Transcript Solution: Germ Theory of Disease

Chapter 19:
Non-additive
representations
Bennett Holman
Foundations of
Measurement
What is essential nonadditivity
 The fact that a representation is nonadditive is not suf ficient
to infer that an additive representation does not exist.
 Nonessential nonadditive structures - Suppose that one is
looking at the probability (Q) that a patient will correctly
guess they are receiving drugs in a double blind trial and
there is an interaction between prior experience with the drug
(E) and severity of side ef fects (S).
 At very low levels of side effects,
experience may not contribute at all
 At very high levels of side-effects
experience may not be needed
 But at intermediate levels prior experience
may allow a patient to pick up on subtle cues
that would otherwise be missed
What is essential nonadditivity
 If in the case above suppose we use a logit transformation
Q log (Q/(1-Q)) that eliminates the interaction, the
nonadditivity of the original values would be nonessential.
 It would be a matter of convention whether we
used h(Q) = x(E) + y(S)
or Q = h^-1 [x(E)] ◙ h^-1 [y(S)]
where u ◙ v = h^-1 [h(u) ◙ h (v)]
 If there exists a monotonic transformation h such that h(Q) =
x(E) + y(S) then the representation is not essentially
nonadditive
Essential nonadditivity
 Suppose that correctly guessing was instead Q = RA
Where R is the probability of recognizing the presence of side
ef fects
and A is the conditional probability of attributing the side
ef fect to the drug when it is detected
 If R and A are both dependent on E and S, Q may be
essentially nonadditive even if R and A are both additive
 If: log (R/(1-R) = f(E) + g(S), and
log (A/(1- A) = k(E) + l(S)
Then there does not exist a monotone transformation h such
that
h(Q) = f(E) + g(S)
Breaking down: decomposable
 We say that Q is decomposable, but essentially nonadditive
Q = H [ f(E) , g(S)]
If k and l are monotonically increasing
functions of f and g respectively, our
example would be decomposable
 More generally, if an observed measure
depends, monotonically on several
unobservable variables, each of which
depends on the same two empirically
specifiable variables, with all the
dependencies covarying monotonically, then the overall
relation will satisfy decomposability
Nonadditive representations aren’t so
strange
 Nonassociativity - x ◙ (y ◙ z) ≠ (x ◙ y) ◙ z
 E.g. averaging: Let x ◙ y = (x + y)/2
 Using the above example with x = 10, y = 30 and z = 50
10 ◙ (30 ◙ 50) ≠ (10 ◙ 30) ◙ 50
10 ◙ (40) ≠ (20) ◙ 50
25 ≠ 35
General binary operations
 Examples: Let x ◙ y = rx + sy + t
 If r + s = 1 and t = 0 this is the weight average
 x ◙ y is never associative and only commutative
if r = s = ½
 One consequence of considering arbitrary
binary operations is that finding a
representation can be seen as a process
of discover y
Whereas the uniqueness problem is best
conceptualized as a process of
scale construction
Def 1: Concatenation Structures (p. 26)
 A = < A , ≥ , ○ > is a concatenation structure if f the following
conditions are satisfied:
 Weak order: ≥ is a weak order on A
 Local definability: if (a ○ b) is defined and a ≥ c and b ≥ d, then (c ○ d) is
defined
 Monotonicity: i. If (a ○ c) and (b ○ c) are defined, then a ≥ b iff (a ○ c) ≥ (b
○ c)
ii. If (c ○ a) and (c ○ b) are defined, then a ≥ b iff (c ○ a) ≥ (c
○ b)
 Compared to an extensive structure, a concatenation structure
preser ves:
 The transitivity and connectedness of ≥
 Monotonicity of ○ with respect to ≥
 Structural conditions assuring us that (a ○ b) is
defined for sufficiently small elements a and b
 A concatenation structure is not necessarily:
 Associative
 Positive
 Archimedean
Def 2: Our vocabulary (p. 26-7)
 Let A = < A , ≥ , ○ > be a concatenation structure, A is
said to be:
 Closed iff (a ○ b) is defined for a, b
A
 Positive iff whenever (a ○ b) is defined, (a ○ b) is strictly
greater than a or b
 Negative iff whenever (a ○ b) is defined, (a ○ b) is strictly less
than a or b
 Idempotent iff a ~ (a ○ a) whenever (a ○ a) is defined
 Intern iff whenever a > b and (a ○ b) or (b ○ a) is defined, a >
(a ○ b) > b and a > (b ○ a) > b
 Intensive iff it is both intern and idempotent (e.g. average)
 Associative iff whenever one of (a ○ b) ○ c or a ○ (b ○ c) is
defined, the other expression is defined and (a ○ b) ○ c ~ a
○ (b ○ c)
Def 2 (cont.): so many new words!
