meow

2 files changed, 82 insertions, 7 deletions

diff --git a/mathematics/algebra.md b/mathematics/algebra.md
index 5838147..5434ff9 100644
--- a/mathematics/algebra.md
+++ b/mathematics/algebra.md
@@ -4,3 +4,78 @@ 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:
+- 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:
+  - commutativity: $∀a,b : a⋆b=b⋆a$
+
+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)
+  - 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)
+  - multiplicative identity: $∃1, ∀a : 1×a = a×1 = a$
+- The *distributive laws* hold for + and ×:
+  - $∀a,b,c : (a+b) × c = (a×c)+(b×c)$
+  - $∀a,b,c : a × (b+c) = (a×b) + (a×c)$
+- An Abelian or **commutative ring** satisfies an additional axiom:
+  - commutativity (of ×): $∀a,b : a×b=b×a$
+
+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:
+- $(V, +)$ is a *commutative group*:
+  - 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: $∀u,v : u+v=v+u$
+- $(V, )$ is a *scalar operation*:
+  - scalar identity: $∃1 ∈ F : 1v = v1 = v$
+  - commutativity: $∀a,b ∈ F, ∀v ∈ V (ab)v = a(bv)$
+- The *distributive laws* hold:
+  - $∀a ∈ F, ∀u,v ∈ V : a(u+v) = au+av$
+  - $∀a,b ∈ F, ∀v ∈ V : (a+b)v = av + bv$
+
+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:
+  - $∀a,b : a ∧ b = b ∧ a$
+  - $∀a,b : a ∨ b = b ∨ a$
+- associativity:
+  - $∀a,b,c : a ∧ (b ∧ c) = (a ∧ b) ∧ c$
+  - $∀a,b,c : a ∨ (b ∨ c) = (a ∨ b) ∨ c$
+- absorption:
+  - $∀a,b : a ∧ (a ∨ b) = a$
+  - $∀a,b : a ∨ (a ∧ b) = a$
+- idempotence:
+  - $∀a : a ∧ a = a$
+  - $∀a : a ∨ a = a$
+
+## group theory
+
+## ring theory
+
+## galois theory
+
+## linear algebra
+
+## order theory
+
+a lattice may alternatively be defined as...
diff --git a/mathematics/logic.md b/mathematics/logic.md
index e19cb52..0760a8c 100644
--- a/mathematics/logic.md
+++ b/mathematics/logic.md
@@ -18,9 +18,9 @@ and many others.
 
 ## orders of logic
 
-### [propositional logic](https://ncatlab.org/nlab/show/propositional+logic)
+### propositional logic
 
-**propositional logic** or **zeroth-order logic** deals with raw *propositions*.
+[**propositional logic**](https://ncatlab.org/nlab/show/propositional+logic) or **zeroth-order logic** deals with raw *propositions*.
 **propositions** are statements that *reduce* to a **truth value**.
 truth values are classically either true or false. in non-classical logics, this can differ.
 
@@ -28,11 +28,11 @@ the basic foundations of propositional logic are as follows:
 
 notation | definition
 ---------|--------------
+0        | *false*
+1        | *true*
 p        | a *proposition*
 ¬p       | *not* p
 p → q   | *if* p *then* q, p *implies* q
-0        | *false*
-1        | *true*
 
 several logical connectives are *derivable* from the above:
 
@@ -40,17 +40,17 @@ notation | derivation | definition
 ------|---------------------|----
 p ∨ q | ¬p → q             | p *or* q, *disjunction*
 p ∧ q | ¬(p → ¬q)          | p *and* q, *conjunction*
-p → q | ¬p ∨ q             | p *implies* q, (material) *implication* (again)
+p → q | ¬p ∨ q             | p *implies* q, (material) *implication*
 p ↔ q | (p → q) ∧ (q → p) | p *if and only if* q, p *iff* q
 p ⊕ q | (p ∨ q) ∧ ¬(p ∧ q) | p *exclusively or* q, p *xor* q
 p ↑ q | ¬(p ∧ q)           | p *not both* q, p *nand* q
 p ↓ q | ¬(p ∨ q)           | *neither* p *nor* q, p *nor* q
 
 note that several of these definitions are circular.
-our choice in $¬$ and $→$ as the primitive connectives is thus arbitrary.
+our choice in ¬ and → as the primitive connectives is thus arbitrary.
 interestingly, ↑ and ↓ are *functionally complete*: we may define all other connectives in terms of them.
 
-<details>
+<details markdown="block">
 <summary>aside: nand and nor</summary>
 
 notation | definition