On Thursday 18 September 2003 12:15 pm, Marc LeBrun wrote:
Let some set be closed under an associative multiplication operator.
If it commutes then the product of distinct squares is also square: aabb = abab.
Contrarily, can you construct a non-commuting system where this is usually false?
That is, where non-trivial products of squares are never square?
(Trivializers include null/idem-potent elements, perhaps all self-squares a=aa).
For example is there a simple model using 2x2 matrices over some Zn?
Actually there is an easy example from computer science. The concatenation operator is associative and non-commutative. and closed over the set of all finite strings of characters. Let a, b, c be strings as in a = "aA", b = "bB" and c = "cC". Where: ab = "aAbB" abc = "aAbBcC" etc. It is clear that (ab)c = a(bc), that is concatenation is associative, but ab is not usually equal to ba, so is concatenation is non-commutative. The product of the squares aabb is also usually not equal to abab. aabb is rarely a square. Regards Otto oto@olympus.net
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