 Bisymmetric iff A is closed and (a ○ b) ○ (c ○ d) ~ (a ○ c) ○ (b ○
d)
 Autodistributive iff A is closed and (a ○ b) ○ c ~ (a ○ c) ○ (b ○ c)
and
c ○ (a ○ b) ~ (c ○ a) ○ (c ○ b)
 Halvable iff A is positive and, for each a
A, there exists a b
A such that (b ○ b) is defined and a ~ (b ○ b)
--- Question: if A is idempotent, why is it not halvable
 Restrictedly solvable iff whenever a > b there exists a c
A such
that either (b ○ c) is defined and a ≥ ( b ○ c) > b or (a ○ c) is defined
and a > ( a ○ c) ≥ b
 Solvable iff given a and b there exists c and d such that (a ○ c) ~ b ~
(d ○ a)
 Dedekind complete iff < A, ≥ > is Dedekind complete, i.e. every
nonempty subset of A that has an upper bound has a least upper
bound in A
 Continuous iff the operation ○ is continuous as a function of two
Let’s use our new words
 S’pose
x ◙ y = x + y,
if x or y is less than 3, and
x ◙ y = xy
other wise
The structure < RE+, ≥ , ◙ > is
Discontinuous: Let x = 4, as y > 3 approaches 3, (x ◙ y) approaches
12, but as
y < 3 approaches 3, (x ◙ y) approaches 7
Nonassociative 4 ◙ (2 ◙ 2) ≠ (4 ◙ 2) ◙ 2
4 ◙ (4) ≠ (6) ◙ 2, (16 ≠ 8)
Closed (x ◙ y) is always defined
Positive (x ◙ y) is strictly greater than x or y
Restrictively solvable because given any a > b, there is always
some c > 0 that I can add to b such that a > c ◙ b > b
Since the ordering is the usual one, it is Dedekind complete
Let’s use our new words
However it is not solvable, there exists an a and b such that no
c and d satisfy (a ○ c) ~ b ~ (d ○ a)
Given a = 6 and b = 10 there does not exist a c and d such
that
(6 ○ c) ~ 10 ~ (d ○ 6). As c approaches 3, 6 ○ c approaches
18 or 9
It is not halvable as values between less than 9 and greater
than or equal to 6 can not be obtained by (a ○ a)
It is not Bisymmetric
(a ○ b) ○ (c ○ d) ~ (a ○ c) ○ (b
○ d)
(4 ○ 2) ○ (5 ○ 6) ≠ (4 ○ 5) ○ (2 ○
6)
6 ○ 30 ≠ 20 ○ 8,
180 ≠ 160
It is not Autodistributive (a ○ b) ○ c ~ (a ○ c) ○ (b ○ c)
Real Examples
 x ◙ y = x + y + 2c(xy) 1/2 , where c is a constant between -1 and 1 .
This is the variance of the sums of two random variables whose
respective variances are x and y and who correlation is c.
 If we consider non negative values of c: It is positive, closed,
nonassociative (except for c = 0 or 1) , generally not bisymmetric,
never autodistributive, halvable, continuous and Dedekind complete,
restrictedly solvable, but not solvable.
Negative examples: gambling
choices
Fails because
actual preferences
violate
transitivity
Sensory thresholds: fail to be
monotonic and locally
definable.
Lesson: Just because we can
concatenate physically
doesn’t mean
the
underlying structure will satisfy
definition 1 
Archimedean sequences: You can get there
from here (even if “there” is ver y far away and you
take small steps)
 Standard sequences- a,
a○ a
(a ○ a) ○ a,
((a ○ a) ○ a) ○ a
 Problems: if ○ is idempotent, e.g. the
average, we get nowhere, a ○ a = a
 For nonassociative concatenation
operations x ○ y ≠ y ○ x
 Different constructions of equally spaced sequences that are
equivalent in associative structures are no longer equivalent in more
general structures
Solutions
 Let x ◙ y = x + y/2. Note that depending on how we decide to
concatenate will determine whether the sequence is
Archimedean.

S’pose
a= 1
a= 1
a ◙ a = 1 .5
a ◙ a = 1 .5
(a ◙ a) ◙ a =2
a ◙ (a ◙ a) = 1 .75
((a ◙ a) ◙ a) ◙ a =2.5 a ◙ (a ◙ (a ◙ a)) = 1 .875
unbounded
bounded by 2
 Solution: arbitrarily choose a rule on how to branch, in this
case the choice of the right side branching would necessitate
a nonstandard Archimedean axiom.
Alternative #1: difference sequence
 S’pose we are take ◙ to be the mathematical average and are
structure to be the positive integers.
 We can construct a dif ference sequence if there exists a b,c in
A
(s.t. b and c are distinct) such that FOR ALL j, j + 1 , a j+1 ◙ b
~ aj ◙ c
 This captures the notion of equivalent spacing
 Here any b, s.t. b = c + 1 , will give us the correct spacing.
