From 19ab14173c4242792cee0bd9b75d81c788e0fb70 Mon Sep 17 00:00:00 2001
From: JJ
Date: Thu, 18 Jan 2024 14:00:05 -0800
Subject: meow

---
 mathematics/algebra.md | 17 ++++++++++++-----
 1 file 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...
-- 
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