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I am working on my functional composition, which has the associative property, to show if a given pair is a semigroup or not. I believe all Semigroups have to have a binary operation that is associative.

Am I correct?

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closed as unclear what you're asking by Daniel, user223391, hardmath, Batominovski, graydad Aug 10 '15 at 1:06

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    $\begingroup$ Yes that is part of the definition of a semigroup. $\endgroup$ – David Wheeler May 29 '15 at 22:40
  • $\begingroup$ Ok thank you! Didn't know if there were any special cases. $\endgroup$ – TheCrownedPixel May 29 '15 at 22:41
  • $\begingroup$ Without the requirement that a binary operation be associative, the structure closed under a binary operation is called a magma. This is more general than a semigroup, not a special case of semigroups. $\endgroup$ – hardmath Aug 9 '15 at 23:33
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In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation

-Wikipedia, 2015

If you're interested in a set with a not-necessarily-associative binary operation, you're looking for a "magma".

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  • $\begingroup$ Thank you, I am learning this, so wanted quick reassurance from some better mathematicians. $\endgroup$ – TheCrownedPixel May 29 '15 at 22:41
  • $\begingroup$ Ok, I have read about them, but our course has yet to cover that. $\endgroup$ – TheCrownedPixel May 29 '15 at 22:43
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Associativity is part of the definition of semigroup, so yes, all semigroups are associative.

If you need to show that some set with an operation on it is a semigroup, though, you can't take it for granted. You must instead show that the operation is defined for every pair of elements and that it is indeed associative.

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  • $\begingroup$ Ok, this is where I need to look into this further! $\endgroup$ – TheCrownedPixel May 29 '15 at 22:43

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