Mathematical Objects

A cheat sheet for mathematical objects and structures, useful for quick reference.

Foundations of Sets and Relations

Set

• A collection of distinct objects, often written as $X = \{x \mid P(x)\}$, where $P(x)$ is a property.
• No additional structure.

Map / Function

• A mapping $f: X \to Y$ assigns each $x \in X$ to a unique $y \in Y$.

Relation

• A subset $R \subseteq X \times X$, such as equivalence relations or partial orders.

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Basic Algebraic Structures

Semigroup

• A set $S$ with a binary operation $\cdot: S \times S \to S$ such that:
  • Associativity: For all $a, b, c \in S$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.

Monoid

• A semigroup with an identity element $e \in M$, satisfying:
  • Identity: $a \cdot e = e \cdot a = a$ for all $a \in M$.

Group

• A monoid where every element $a \in G$ has an inverse $a^{-1}$, such that:
  • Inverse: $a \cdot a^{-1} = a^{-1} \cdot a = e$.
• Denoted as $(G, \cdot, e)$.

Abelian Group

• A group that satisfies the commutative property:
  • Commutativity: $a \cdot b = b \cdot a$.

Ring

• A set $R$ with two operations $+$ and $\cdot$, such that:
  • $(R, +)$ is an Abelian group.
  • $(R, \cdot)$ is a semigroup.
  • Distributivity: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$.

Field

• A ring $F$ where:
  • $(F, +)$ is an Abelian group.
  • $(F^*, \cdot)$ (where $F^* = F \setminus \{0\}$) is an Abelian group.
• Distributive property: As in a ring.

Module

• For a ring $R$, an $R$-module $M$ satisfies:
  • $(M, +)$ is an Abelian group.
  • Scalar multiplication $R \times M \to M$ satisfies:
    • $r(m_1 + m_2) = rm_1 + rm_2$.
    • $(r_1 + r_2)m = r_1m + r_2m$.
    • $(r_1r_2)m = r_1(r_2m)$.

Vector Space

• A special case of a module where the scalar ring is a field $F$.
• Follows the same axioms as modules but includes scalar inverses.

Algebra

• A vector space $A$ equipped with a bilinear multiplication $\cdot$, satisfying:
  • Associativity: $(ab)c = a(bc)$.

Lattice

• A partially ordered set $(L, \leq)$ where every two elements $a, b$ have:
  • Supremum (join): $a \vee b = \sup(a, b)$.
  • Infimum (meet): $a \wedge b = \inf(a, b)$.

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Topological Structures

Topological Space

• A set $X$ with a topology $\mathcal{T} \subseteq \mathcal{P}(X)$ such that:
  • The empty set and the whole set are in $\mathcal{T}$.
  • Arbitrary unions of sets in $\mathcal{T}$ are also in $\mathcal{T}$.
  • Finite intersections of sets in $\mathcal{T}$ are also in $\mathcal{T}$.

Metric Space

• A set $M$ with a metric $d: M \times M \to \mathbb{R}$, satisfying:
  • $d(x, y) \geq 0$, and $d(x, y) = 0 \iff x = y$.
  • $d(x, y) = d(y, x)$.
  • $d(x, z) \leq d(x, y) + d(y, z)$ (triangle inequality).

Manifold

• A topological space $M$ locally homeomorphic to $\mathbb{R}^n$, satisfying Hausdorff and second countability properties.

Fiber Bundle

• A continuous map $\pi: E \to B$ where each $\pi^{-1}(b)$ is homeomorphic to a fiber $F$.

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Analytical and Functional Structures

Normed Space

• A vector space $V$ with a norm $\|\cdot\|: V \to \mathbb{R}$ satisfying:
  • $\|x\| \geq 0$, and $\|x\| = 0 \iff x = 0$.
  • $\|cx\| = |c| \|x\|$, for scalar $c$.
  • $\|x + y\| \leq \|x\| + \|y\|$ (triangle inequality).

Banach Space

• A normed space that is complete with respect to the metric induced by its norm.

Hilbert Space

• A Banach space where the norm arises from an inner product $\langle \cdot, \cdot \rangle$:
  • $\|x\| = \sqrt{\langle x, x \rangle}$.

Measure Space

• A triple $(X, \Sigma, \mu)$, where:
  • $X$ is a set.
  • $\Sigma$ is a σ-algebra on $X$.
  • $\mu: \Sigma \to [0, \infty]$ is a measure.

Sobolev Space

• Defined as $W^{k, p}(\Omega) = \{ f \in L^p(\Omega) \mid D^\alpha f \in L^p(\Omega), \forall |\alpha| \leq k \}$, where $D^\alpha f$ denotes weak derivatives.

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Categorical Structures

Category

• Defined by:
  • Objects: $\text{Ob}(\mathcal{C})$.
  • Morphisms: $\text{Hom}(A, B)$, satisfying:
    • Composition: If $f: A \to B$ and $g: B \to C$, then $g \circ f: A \to C$.
    • Identity: For each object $A$, there exists $\text{id}_A$ such that $f \circ \text{id}_A = f$ and $\text{id}_B \circ f = f$.

Functor

• A map between categories $F: \mathcal{C} \to \mathcal{D}$ preserving composition and identities.