An ordered structure of pseudo-BCI-algebras
Mathematica Bohemica (2016)
- Volume: 141, Issue: 1, page 91-98
- ISSN: 0862-7959
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topChajda, Ivan, and Länger, Helmut. "An ordered structure of pseudo-BCI-algebras." Mathematica Bohemica 141.1 (2016): 91-98. <http://eudml.org/doc/276788>.
@article{Chajda2016,
abstract = {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).},
author = {Chajda, Ivan, Länger, Helmut},
journal = {Mathematica Bohemica},
keywords = {pseudo-BCI-algebra; directoid; antitone mapping; pseudo-BCI-structure},
language = {eng},
number = {1},
pages = {91-98},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An ordered structure of pseudo-BCI-algebras},
url = {http://eudml.org/doc/276788},
volume = {141},
year = {2016},
}
TY - JOUR
AU - Chajda, Ivan
AU - Länger, Helmut
TI - An ordered structure of pseudo-BCI-algebras
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 1
SP - 91
EP - 98
AB - 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).
LA - eng
KW - pseudo-BCI-algebra; directoid; antitone mapping; pseudo-BCI-structure
UR - http://eudml.org/doc/276788
ER -
References
top- Chajda, I., 10.1007/s10773-013-1739-4, Int. J. Theor. Phys. 53 (2014), 3391-3396. (2014) Zbl1302.81032MR3253801DOI10.1007/s10773-013-1739-4
- Chajda, I., Länger, H., On the structure of pseudo-BCK algebras, (to appear) in J. Multiple-Valued Logic Soft Computing.
- Chajda, I., L{ä}nger, H., Directoids. An Algebraic Approach to Ordered Sets, Research and Exposition in Mathematics 32 Heldermann, Lemgo (2011). (2011) Zbl1254.06002MR2850357
- Ciungu, L. C., Non-commutative Multiple-Valued Logic Algebras, Springer Monographs in Mathematics Springer, Cham (2014). (2014) Zbl1279.03003MR3112745
- Dudek, W. A., Jun, Y. B., Pseudo-BCI algebras, East Asian Math. J. 24 (2008), 187-190. (2008) Zbl1149.06010
- Dymek, G., 10.7151/dmgaa.1184, Discuss. Math., Gen. Algebra Appl. 31 (2011), 217-229. (2011) Zbl1258.06014MR2953913DOI10.7151/dmgaa.1184
- Dymek, G., Kozanecka-Dymek, A., Pseudo-BCI-logic, Bull. Sect. Log., Univ. Łódź, Dep. Log. 42 (2013), 33-42. (2013) Zbl1287.03058MR3077651
- Imai, Y., Iséki, K., On axiom systems of propositional calculi. XIV, Proc. Japan Acad. 42 (1966), 19-22. (1966) Zbl0156.24812MR0195704
- Is{é}ki, K., An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966), 26-29. (1966) Zbl0207.29304MR0202571
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