Preliminaries/ Chapter 1: Introduction

Definitions: from Abstract to Linear Algebra

Let A be a set, with a binary function : A A → A defined on it.

1. <A, > is a semigroup if is associative:

(ab)c = a(bc)

2. <A, > is a group if also:

(i) there exists some such that for all a:

a = a = a

(ii) for all a, there is some -a such that:

= a-a = -aa

3. <A, > is an abelian (or commutative) group if also: ab = ba

Let be another binary function defined on A.

4. <A, , > is a ring if <A, > is an abelian group, and also:

(i) is associative: (ab)c = a(bc)

(ii) a(bc) = (ab)ac), and

(ab)c = (ac)bc)

5. The ring <A, , > is a field if <A, > and

<A-{}, > are both abelian groups, the latter

with identity element, where ≠ .

Let V be a set, and let F be a field. Let +: V V → V and ◦: F V → V be two binary functions defined on them.

6. V is a vector space over the field F if <V, +> is an abelian group, and for all a, b ϵ F, u, v ϵ V:

a◦(u + v) = (a◦u) + (a◦v)

(ab)◦u = (a◦u) + (b◦u)

(ab)◦u = a◦(b◦u)

◦u = u

(p. 8) Homomorphism: φ sends empirical domain A into R in such a way that ≥ and + preserve the properties of and ○≿

Isomorphism: a 1-1 homomorphism.

(N.b. These defs are a little different from logic, which differ from logic.)

HomomorphismIsomorphism

HomomorphismIsomorphism

HomomorphismIsomorphism

HomomorphismIsomorphism

Homomorphism

Let A be some set.

An equivalence relation on A is any (binary) reflexive (a~a), symmetric (if a~b, then b~a), and transitive (if a~b, and b~c, then a~c relation.

Let [a]~ = {b ϵ A: a~b}

The quotient set of A wrt ~ is A/~ = {[a]~ : a ϵ

A}

Proposition. The following are equivalent:

(i) [a]~ = [b]

~

(ii) [a]~ ∩ [b]

~ is nonempty

(iii) a ~ b

A partition of A is any collection P = {pi : i ϵ I}

of nonempty subsets of A such that: (i) UP = A, and (ii) p

i∩p

j = (i ≠ j).

Proposition. Any partition P is the quotient set of the relation: a~b iff a, b ϵ p

i, for some p

i ϵ

P.

Proposition. The quotient set of any equivalence relation is a partition.

Proposition. There is a bijection from equivalence relations on A to partitions of A that maps the former onto their quotient sets.

Let q(a) = [a]~

Let φ: A → R be such that

(i) if a ~ b, then φ(a) = φ(b)

Proposition. There exists a unique surjection

ψ: A/~ → Range(φ), where φ = ψ○q. ψ is an injection iff φ also observes:

(ii) if φ(a) = φ(b), then a ~ b

. . . . a b c ~ d ~ e f . . .≻ ≻ ≻ ≻

Weak Order: is transitive and connected ≿(total)

Allowed: c ~ d ~ e but c ≠ d = e

Simple Order: antisymmetric weak order

. . . . a b c ~ d ~ e f . . .≻ ≻ ≻ ≻If x ~ y, then x = y

. . . . a b c f ≻ ≻ ≻ ≻ . . .When order is preserved, a b iff φ(a) ≥ φ(b), ≿weak orders may be treated as simple orders

by using quotient sets:

a = [a]~ = {b : a ~ b}

Order is then given as:

a b≿ iff a' b' for≿ some a' ϵ a, b' ϵ b

iff a' b' for ≿ every a' ϵ a, b' ϵ b

a b≻ iff a' b', for every a' ϵ ≻ a, b' ϵ b

a ~ b iff a = b

Three ways to assign numbers to things

1. Ordinal measurement

a b iff φ(a) ≥ φ(b)≿2. Counting of units

Standard sequences

3. Solving inequalities

b ~ a○a, and c a○b might imply: ≿ φ(c)/φ(a) ≥ 3

φ is ordinal, additive

Chapter 2: Construction of Numerical Functions

1. Ordinal Measurement

a b iff φ(a) ≥ φ(b)≿

Ordering Theorems for a simple order <A, >≿Desideratum: φ:A → R such that a b iff φ(a) ≥ φ(b) ≿Theorem 1. If A is countable, we have such a φ.

Def. B A is order dense in A iff for any a b ≻there is c ϵ B: a c b≿ ≿

Theorems 2, 3. There is a denumerable order dense B A iff φ exists and is 1-1. φ is unique up to monotonically strictly increasing transformations.

2. Counting of units

Additive representations

φ(a○b) = φ(a) + φ(b)

Ordered Semigroup<A, , (B = A), ○ >≿

1. <A, > is a simple order≿2. [ok]

3. If a b, then c○a c○b≿ ≿4. If a b, then a○c b○c≿ ≿5. (a○b)○c = a○(b○c)

6. a○b a [≻ pos.]

