of computation of Iséki's BCK-algebras of small orders, which may be used also to computation of Komori's BCC-algebras. In detail we describe the connection of initial segments and some types of ideals with congruences in BCC-algebras. Finally, we present a method of fuzzification of ideals in BCC-algebras.
Keywords	non-classical logic algebras, Leśniewski equivalential calculus, groupoids, BCK-algebras, BCC-algebras, Iséki BCK-algebras, Komori BCC-algebras, ideals in BCC-algebras,
DOI	doi:10.30970/ms.14.1.3-18
Reference	1. Y. Arai, On axiom systems of propositional calculi. XVII, Proc. Japan Acad. 42 (1966), 351–354. 2. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96. 3. W. M. Bunder, BCK and related algebras and their corresponding logics, The Journal of non-classical logic, 7 (1983), 15–24. 4. W. A. Dudek, On BCC-algebras, Logique et Analyse, 129–130 (1990), 103–111. 5. W. A. Dudek, The number of subalgebras of finite BCC-algebras, Bul. Inst. Math. Acad. Sinica 20 (1992), 129–135. 6. W. A. Dudek, On proper BCC-algebras, Bul. Inst. Math. Acad. Sinica 20 (1992), 137–150. 7. W. A. Dudek, On constructions of BCC-algebras, Selected Papers on BCK- and BCI-algebras 1 (1992), 93–96. 8. W. A. Dudek, On the axioms system for BCI-algebras, Matematychni Studii 3 (1994), 5–9. 9. W. A. Dudek, Algebras inspired by logics, Abstract of Talks of the International Conference on Mathematical Logic, Novosibirsk 1994, 45–46. 10. W. A. Dudek, Algebras connected with the equivalential calculus, Math. Montisnigri 4 (1995), 13–18. 11. W. A. Dudek, Algebras motivated by the equivalential calculus, Rivista Mat. Pura et Appl. 17 (1996), 107–112. 12. W. A. Dudek, Remarks on the axioms system for BCI-algebras, Prace Naukowe WSP w Czestochowie, ser. Matematyka, 2 (1996), 46–61. 13. W. A. Dudek, A computer method of computation of BCK and BCI-algebras of small orders, Proceedings of the XI Conference on Applied Mathematics, PRIM'96, Novi Sad 1997, 41–64. 14. W. A. Dudek, On embedding Hilbert algebras in BCK-algebras, Math. Moravica 3 (1999), 25–28. 15. W. A. Dudek, Algebras inspired by the equivalential calculus, Italian J. Pure and Appl. Math. 8 (1999), (in print) 16. W. A. Dudek and Y. B. Jun, Fuzzy BCC-ideals in BCC-algebras, Math. Montisnigri 10 (1999), 21–30. 17. W. A. Dudek, Y. B. Jun and S. M. Hong, On fuzzy topological BCC-algebras, Discussiones Math., General Algebra and Appl. 20 (2000), ( in print). 18. W. A. Dudek, Y. B. Jun and Z. Stojaković, On fuzzy ideals in BCC-algebras, Fuzzy Sets and Systems ( in print). 19. W. A. Dudek, R. Rousseau, Set-theoretic relations and BCH-algebras with trivial structure, Zbornik Radova Prirod.-Mat. Fak. Univ. u Novom Sadu 25.1 (1995), 75–82. 20. W. A. Dudek, J. Thomys, On decompositions of BCH-algebras, Math. Japonica 35 (1990), 1131–1138. 21. W. A. Dudek and X. H. Zhang, On atoms in BCC-algebras, Discussiones Math., Algebra and Stoch. Methods 15 (1995), $81-85.$ 22. W. A. Dudek, X. H. Zhang, Initial segments in BCC-algebras, International Algebraic Conference, Slovyans'k 1997, 29–30. 23. W. A. Dudek, X. H. Zhang, On ideals and congruences in BCC-algebras, Czech. Math. J. 48 (123) (1998), 21–29. 24. I. Fleischer, Subdirect product of totally ordered BCK-algebras, J. Algebra 111 (1987), 384–387. 25. H. G. Forder, J. A. Kalman, Implication in equational logic, Math. Gazette 46 (1962), 122–126. 26. K. Iséki, On axiom systems of propositional calculi. XV, Proc. Japan Acad. 42 (1966), 217–220. 27. K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica 23 (1978), 1–26. 28. K. Iséki, S. Tanaka, Ideal theory of BCK-algebras, Math. Japonica 21(1976), 351–366. 29. Z. Jordan, The development of mathematical logic and logical positivism in Poland between the two wars, being Polish Science and Learning No 6 (Oxford University Press, 1945). 30. Y. Komori, The variety generated by BCC-algebras is finitely based, Reports Fac. Sci. Shizuoka Univ. 17 (1983), 13–16. 31. S. Leśniewski, Grundrüge eines neuen Systems der Grundlagen der Mathematik, Fund. Math. 14 (1929), 1–81. 32. H.Rasiowa, An algebraic approach to non-classical logics, North-Holland, Amsterdam 1974. 33. B. Sobociński, An investigation of protothetic, Éditions de l'Institut d'Études Polonaises en Belgique (Brussels, 1949). 34. S. Tanaka, On axiom systems of propositional calculi. XVIII, Proc. Japan Acad. 42 (1966), 355–357. 35. T. Traczyk, On the variety of bounded commutative BCK-algebras, Math. Japonica 24 (1979), 283–294. 36. A. Wroński, BCK-algebras do not form a variety, Math. Japonica 28 (1983), 211–213. 37. Y. Yu, J. N. Mordeson and S. C. Cheng, Elements of L-algebra, Lecture Notes in Fuzzy Math. and Computer Sci., Creighton Univ., Omaha, Nebraska (USA) 1994. 38. L. A. Zadeh, Fuzzy sets, Inform. control. 8 (1965), 338–353.
Pages	3-18
Volume	14
Issue	1
Year	2000
Journal	Matematychni Studii
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