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Diffstat (limited to 'mathematics/algebra.md')
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1 files changed, 12 insertions, 5 deletions
diff --git a/mathematics/algebra.md b/mathematics/algebra.md index 5047970..4b9a097 100644 --- a/mathematics/algebra.md +++ b/mathematics/algebra.md @@ -6,14 +6,13 @@ title: mathematics/algebra # algebra modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, vector spaces, lattices, and the like. - these structures are very general: and so results from abstract algebra can be applied to a wide variety of situations. ## structures 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** $G$ 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: $∀a, ∃a^{-1} : a⋆a^{-1} = e$ @@ -22,7 +21,7 @@ A **group** $G$ is a set with a single binary operation ⋆ satisfying the follo 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** $R$ 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: $∃0, ∀a : 0+a = a+0 = a$ @@ -39,7 +38,7 @@ A **ring** $R$ is a set with two binary operations + and × satisfying the follo A **field** is a *commutative ring* where $0 ≠ 1$ and all elements sans $0$ have an inverse $a^{-1}$ under multiplication. -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** $V$ 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: $∃0, ∀v: 0+v = v+0 = v$ @@ -55,7 +54,7 @@ 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: -- commutativity: +- commutativity: - $∀a,b : a ∧ b = b ∧ a$ - $∀a,b : a ∨ b = b ∨ a$ - associativity: @@ -70,12 +69,20 @@ A **lattice** $L$ is a set with two binary operations ∧ and ∨ satisfying the ## group theory +... + ## ring theory +... + ## galois theory +... + ## linear algebra +... + ## order theory a lattice may alternatively be defined as... |