7. If a b, then for some c, a b○c [≻ ≿ reg.]

8. {n: b na} is finite [≻ Arch.]

Ordered Local Semigroup<A, , B, ○ >≿

1. <A, > is a simple order≿2. If a○b exists, and a c, b d, then c○d exists ≿ ≿3. If c○a exists, and a b, then c○a c○b≿ ≿4. If a○c exists, and a b, then a○c b○c≿ ≿5. (a○b), (a○b)○c exist iff (b○c), a○(b○c) do, in

which case: (a○b)○c = a○(b○c)

6. If a○b exists, then a○b a [≻ pos.]

7. If a b, then for some c, b○c exists, and a ≻ ≿b○c [reg.]

8. {n: na exists and b na} is finite [≻ Arch.]

Theorem 4. Let <A, , B, ○ > be a positive, ≿regular, Archimedean ordered local semigroup.

There is a φ: A → R+ such that:

(i) a b iff φ(a) ≥ φ(b)≿ (ii) if a○b exists, then φ(a○b) = φ(a) + φ(b)

If φ': A → R+ also satisfies (i) and (ii), then

φ'(a) = βφ(a),

for some β > 0, and all nonmaximal a in A.

Theorem 4'. Set: φ as in Theorem 4. the l.u.b. of Range(φ), A' the nonmaximal elements of A, and B' the set of nonmaximal concatenations.

Then φ is an isomorphism of <A', , B', ○ > ≿into <R, ≥ , R , +>.

<A, , ○> is a ≿ simply ordered group iff

<A, > is a simple order≿<A, ○> is a group

If a b, then a○c b○c and c○a c○b. ≿ ≿ ≿<A, , ○> is also ≿ Archimedean if (with the

identity element e) a e, then na b, for ≻ ≻some n.

Theorem 5 (Holder's Theorem) An Archimedean simply ordered group is isomorphic to a subgroup of <R, ≥, +>, and the isomorphism is unique up to scaling by a positive constant.

Ordered Local Semiring<A, , B, ≿ >

1. <A, , B, ≿ > is a simple order

2. <A, , B≿ *, > is a simple order, using the weaker associativity axiom:

If ab and bc exist, then (ab)c exists iff a(bc) does, in which case, they are identical.

3. If (ab)c exists, then so does (ac)(bc), and they are identical.

If a(bc) exists, then so does (ab)(ac), and they are identical.

4.For any a, there exists some a(bc)

Theorem 6. Let <A, , B, ≿ > be a regular, positive, Archimedean ordered semiring. Then there is a unique φ: A → R+ such that

1. a b iff φ(a) ≥ φ(b)≿2. If ab exists then φ(ab) = φ(a) + φ(b)

3. If ab exists, then φ(ab) = φ(a)φ(b)

Archimedean Ordered Ring<A, , ≿ >

1. <A, > is a ring with zero element θ;

2. <A, , ≿ > is an Archimedean ordered group;

3. If a θ, and b c, then a≻ ≻ b a≻ c and ba ≻ca.

Corollary. An Archimedean ordered ring is isomorphic to a subring of < R, ≥, +, >. This isomorphism is unique.

3. Solving inequalities

a1○a

5 a≻

3○a

4 a≻

1○a

2 a≻

5 a≻

4 a≻

3

a≻2 a≻

1

x1 + x

5 – x

3 – x

4 > 0

x3 + x

4 – x

1 – x

2 > 0

x1 + x

2 – x

5 > 0

x5 – x

4 > 0

x4 – x

3 > 0

x3 – x

2 > 0

x2 – x

1 > 0

1 0 -1 -1 1

-1 -1 1 1 0

1 1 0 0 -1

0 0 0 -1 1

0 0 -1 1 0

0 -1 1 0 0

-1 1 0 0 0

x1

x2

x3

x4

x5

Ax > 0

Ax '>' 0, Bx = 0

Theorem 7. There is a solution x to the above inequalities iff the polyhedron (in Rn) whose corners are the m' row vectors of A does not intersect the subspace spanned by the row vectors of B.

'1 01

mixan

iiij

''1 01

mixbn

iiij

Theorem 7. Let A and B be m' by n and m'' by n matrices, respectively. There exists an x ϵ Rn such that Bx = 0 and the m' elements of Ax are positive

if and only if

there does not exist a pair λ ϵ Rm', μ ϵ Rm'' such that (i) AT λ = BTμ, (ii) λ

i > 0, and (iii) 1Tλ = 1.

Lemma 7. Suppose the m row vectors of A are linearly independent. Then for any t ϵ Rm, there is some x ϵ Rn such that Ax = t.

Lemma 8. There exists an x ϵ Rn such that (i) the m elements of Ax are nonnegative, and (ii) zTx < 0.

if and only if

There does not exist a y ϵ Rm such that (i) the m elements of y are nonnegative, and (ii) ATy = z.