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A **monoid** is a structure `(S, *, z)` consisting of an associative binary operation `*` over some set `S`, which is closed under `*`, and which contains an identity element `z` for `*`. That is:
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for all `s1`, `s2`, `s3` in `S`:
(i) `s1*s2` etc are also in `S`
(ii) `(s1*s2)*s3` = `s1*(s2*s3)`
(iii) `z*s1` = `s1` = `s1*z`
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Some examples of monoids are:
To have a category, the elements and morphisms have to satisfy some constraints:
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-(i) the class of morphisms has to be closed under composition: where `f:C1->C2` and `g:C2->C3`, `g o f` is also a morphism of the category, which maps `C1->C3`.
-(ii) composition of morphisms has to be associative
+(i) the class of morphisms has to be closed under composition: where `f:C1->C2` and `g:C2->C3`, `g o f` is also a morphism of the category, which maps `C1->C3`.<BR>
+(ii) composition of morphisms has to be associative<BR>
(iii) every element `E` of the category has to have an identity morphism 1<sub>E</sub>, which is such that for every morphism `f:C1->C2`: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>
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