On the axioms system for BCI-algebras

W.A.~Dudek

BCI-algebras and related systems such as BCK-algebras, BCC-algebras, BCH-algebras etc. are motivated by implicational logic and by propositional calculi. We prove that the class of BCI-algebras forms a quasivariety of groupoids $(G,\bullet,0)$ determined by the following independent axioms system: (1) $((xy)(xz))(zy)=0$, (4) $xy=yx=0$ implies $x=y$, (6) $x0=x$. The class of all BCK-algebras (connected with BCC-logic) is a quasivariety defined by an independent axioms system: (1), (4), (6) and $0x=0$. The class of BCK-algebras satisfying (9) $x(xy)=y(yx)$ is a variety defined by (1), (6) and (9).

1. Bunder W.M. BCK and related algebras and their corresponding logics // The Journal of Non-classical Logic. 1983. V.7. P.15–24.

2. Cornish W.H. A multipler approach to implicative BCK-algebras // Math. Seminar Notes. 1980. V.8. P.157–169.

3. Cornish W.H. A 3-distributive 2-based variety // Math. Seminar Notes. 1980. V.8. P.381–387.

4. Cornish W.H. Two independent varieties of BCI-algebras // Math. Seminar Notes. 1980. V.8. P.413–420.

5. Dudek W.A. On BCC-algebras // Logique et Analyse. 1990. V.129–139. P.103–111.

6. Dudek W.A. Algebraic characterization of the equivalential calculus // Rivista di Mat. Pura ed Appl (Udine). V.17 (in print).

7. Dudek W.A., Thomys J. On decompositions of BCH-algebras // Math. Japonica. 1990. V.35. P.1131–1138.

8. Fleischer I. Subdirect product of totally ordered BCK-algebras // J. Algebra. 1987. V.111. P.384–387.

9. Iséki K., Tanaka S. An introduction to the theory of BCK-algebras // Math. Japonica. 1978. V.23. P.1–26.

10. Pa asiński M. Some remarks on BCK-algebras // Math. Seminar Notes. 1980. V.8. P.137–144.

11. Shi L.H. An axioms system of BCI-algebras // Math. Japonica. 1990. V.30. P.351–352.

12. Traczyk T. On the variety of bounded commutative BCK-algebras // Math. Japonica. 1979. V.24. P.283–294.

13. Yutani H. On a system of axioms of a commutative BCK-algebras // Math. Seminar Notes. 1977. V.5. P.255-256. Institute of Mathematics, Technical University, Wybrze ze Wyspiańskiego 27, 50-370 Wroc aw, Poland e-mail: wad2plwrtu11.bitnet

	On the axioms system for BCI-algebras
Author	W.A.~Dudek wad2@plwrtu11.bitnet Institute of Mathematics, Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Abstract	BCI-algebras and related systems such as BCK-algebras, BCC-algebras, BCH-algebras etc. are motivated by implicational logic and by propositional calculi. We prove that the class of BCI-algebras forms a quasivariety of groupoids $(G,\bullet,0)$ determined by the following independent axioms system: (1) $((xy)(xz))(zy)=0$, (4) $xy=yx=0$ implies $x=y$, (6) $x0=x$. The class of all BCK-algebras (connected with BCC-logic) is a quasivariety defined by an independent axioms system: (1), (4), (6) and $0x=0$. The class of BCK-algebras satisfying (9) $x(xy)=y(yx)$ is a variety defined by (1), (6) and (9).
Keywords	BCI-algebras, BCK-algebras, BCC-algebras, BCH-algebras, implicational logic, propositional calculi
DOI	doi:10.30970/ms.3.1.5-9
Reference	1. Bunder W.M. BCK and related algebras and their corresponding logics // The Journal of Non-classical Logic. 1983. V.7. P.15–24. 2. Cornish W.H. A multipler approach to implicative BCK-algebras // Math. Seminar Notes. 1980. V.8. P.157–169. 3. Cornish W.H. A 3-distributive 2-based variety // Math. Seminar Notes. 1980. V.8. P.381–387. 4. Cornish W.H. Two independent varieties of BCI-algebras // Math. Seminar Notes. 1980. V.8. P.413–420. 5. Dudek W.A. On BCC-algebras // Logique et Analyse. 1990. V.129–139. P.103–111. 6. Dudek W.A. Algebraic characterization of the equivalential calculus // Rivista di Mat. Pura ed Appl (Udine). V.17 (in print). 7. Dudek W.A., Thomys J. On decompositions of BCH-algebras // Math. Japonica. 1990. V.35. P.1131–1138. 8. Fleischer I. Subdirect product of totally ordered BCK-algebras // J. Algebra. 1987. V.111. P.384–387. 9. Iséki K., Tanaka S. An introduction to the theory of BCK-algebras // Math. Japonica. 1978. V.23. P.1–26. 10. Pa asiński M. Some remarks on BCK-algebras // Math. Seminar Notes. 1980. V.8. P.137–144. 11. Shi L.H. An axioms system of BCI-algebras // Math. Japonica. 1990. V.30. P.351–352. 12. Traczyk T. On the variety of bounded commutative BCK-algebras // Math. Japonica. 1979. V.24. P.283–294. 13. Yutani H. On a system of axioms of a commutative BCK-algebras // Math. Seminar Notes. 1977. V.5. P.255-256. Institute of Mathematics, Technical University, Wybrze ze Wyspiańskiego 27, 50-370 Wroc aw, Poland e-mail: wad2plwrtu11.bitnet
Pages	5-9
Volume	3
Issue	1
Year	1994
Journal	Matematychni Studii
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