Power associativity: Difference between revisions
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In [[abstract algebra]], '''power-associativity''' is a weak form of [[associative|associativity]] |
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A [[groupoid]] (in the sense of a [[set]] with a [[binary operation]] on it) is said to be power-associative if every element generates a [[semigroup|subsemigroup]]. |
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This is equivalent to saying that ''x''<sup>m</sup>''x''<sup>n</sup> = ''x''<sup>m+n</sup> for every ''x'' in the groupoid. |
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Similarly, and more commonly, an [[algebra]] ''A'' is said to be power-associative if the subalgebra generated by any element of ''A'' is associative. |
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This is equivalent to the multiplicative groupoid of ''A'' being power-associative. |
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Every [[associative algebra]] is obviously power-associative, and so too are the [[octonions]] and the [[sedenions]]. |
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Revision as of 15:43, 25 February 2002
In abstract algebra, power-associativity is a weak form of associativity
A groupoid (in the sense of a set with a binary operation on it) is said to be power-associative if every element generates a subsemigroup. This is equivalent to saying that xmxn = xm+n for every x in the groupoid.
Similarly, and more commonly, an algebra A is said to be power-associative if the subalgebra generated by any element of A is associative. This is equivalent to the multiplicative groupoid of A being power-associative. Every associative algebra is obviously power-associative, and so too are the octonions and the sedenions.