<pre>
for all s1, s2, s3 in S:
- (i) s1⋆s2 etc are also in S
- (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
+ (i) s1⋆s2 etc are also in S
+ (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
(iii) z⋆s1 = s1 = s1⋆z
</pre>
To have a category, the elements and morphisms have to satisfy some constraints:
<pre>
- (i) the class of morphisms has to be closed under composition:
+ (i) the class of morphisms has to be closed under composition:
where f:C1→C2 and g:C2→C3, g ∘ f is also a
morphism of the category, which maps C1→C3.
- (ii) composition of morphisms has to be associative
+ (ii) composition of morphisms has to be associative
(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> ∘ f = f = f ∘ 1<sub>C1</sub>
+ morphism 1<sub>E</sub>, which is such that for every morphism f:C1→C2:
+ 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
</pre>
These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `E` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism.