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diff --git a/mathematics/algebra.md b/mathematics/algebra.md index 0ba7847..3717445 100644 --- a/mathematics/algebra.md +++ b/mathematics/algebra.md @@ -11,7 +11,7 @@ Modern algebra is the study of **algebraic structures**: groups, rings, fields, An **algebraic structure** is a set with a collection of *operations* and a finite set of *axioms* those operations must satisfy. -A **group** $G$ is a set with a single binary operation $⋆$ satisfying the following axioms: +A [**group**](group-theory) $G$ is a set with a single binary operation $⋆$ satisfying the following axioms: - associativity: $∀a,b,c : (a⋆b)⋆c = a⋆(b⋆c)$ - identity: $∃e, ∀a : e⋆a = a⋆e = a$ - inverse: $∀a, ∃a^{-1} : a⋆a^{-1} = e$ @@ -20,7 +20,7 @@ A **group** $G$ is a set with a single binary operation $⋆$ satisfying the fol A **monoid** is a group without an inverse operation. -A **ring** $R$ is a set with two binary operations $+$ and $×$ satisfying the following axioms: +A [**ring**](ring-theory) $R$ is a set with two binary operations $+$ and $×$ satisfying the following axioms: - $(R, +)$ is a *commutative group*: - associativity: $∀a,b,c : (a+b)+c = a+(b+c)$ - additive identity: $∃0, ∀a : 0+a = a+0 = a$ @@ -37,7 +37,7 @@ A **ring** $R$ is a set with two binary operations $+$ and $×$ satisfying the f A **field** is a *commutative ring* where all elements sans $0$ have an inverse $a^{-1}$ under multiplication. Subsequently, $0 ≠ 1$. A field may be also thought of as a set on which addition, subtraction, multiplication, and division are defined and behave as they do on $ℝ$. -A **vector space** $V$ over a field $F$ of scalars is a set with a binary operation $+$ and a binary function satisfying the following axioms: +A [**vector space**](linear-algebra) $V$ over a field $F$ of scalars is a set with a binary operation $+$ and a binary function satisfying the following axioms: - $(V, +)$ is a *commutative group*: - associativity: $∀u,v,w : (u+v)+w = u+(v+w)$ - additive identity: $∃0, ∀v: 0+v = v+0 = v$ @@ -52,7 +52,7 @@ A **vector space** $V$ over a field $F$ of scalars is a set with a binary operat A **module** $M$ is a generalization of a *vector space* to function over a ring $R$ instead of a field. -A **lattice** $L$ is a set with two binary operations $∧$ and $∨$ satisfying the following axioms: +A [**lattice**](order-theory) $L$ is a set with two binary operations $∧$ and $∨$ satisfying the following axioms: - commutativity: - $∀a,b : a ∧ b = b ∧ a$ - $∀a,b : a ∨ b = b ∨ a$ @@ -66,14 +66,4 @@ A **lattice** $L$ is a set with two binary operations $∧$ and $∨$ satisfying - $∀a : a ∧ a = a$ - $∀a : a ∨ a = a$ -## [group theory](group-theory) - -## [ring theory](ring-theory) - -## [galois theory](galois-theory) - -## [linear algebra](linear-algebra) - -## [order theory](order-theory) - -## [coding theory](coding-theory) +An **algebra** $A$ over a field $F$ is a *vector space* equipped with an additional *bilinear product*. It is also common to consider algebras over a *ring* (and thus $A$ as a *module* with an additional product). |