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| {{Unreferenced|date=April 2012}}
| | Irwin Butts is what my spouse loves to call me though I don't truly like becoming called like that. For many years he's been working as a receptionist. North Dakota is her beginning place but she will have to transfer one day or another. What I adore performing is to gather badges but I've been taking on new things recently.<br><br>my webpage - [http://raybana.com/chat/pg/profile/GIsaachse at home std testing] |
| In [[mathematics]], it can be shown that there exist '''[[magma (algebra)|magmas]] that are [[commutative]] but not [[associative]]'''. A simple example of such a magma is given by considering the children's game of [[rock, paper, scissors]].
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| ==A commutative non-associative magma==
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| Let <math>M := \{ r, p, s \}</math> and consider the [[binary operation]] <math>\cdot : M \times M \to M</math> defined, loosely inspired by the [[rock-paper-scissors]] game, as follows:
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| :<math>r \cdot p = p \cdot r = p</math> "paper beats rock";
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| :<math>p \cdot s = s \cdot p = s</math> "scissors beat paper";
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| :<math>r \cdot s = s \cdot r = r</math> "rock beats scissors";
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| :<math>r \cdot r = r</math> "rock ties with rock";
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| :<math>p \cdot p = p</math> "paper ties with paper";
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| :<math>s \cdot s = s</math> "scissors tie with scissors".
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| By definition, the magma <math>(M, \cdot)</math> is commutative, but it is also non-associative, as the following shows:
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| :<math>r \cdot (p \cdot s) = r \cdot s = r</math>
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| but | |
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| :<math>(r \cdot p) \cdot s = p \cdot s = s.</math>
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| ==A commutative non-associative algebra==
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| Using the above example, one can construct a commutative non-associative [[algebra over a field]] <math>K</math>: take <math>A</math> to be the three-dimensional [[vector space]] over <math>K</math> whose elements are written in the form
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| :<math>(x, y, z) = x r + y p + z s</math>, | |
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| for <math>x, y, z \in K</math>. Vector addition and scalar multiplication are defined component-wise, and vectors are multiplied using the above rules for multiplying the elements <math>r, p</math> and <math>s</math>. The set
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| :<math>\{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}</math> i.e. <math>\{ r, p, s \}</math>
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| forms a [[basis (linear algebra)|basis]] for the algebra <math>A</math>. As before, vector multiplication in <math>A</math> is commutative, but not associative.
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| [[Category:Non-associative algebra]]
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Irwin Butts is what my spouse loves to call me though I don't truly like becoming called like that. For many years he's been working as a receptionist. North Dakota is her beginning place but she will have to transfer one day or another. What I adore performing is to gather badges but I've been taking on new things recently.
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