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Difference between revisions of "BCH-algebra"

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A variant of a [[BCI-algebra|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101701.png" /> with a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101702.png" /> and a [[Binary relation|binary relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101703.png" /> satisfying the following axioms:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101704.png" />;
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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 relation|binary relation]] $*$ satisfying the following axioms:
  
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2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101706.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101707.png" />;
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1) $x * x = 0$;
  
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3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110170/b1101708.png" />. 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.
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2) if $x * y = 0$ and $y * x = 0$, then $x = y$;
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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====
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<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>
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<table>
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<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>
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<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>
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<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>
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</table>

Revision as of 20:24, 12 December 2014


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 relation $*$ 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
[a2] Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" Proc. Japan Acad. Ser. A Math. Sci. , 42 (1966) pp. 19–22
[a3] K. Iséki, "An algebra related with a propositional calculus" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 26–29
How to Cite This Entry:
BCH-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BCH-algebra&oldid=35595
This article was adapted from an original article by C.S. Hoo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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