Solution 2: Regular sequences
 While a dif ference sequence will be suf ficient for solvable
concatenation structures, they may not exist otherwise.
 We can weaken this notion to create a regular sequence if
there exists a b,c in A with c > b such that FOR ALL j, j + 1 ,
a j+1 ◙ b > a j ◙ c and b ◙ a j+1 > c ◙ a j
Theorem 1 (p. 37)
 S’pose A is a Dedekind complete concatenation structure
i. If A is left-solvable in the sense that for b > a there
exists a c s.t.
b = c ○a, then it is Archimedean in the in the standard
sequence
ii. If A is solvable, then it is Archimedean in dif ference
sequences
 Upshot: For structures that are Dedekind complete, solvability
insures Archimedean properties.
 It is usually possible to show that structural and Archimedean
properties
follow from the topological and universal
axioms
Representations of PCSs, Def 3 (p. 38)
 S’pose A = < A , ≥ , ○ > is a concatenation structure.
 1. A is a PCS iff it is positive, restrictedly solvable, and
Archimedean in standard sequences
 2. An Associative PCS is said to be extensive
 3. a PCS in which A is a subset of RE + and ≥ is the usual ordering ≥
of RE + is said to be a numerical PCS
Definition 4 (p. 38)
 Let A = < A , ≥ , ○ > and A ‘ = < A’, ≥ ’ , ○’ > be PCSs, let φ be
a function from A into A’. φ is a homomorphism of A into A’ if f
the following hold
 1. φ preserves the order of ≥
 2. φ preserves the results of ○
So φ (a) ○’ φ (b) = φ (a○ b)
 If x ◙ y = x + y + c(xy) 1/2 is a numerical PCS for c ≥ 0 where x
and y are positive as is x ◙’ y = (x 2 + y 2 + cxy) 1/2
 The two structures are related by the homomorphism x  x 1/2
 If ◙ interpreted as the addition formula for variances, then ◙’ is the
corresponding formula for standard deviations.
 This is awesome!
Theorem 2: Uniqueness and construction
 Since homomorphisms preserve ordering, and concatenation,
they are one point unique.
 If φ and ψ are two homomorphisms of A into A ‘if they
agree on one point, then they agree on all points (except
maybe a maximal point) because the nonmaximal points are
tightly coupled to each other by concatenation
 This also means that we can order homomorphisms because
if φ(a) > ψ(a) for any a (nonmaximal), it will be true for all a
Theorem 3: Anything you can do I can do better
(well maybe not better, but just as well… so long as
there is a suitable strictly increasing function that
relates us)
 S’pose A = < A , ≥ , ○ > is a PCS
1 . There exists a numerical PCS such that there is a homomorphism
of the PCS into the numerical PCS
2. All such homomorphisms can be obtained be a strictly
increasing function h
from φ (A) onto φ’(A) such that for all a
A
φ’ (a) = h[φ(a)]
and the operations ◙ and ◙’ are related as follows
x ◙’ y = h -1 {h(x) ◙ h(y)}
Theorem 3 says that all PCSs the conditions for ordinal representation
are met and the objects in A can be given numerical labels that
preser ve order. Fur ther if it can be done at all, it can be done in
many ways which are just as good and any two sets of labels can be
related by a strictly increasing function.
Thus for positive operations associativity can be dropped and with a
slight modification of the Archimedean axiom we can prove that
numerical relations exist
Pandering to Jenny
Automorphism groups of PCSs
 We have shown that PCSs are one point unique, but have not
characterized the class of admissible transformations
 This is made dif ficult since we do not have a canonical
numerical operation (i.e. +) and we need a characterization
that is intrinsic to the structure itself
 Fortunately if φ and ψ are two isomorphisms from a totally
ordered PCS onto the same numerical PCS and if h is an
increasing function from φ into ψ (theorem 3) then φ and ψ
are two homomorphisms such that
φ -1 ψ is an automorphism (that is an isomorphism of A
onto itself
Ordering groups: Theorem 4 (p. 45)
 Theorem 4: The automorphism group of a PCS is and
Archimedean ordered group
 Remember that for and to automorphism the order is preserved, so if
α(a) > β(a) for some nonmaximal a, then it will be true for all a
 We can use this fact to define an ordering on automorphism groups!
