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My book defines semigroups, monoids and groups in a traditional way. Where semigroups are defined with associativity, monoids with associativity and identity and groups with associativity, identity and inverses (where these objects all include the closure property).

Definitionally, it is clear to me not all semigroups are groups, and not all monoids are groups, as groups have more structure.

But I have a few questions around the definitions that the book doesn't make clear.

Question 1: are all groups considered semigroups and are all groups considered monoids? Technically the group definition satisfies the properties of semigroups and monoids, but it isn't clear if they are considered distinctly different mathematical objects.

Question 2: If Question 1) is true, then would this also imply all monoids are semigroups?

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  • $\begingroup$ In mathematics, you aren't limited to being one kind of thing. A field, for example, is simultaneously an algebra, ring, vector space, abelian group, monoid (in two different ways) and semigroup (in two different ways!) under appropriate specification of binary operations. $\endgroup$ – rschwieb Feb 8 at 16:45
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Yes, of course. Groups are semigroups and groups are monoids and monoids are semigroups. An object that satisfies all the requirements to be a member of a particular category, and has additional structure, is still a member of that category.

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Of course, groups are monoids are semigroups, since we tack on additional requirements in moving from one to the other. That is, groups satisfy all the properties of a monoid, and monoids those for a semigroup. So yes.

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