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. is a semigroup if  is associative: (a  b)  c = a  (b  c) 2. 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. is an abelian (or commutative) group if also: a  b = b  a

Let  be another binary function defined on A. 4. is a ring if 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 is a field if and 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 is an abelian group, and for all a, b F, u, v V:  a◦(u + v) = (a◦u) + (a◦v) (a  b)◦u = (a◦u) + (b◦u) (a  b)◦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.)

Homomorphism Isomorphism

Homomorphism Isomorphism

Homomorphism Isomorphism

Homomorphism Isomorphism

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 A: a~b} The quotient set of A wrt ~ is A/~ = {[a] ~ : a 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 = {p i : i 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 p i, for some p i 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' a, b' b iff a' ≿ b' for every a' a, b' b a ≻ biff a' ≻ b', for every a' a, b' b a ~ biff 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 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 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 1. 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 1. 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 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 into.

is a simply ordered group iff is a simple order is a group If a ≿ b, then a○c ≿ b○c and c○a ≿ c○b. 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, and the isomorphism is unique up to scaling by a positive constant.

Ordered Local Semiring 1. is a simple order 2. 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)

Theorem 6. Let 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)

Archimedean Ordered Ring 1. is a ring with zero element θ; 2. 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. This isomorphism is unique.

3. Solving inequalities

a 1 ○a 5 ≻ a 3 ○a 4 ≻ a 1 ○a 2 ≻ a 5 ≻ a 4 ≻ a 3 ≻ a 2 ≻ a 1 x 1 + x 5 – x 3 – x 4 > 0 x 3 + x 4 – x 1 – x 2 > 0 x 1 + x 2 – x 5 > 0 x 5 – x 4 > 0 x 4 – x 3 > 0 x 3 – x 2 > 0 x 2 – x 1 > x1x1 x2x2 x3x3 x4x4 x5x5 Ax > 0

Ax '>' 0, Bx = 0 Theorem 7. There is a solution x to the above inequalities iff the polyhedron (in R n ) whose corners are the m' row vectors of A does not intersect the subspace spanned by the row vectors of B.

Theorem 7. Let A and B be m' by n and m'' by n matrices, respectively. There exists an x R n such that Bx = 0 and the m' elements of Ax are positive if and only if there does not exist a pair λ R m', μ R m'' such that (i) A T λ = B T μ, (ii) λ i > 0, and (iii) 1 T λ = 1.

Lemma 7. Suppose the m row vectors of A are linearly independent. Then for any t R m, there is some x R n such that Ax = t. Lemma 8. There exists an x R n such that (i) the m elements of Ax are nonnegative, and (ii) z T x < 0. if and only if There does not exist a y R m such that (i) the m elements of y are nonnegative, and (ii) A T y = z.