I am currently learning category theory and a saying that I see a lot is that X is the algebra of something (e.g. Monoid is an algebra of something). Can someone explain to me what that means? Thanks!
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$\begingroup$ Do you see algebra or algebraic structure? $\endgroup$– plopDec 1, 2020 at 21:19
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$\begingroup$ @plop Algebra, like "monoid is the algebra whose universe is the set of natural numbers $\mathbb{N}$ and whose operation is $+$" $\endgroup$– thoughtpoliceDec 2, 2020 at 0:38
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2$\begingroup$ Well, it is a bit inappropriate. Algebra is a concept that is normally used to mean something else that is not what they mean there, and what they mean to mean there is normally called algebraic structure. Algebra is normally used as a short for the concept of algebra over a field or some analogous concept in which there are two operations. The use in that quote is instead the concept of algebraic structure. $\endgroup$– plopDec 2, 2020 at 14:40
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1$\begingroup$ Google matched your quote to this page. They say that a typical example of a monoid is the algebraic structure formed by the set of natural numbers together with their usual sum as the monoid operation. Here, algebraic structure, means just that, a set $\mathbb{N}$ and internal operation $+:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ such that it satisfies the axioms of a monoid. These are that the operation has an identity (the number $1$) and it is associative (that $(a+b)+c=a+(b+c)$). $\endgroup$– plopDec 2, 2020 at 14:44
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