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: (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

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  ≠ .

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

(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 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) 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.

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 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.

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 , +>.

<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 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 <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)

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.

3. Solving inequalities

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

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.

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.

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.