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I need the name of the algebraic structure that has an identity element, is associative and closed under some operation $\circ$. It is a monoid associated with a set which it is closed in, or a group without inverses.

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    $\begingroup$ you are talking about a Monoid $\endgroup$ – Hector Blandin Nov 2 '17 at 2:20
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    $\begingroup$ In other words, a monoid or a semigroup with an identity element. $\endgroup$ – Hector Blandin Nov 2 '17 at 2:21
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The structure you're describing is a monoid, but you seem to already know that.

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When I was young we called it a semigroup with an identity element.

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    $\begingroup$ Did you require homomorphisms to preserve the identity? If not, then that's the right name for the structure. If yes, then you were actually studying monoids, rather than studying the subclass of semigroups that have an identity. $\endgroup$ – user14972 Nov 2 '17 at 19:58

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