(In Progress Eternally)
algebra
A binary operation on a set \(S\) is a pairwise operation \(S \times S \rightarrow S\). The set is closed under the operation.
A semigroup is a set of elements and an associative binary operation operating on the set.
A monoid is a semigroup with an identity element.
A group is a set of monoid such that for each element there exists an inverse element
A commutative group (also known as an abelian group) is a group whose operation is commutative.
A ring is a set of elements and two binary operations \((+)\) and \((\times)\) such that:
- it forms an abelian group under \((+)\)
- it forms a monoid under \((\times)\)
- \((\times)\) distributes over \((+)\)
A field is a ring such that:
- for each nonzero element (where \(0\) is the additive identity) there exists an inverse element
- \((\times)\) is commutative
The integers \(\mathbb{Z}\) are the cannonical example of a ring. The integers are not a field because some inverses are not contained in the set (e.g. 1/5).
topology
A topological space is a set equipped with a topology.
A topology \(T\) on a set \(X\) is a collection of subsets of \(X\) such that:
- the empty set and \(X\) are elements of \(T\)
- arbitrary unions are elements of \(T\)
- finite intersections are elements of \(T\)
The elements of \(T\) are defined to be open sets. A subset that is a complement of a closed set is an open set.
Why are the intersections finite and not arbitrary? How do we fit a counterexample into the frame of the subset definition?
Using De Morgan's laws for set complements (union of complements is complement of intersection; intersection of complements is complement of union), we can rewrite these properties:
- the empty set and \(X\) are closed
- the arbitrary intersection of closed sets is closed (i.e. the complement of the arbitrary intersection of closed sets is open)
- the finite union of closed sets is closed (i.e. the complement of the finite union of closed sets is open)
There is also a definition based on neighborhoods of points.
category theory
A category consists of:
- a collection of objects
- for each pair of objects, a collection of morphisms (arrows) from one to the other
- for each pair of compatible morphisms, a binary operation called composition
such that:
- if \(u: a \rightarrow b\) and \(v: b \rightarrow c\) then there is a morphism \(u \circle v: a \rightarrow c\)
- composition of morphisms is associative
- for each object \(a\), there is an identity morphism \(I_a\) such that for any morphism \(u: a \rightarrow b\), \(I_a \circle u = u \circle I_b = u\)