Kleinfeld function: Difference between revisions

Revision as of 18:57, 15 May 2012

Definition

Suppose $R$ is a non-associative ring (i.e., a not necessarily associative ring) with multiplication $*$ . Denote by $a$ the associator of $R$ , i.e., $a (x, y, z) : = ((x * y) * z) - (x * (y * z))$ . The Kleinfeld function of $R$ is a function $f$ from $R^{4}$ to $R$ defined as follows:

$f (w, x, y, z) : = a (w * x, y, z) - (x * a (w, y, z)) - (a (x, y, z) * w)$

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	Suppose <math>R</math> is a [[non-associative ring]] (i.e., a not necessarily associative ring) with multiplication <math></math>. Denote by <math>a</math> the [[defining ingredient::associator]] of <math>R</math>, i.e., <math>a(x,y,z) := ((x y) * z) - (x * (y * z))</math>. The '''Kleinfeld function''' of <math>R</math> is a function <math>f</math> from <math>R^4</math> to <math>R</math> defined as follows:		Suppose <math>R</math> is a [[non-associative ring]] (i.e., a not necessarily associative ring) with multiplication <math></math>. Denote by <math>a</math> the [[defining ingredient::associator]] of <math>R</math>, i.e., <math>a(x,y,z) := ((x y) * z) - (x * (y * z))</math>. The '''Kleinfeld function''' of <math>R</math> is a function <math>f</math> from <math>R^4</math> to <math>R</math> defined as follows:

	<math>f(w,x,y,z) := a(w * x, y, z) - x * a(w,y,z) - a(x,y,z) * w</math>		<math>f(w,x,y,z) := a(w * x, y, z) - (x * a(w,y,z)) - (a(x,y,z) * w)</math>