Preliminaries/ Chapter 1: Introduction

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Preliminaries/ Chapter 1 Introduction
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Definitions from Abstract to Linear Algebra
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• Let A be a set, with a binary function ? A? A ?
  A defined on it.
• 1. ltA, ?gt is a semigroup if ? is associative
• (a?b)?c a?(b?c)
• 2. ltA, ?gt 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. ltA, ?gt is an abelian (or commutative) group if
  also a?b b?a

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• Let ? be another binary function defined on A.
• 4. ltA, ?, ?gt is a ring if ltA, ?gt 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 ltA, ?, ?gt is a field if ltA, ?gt and
• ltA-?, ?gt are both abelian groups, the
  latter
• with identity element, where ? ? ?.

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• 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 ltV, gt
  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

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• (p. 8) Homomorphism f 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.)

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Homomorphism Isomorphism
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Homomorphism Isomorphism
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Homomorphism Isomorphism
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Homomorphism Isomorphism
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Homomorphism
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• Let A be some set.
• An equivalence relation on A is any (binary)
  reflexive (aa), symmetric (if ab, then ba),
  and transitive (if ab, and bc, then ac
  relation.
• Let a b ? A ab
• The quotient set of A wrt is A/ a a ?
  A
• Proposition. The following are equivalent
• (i) a b
• (ii) a n b is nonempty
• (iii) a b

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• A partition of A is any collection P pi i ?
  I of nonempty subsets of A such that (i) UP
  A, and (ii) pinpj ??(i ? j).
• Proposition. Any partition P is the quotient set
  of the relation ab iff a, b ? pi, for some pi ?
  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.

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• Let q(a) a
• Let f A ? R be such that
• (i) if a b, then f(a) f(b)
• Proposition. There exists a unique surjection
• ? A/ ? Range(f), where f ??q. ? is an
  injection iff f also observes
• (ii) if f(a) f(b), then a b

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• . . . . 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

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• . . . . a ? b ? c ? f ? . . .
• When order is preserved, a ? b iff f(a) f(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 ? b iff a' ? b', for every a' ? a, b' ? b
• a b iff a b

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Three ways to assign numbers to things

• 1. Ordinal measurement
• a ? b iff f(a) f(b)
• 2. Counting of units
• Standard sequences
• 3. Solving inequalities
• b a?a, and c ? a?b might imply
• f(c)/f(a) 3
• f is ordinal, additive

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Chapter 2 Construction of Numerical Functions
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1. Ordinal Measurement

• a ? b iff f(a) f(b)

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Ordering Theorems for a simple order ltA, ?
gtDesideratum fA ? R such that
a ? b iff f(a) f(b)

• Theorem 1. If A is countable, we have such a f.
• 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 f exists and is 1-1. f is unique up to
  monotonically strictly increasing
  transformations.

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2. Counting of units

• Additive representations
• f(a?b) f(a) f(b)

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Ordered Semigroup

• ltA, ? , (B A), ? gt
• 1. ltA, ? gt 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.

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Ordered Local Semigroup

• ltA, ? , B, ? gt
• 1. ltA, ? gt 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.

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• Theorem 4. Let ltA, ?, B, ? gt be a positive,
  regular, Archimedean ordered local semigroup.
• There is a f A ? R such that
• (i) a ? b iff f(a) f(b)
• (ii) if a?b exists, then f(a?b) f(a)
  f(b)
• If f' A ? R also satisfies (i) and (ii), then
• f'(a) ßf(a),
• for some ß gt 0, and all nonmaximal a in A.

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• Theorem 4'. Set
• f as in Theorem 4.
• ? the l.u.b. of Range(f),
• A' the nonmaximal elements of A, and
• B' the set of nonmaximal concatenations.
• Then f is an isomorphism of ltA', ?, B', ? gt into
  ltR, , R? , gt.

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• ltA, ? , ?gt is a simply ordered group iff
• ltA, ?gt is a simple order
• ltA, ?gt is a group
• If a ? b, then a?c ? b?c and c?a ? c?b.
• ltA, ? , ?gt 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 ltR, , gt, and the isomorphism is unique up to
  scaling by a positive constant.

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Ordered Local Semiring

• ltA, ? , B, ???????? gt
• 1. ltA, ?, B, ? gt is a simple order
• 2. ltA, ?, B, ? gt 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)

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• Theorem 6. Let ltA, ? , B, ???????? gt be a
  regular, positive, Archimedean ordered semiring.
  Then there is a unique f A ? R such that
• 1. a ? b iff f(a) f(b)
• 2. If a?b exists then f(a?b) f(a) f(b)
• 3. If a?b exists, then f(a?b) f(a)f(b)

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Archimedean Ordered Ring

• ltA, ? , ???? gt
• 1. ltA, ???? gt is a ring with zero element ?
• 2. ltA, ? , ??gt is an Archimedean ordered group
• 3. If a ? ?, and b ? c, then a?b ? a?c and b?a ?
  c?a.
• Corollary. An Archimedean ordered ring is
  isomorphic to a subring of lt R, , , ? gt. This
  isomorphism is unique.

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3. Solving inequalities
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a1?a5 ? a3?a4 ? a1?a2 ? a5 ? a4 ? a3 ? a2 ? a1
Ax gt 0
x1 x5 x3 x4 gt 0
x3 x4 x1 x2 gt 0
x1 x2 x5 gt 0
x5 x4 gt 0
x4 x3 gt 0
x3 x2 gt 0
x2 x1 gt 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
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• Ax 'gt' 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.

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

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• Lemma 7. Suppose the m row vectors of A are
  linearly independent. Then for any t ? Rm, there
  is some x ? Rn such that Ax t.
• Lemma 8. There exists an x ? Rn such that (i) the
  m elements of Ax are nonnegative, and (ii) zTx lt
  0.
• if and only if
• There does not exist a y ? Rm such that (i) the m
  elements of y are nonnegative, and (ii) ATy z.