Difference between revisions of "BCH-algebra"
From Encyclopedia of Mathematics
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| − | + | A variant of a [[BCI-algebra]]. One can define it by taking some of the axioms for a BCI-algebra and some of the important properties of a BCI-algebra. Specifically, a BCH-algebra is a non-empty set $X$ with a constant $0$ and a [[binary operation]] $*$ satisfying the following axioms: | |
| − | + | 1) $x * x = 0$; | |
| − | + | 2) if $x * y = 0$ and $y * x = 0$, then $x = y$; | |
| + | |||
| + | 3) $(x*y)*z = (x*z)*y$. | ||
| + | |||
| + | Clearly a BCI-algebra is a BCH-algebra; however, the converse is not true. While some work has been done on such algebras, generally they have not been as extensively investigated as BCI-algebras. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Qing-ping Hu, Xin Li, "On BCH-algebras" ''Math. Seminar Notes (Kobe University)'' , '''11''' (1983) pp. 313–320</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" ''Proc. Japan Acad. Ser. A Math. Sci.'' , '''42''' (1966) pp. 19–22</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Iséki, "An algebra related with a propositional calculus" ''Proc. Japan Acad. Ser. A, Math. Sci.'' , '''42''' (1966) pp. 26–29</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> Qing-ping Hu, Xin Li, "On BCH-algebras" ''Math. Seminar Notes (Kobe University)'' , '''11''' (1983) pp. 313–320 {{ZBL|0579.03047}}</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" ''Proc. Japan Acad. Ser. A Math. Sci.'' , '''42''' (1966) pp. 19–22 {{DOI|10.3792/pja/1195522169}} {{MR|0195704}} {{ZBL|0156.24812}}</TD></TR> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Iséki, "An algebra related with a propositional calculus" ''Proc. Japan Acad. Ser. A, Math. Sci.'' , '''42''' (1966) pp. 26–29 {{DOI|10.3792/pja/1195522171}} {{MR|0202571}} {{ZBL|0207.29304}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 07:03, 1 May 2016
A variant of a BCI-algebra. One can define it by taking some of the axioms for a BCI-algebra and some of the important properties of a BCI-algebra. Specifically, a BCH-algebra is a non-empty set $X$ with a constant $0$ and a binary operation $*$ satisfying the following axioms:
1) $x * x = 0$;
2) if $x * y = 0$ and $y * x = 0$, then $x = y$;
3) $(x*y)*z = (x*z)*y$.
Clearly a BCI-algebra is a BCH-algebra; however, the converse is not true. While some work has been done on such algebras, generally they have not been as extensively investigated as BCI-algebras.
References
| [a1] | Qing-ping Hu, Xin Li, "On BCH-algebras" Math. Seminar Notes (Kobe University) , 11 (1983) pp. 313–320 Zbl 0579.03047 |
| [a2] | Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" Proc. Japan Acad. Ser. A Math. Sci. , 42 (1966) pp. 19–22 DOI 10.3792/pja/1195522169 MR0195704 Zbl 0156.24812 |
| [a3] | K. Iséki, "An algebra related with a propositional calculus" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 26–29 DOI 10.3792/pja/1195522171 MR0202571 Zbl 0207.29304 |
How to Cite This Entry:
BCH-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BCH-algebra&oldid=17336
BCH-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BCH-algebra&oldid=17336
This article was adapted from an original article by C.S. Hoo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article