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Diffstat (limited to 'mathematics/algebra.md')
| -rw-r--r-- | mathematics/algebra.md | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/mathematics/algebra.md b/mathematics/algebra.md index 5434ff9..5047970 100644 --- a/mathematics/algebra.md +++ b/mathematics/algebra.md @@ -14,7 +14,7 @@ these structures are very general: and so results from abstract algebra can be a 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: -- associativity: $∀a,b,c : (a⋆b)⋆c = a⋆(b⋆c) +- associativity: $∀a,b,c : (a⋆b)⋆c = a⋆(b⋆c)$ - identity: $∃e, ∀a : e⋆a = a⋆e = a$ - inverse: $∀a, ∃a^{-1} : a⋆a^{-1} = e$ - An Abelian or **commutative group** satisfies an additional axiom: @@ -24,12 +24,12 @@ A **monoid** is a group without an inverse operation. 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) + - associativity: $∀a,b,c : (a+b)+c = a+(b+c)$ - additive identity: $∃0, ∀a : 0+a = a+0 = a$ - additive inverse: $∀a, ∃-a : a+(-a) = 0$ - commutativity: $∀a,b : a+b=b+a$ - $(R, ×)$ is a *monoid* - - associativity: $∀a,b,c : (a×b)×c = a×(b×c) + - associativity: $∀a,b,c : (a×b)×c = a×(b×c)$ - multiplicative identity: $∃1, ∀a : 1×a = a×1 = a$ - The *distributive laws* hold for + and ×: - $∀a,b,c : (a+b) × c = (a×c)+(b×c)$ @@ -41,7 +41,7 @@ A **field** is a *commutative ring* where $0 ≠ 1$ and all elements sans $0$ ha 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) + - associativity: $∀u,v,w : (u+v)+w = u+(v+w)$ - additive identity: $∃0, ∀v: 0+v = v+0 = v$ - additive inverse: $∀v, ∃-v: v+(-v) = 0$ - commutativity: $∀u,v : u+v=v+u$ |