Monoids
-------
-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:
+A **monoid** is a structure <code>(S,⋆,z)</code> consisting of an associative binary operation <code>⋆</code> over some set `S`, which is closed under <code>⋆</code>, and which contains an identity element `z` for <code>⋆</code>. That is:
<pre>
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
+ (i) s1⋆s2 etc are also in S
+ (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
+ (iii) z⋆s1 = s1 = s1⋆z
</pre>
Some examples of monoids are:
-* finite strings of an alphabet `A`, with `*` being concatenation and `z` being the empty string
-* all functions `X→X` over a set `X`, with `*` being composition and `z` being the identity function over `X`
-* the natural numbers with `*` being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.)
-* if we let `*` be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item.
+* finite strings of an alphabet `A`, with <code>⋆</code> being concatenation and `z` being the empty string
+* all functions `X→X` over a set `X`, with <code>⋆</code> being composition and `z` being the identity function over `X`
+* the natural numbers with <code>⋆</code> being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.)
+* if we let <code>⋆</code> be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item.
Categories
----------
* Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., sin and cos) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set.
-* any monoid `(S,*,z)` generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where `s3=s1*s2`. The identity morphism for the (single) category element `x` is the monoid's identity `z`.
+* any monoid <code>(S,⋆,z)</code> generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where <code>s3=s1⋆s2</code>. The identity morphism for the (single) category element `x` is the monoid's identity `z`.
* a **preorder** is a structure `(S, ≤)` consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither `x≤y` nor `y≤x`). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that `s1≤s2` and `s2≤s1` but `s1` and `s2` are not identical). Some examples: