1. Preliminaries/ Chapter 1: Introduction

2. Definitions: from Abstract to Linear Algebra

3. Let A be a set, with a binary function : A A → A defined on it.
1. <A, > is a semigroup if  is associative:
(ab)c = a(bc)
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 = -aa
3. <A, > is an abelian (or commutative) group if also: ab = ba

4. Let  be another binary function defined on A.
4. <A, , > is a ring if <A, > is an abelian group, and also:
(i)  is associative: (ab)c = a(bc)
(ii) a(bc) = (ab)ac), and
(ab)c = (ac)bc)
5. The ring <A, , > is a field if <A, > and
<A-{}, > are both abelian groups, the latter
with identity element, where  ≠ .

5. Let V be a set, and let F be a field
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)
(ab)◦u = (a◦u) + (b◦u)
(ab)◦u = a◦(b◦u)
◦u = u

7. (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.)

8. Homomorphism
Isomorphism

9. Homomorphism
Isomorphism

10. Homomorphism
Isomorphism

11. Homomorphism
Isomorphism

15. Homomorphism

16. 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

17. A partition of A is any collection P = {pi : i ϵ I} of nonempty subsets of A such that: (i) UP = A, and (ii) pi∩pj = (i ≠ j).
Proposition. Any partition P is the quotient set of the relation: a~b iff a, b ϵ pi, for some pi 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.

18. 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

19. 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
If x ~ y, then x = y

20. 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

21. 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

22. Chapter 2: Construction of Numerical Functions

23. 1. Ordinal Measurement
a ≿ b iff φ(a) ≥ φ(b)

24. 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.

25. 2. Counting of units
Additive representations
φ(a○b) = φ(a) + φ(b)

26. 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.]

27. 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.]

28. 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.

29. A' the nonmaximal elements of A, and
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 , +>.

30. <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.

31. 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 ab and bc exist, then (ab)c exists iff a(bc) does, in which case, they are identical.
3. If (ab)c exists, then so does (ac)(bc), and they are identical.
If a(bc) exists, then so does (ab)(ac), and they are identical.
4.For any a, there exists some a(bc)

32. 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 ab exists then φ(ab) = φ(a) + φ(b)
3. If ab exists, then φ(ab) = φ(a)φ(b)

33. Archimedean Ordered Ring
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 ba ≻ ca.
Corollary. An Archimedean ordered ring is isomorphic to a subring of < R, ≥, +,  >. This isomorphism is unique.

34. 3. Solving inequalities

35. a1○a5 ≻ a3○a4 ≻ a1○a2 ≻ a5 ≻ a4 ≻ a3 ≻ a2 ≻ a1
Ax >
x1 + x5 – x3 – x4 > 0
x3 + x4 – x1 – x > 0
x1 + x2 – x > 0
x5 – x > 0
x4 – x > 0
x3 – x > 0
x2 – x > 0 1 -1 x1 x2 x3 x4 x5

36. 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.

40. Theorem 7. Let A and B be m' by n and m'' by n matrices, respectively
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) λi > 0, and (iii) 1Tλ = 1.

41. Lemma 7. Suppose the m row vectors of A are linearly independent
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.