Math Defintions

(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 $u: a \rightarrow b$, $I_a \circle u = u \circle I_b = u$

analysis