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pseudo-BCI-algebra; directoid; antitone mapping; pseudo-BCI-structure
In Chajda's paper (2014), to an arbitrary BCI-algebra the author assigned an ordered structure with one binary operation which possesses certain antitone mappings. In the present paper, we show that a similar construction can be done also for pseudo-BCI-algebras, but the resulting structure should have two binary operations and a set of couples of antitone mappings which are in a certain sense mutually inverse. The motivation for this approach is the well-known fact that every commutative BCK-algebra is in fact a join-semilattice and we try to obtain a similar result also for the non-commutative case and for pseudo-BCI-algebras which generalize BCK-algebras, see e.g. Imai and Iséki (1966) and Iséki (1966).
[1] Chajda, I.: A structure of BCI-algebras. Int. J. Theor. Phys. 53 (2014), 3391-3396. DOI 10.1007/s10773-013-1739-4 | MR 3253801 | Zbl 1302.81032
[2] Chajda, I., Länger, H.: On the structure of pseudo-BCK algebras. (to appear) in J. Multiple-Valued Logic Soft Computing.
[3] Chajda, I., L{ä}nger, H.: Directoids. An Algebraic Approach to Ordered Sets. Research and Exposition in Mathematics 32 Heldermann, Lemgo (2011). MR 2850357 | Zbl 1254.06002
[4] Ciungu, L. C.: Non-commutative Multiple-Valued Logic Algebras. Springer Monographs in Mathematics Springer, Cham (2014). MR 3112745 | Zbl 1279.03003
[5] Dudek, W. A., Jun, Y. B.: Pseudo-BCI algebras. East Asian Math. J. 24 (2008), 187-190. Zbl 1149.06010
[6] Dymek, G.: On two classes of pseudo-BCI-algebras. Discuss. Math., Gen. Algebra Appl. 31 (2011), 217-229. DOI 10.7151/dmgaa.1184 | MR 2953913 | Zbl 1258.06014
[7] Dymek, G., Kozanecka-Dymek, A.: Pseudo-BCI-logic. Bull. Sect. Log., Univ. Łódź, Dep. Log. 42 (2013), 33-42. MR 3077651 | Zbl 1287.03058
[8] Imai, Y., Iséki, K.: On axiom systems of propositional calculi. XIV. Proc. Japan Acad. 42 (1966), 19-22. MR 0195704 | Zbl 0156.24812
[9] Is{é}ki, K.: An algebra related with a propositional calculus. Proc. Japan Acad. 42 (1966), 26-29. MR 0202571 | Zbl 0207.29304
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