 Thus the automorphism group of any PCS is isomorphic to the
additive reals
Theorem 5: Continuity
 Theorem 5 assures us that a representation can be selected
that is continuous using the normal topology of subsets of
real numbers (rather than a special (order) topology for each
set of labels
 restricting our attention to order topologies, any order
preserving function is bicontinuous
 That is, if h is a continuous order preserving function so is h -1
 With just one more definition we can take
this topological notion and give an equivalent
algebraic formulation
Definition 5 and Theorem 6: upper and lower
semicontinuity
 Let A = < A , ≥ , ○ > be a PCS with no minimal element
 It is lower semicontinuous if given that (a ○ b) > c we can
concatenate b with an element less than a that would still be
greater than c and similarly for a (i.e. there exists an a’ s.t a
> a’ and (a’ ○ b) > c and
there exists an b’ s.t b > b’ and (a ○ b’) > c
 Upper semicontinuity is defined in essentially the same way
except we have to establish that there exists an a’’ > a
because there may be a maximal element
 Lower and upper continuity are defined in two parts to ensure
that both right and left concatenation are semicontinuous
 Theorem 6 gives us that is continuous if f it satisfies upper
and lower semicontinuity
19.4 completions of total orders and
PCSs
 Prior literature pursued dif ferent goals: one emphasized
algebriac and counting aspects the other tried to achieve
measurement onto real intervals, to permit use of standard
mathematical machinery
 Theorems 7-10 try to steer a course between the two
 If a structure has “holes” so it cannot naturally be mapped
onto a real interval it may nevertheless be possible to plug
these holes with ideal elements
 Doing so allows the use of standard mathematics
+
=
Algebra and topology
 Algebriac theorems usually make use of:
 i. First-order universals (e.g. weak order or monotonicity
 ii. First-order existential (e.g. solvability or closure)
 iii. Second-order axioms (e.g. Archimedean or existance of countable, order dense subsets)
 iv. Higher order axioms (e.g. constraints on automorphism groups
 Measurement onto real inter vals of ten use i. but replace ii. and iii.
With topological assumptions (e.g. continuity, Dedeking
completeness, topological completeness, or topological
connectedness.
 It is usually possible to star t from the toplogical postulates and
show the structural and the Archimedean proper ties (see theorem
1) but not conver sely
 Here we look at how to move the other way. Narens & Luce (1976)
proposed to find algebraic conditions on a structure that made it
densely embeddable in a Dedekind -complete structure (similar to
the embedding of the rationals into the reals
Characterizing simple orders (p. 50)
 Quick definition: a set is simply ordered
 If a ≤ b and b ≤ a then a = b (antisymmetry);
 If a ≤ b and b ≤ c then a ≤ c (transitivity);
 a ≤ b or b ≤ a (totality).
 An order is dense if, for all x and y in X for which x < y, there is a z
in X such that
x < z < y.
 For a simple order to be order -isomorphic to inter vals in Re three
conditions must be satisfied: (Theorem 7)
1: The simple order must have a countable order -dense subset
-guarantees the existence of a continuous isomorphism into
Re
2: There must not be gaps.
3: There must be no “holes”, i.e. the simple order must be
Dedekind complete
-combined these assure us that the simple order is connected
How can you have a hole with no gap?
 A gap occurs when given a > c there exists no b
such that a>b>c
 So the integers have gaps but no holes
 The rationals have holes but no gaps
 Lexigraphic ordering of a plane has neither,
but has no countable order -dense subset
 Theorem 7 shows that Dedekind complete structures map
onto the reals, but it remains to be shown which PCSs can be
densely embedded in Dedekind complete structures.
 A PCS with no minimal element may have no gap, but if it has
a hole trying to fill it may result in a gap when the
concatenation is discontinous
Definition 6: Completion
 Let A = < A , ≥ > be a total order without gaps. A completion
of A is a pair <A , φ > such that
 The total order is a topologically connected simple -order
 φ is an isomorphism from A onto A
 φ(A) is order dense in A
 φ maps the extremum of A onto the extremum of A
 Theorem 8 gives us:
 The existence of a completion if A is a gapless simple
order
 Extensions of homomorphisms on the algebraic structure
 Uniqueness of the completion up to an isomorphism
I think Jeremy taught a class on this… why
didn’t I take it…
 The construction of a completion is exactly the same as
Dedekind’s construction of the reals from the rationals
 Every real number is taken to be the set of all the smaller
rationals
 √2 is identified with the set of all rationals such that r 2 < 2
 The subsets that will be identified as non -maximal elements
are called cuts
 A cut is a nonempty subset that has the following properties:
 It has a nonempty complement
 Every element in the comlement is larger than every element in the
subset
 The complement has no minimum (i.e. no gaps)
I think Jeremy taught a class on this… why
didn’t I take it…
 All of the holes in A are filled in the completion by the set of
all the elements “below the hole”
 The homomorphism φ maps each element to the
corresponding cut
 The ordering of the cuts is just set inclusion since for a > b
cut φ(a) includes the cut φ(b)
 Theorem 9 looks to give us a Dedekind completion if we can
have a similar ordering defined by inclusion
 Theorem 10 asserts that if a PCS is strictly ordered, closed
and gapless there is at most one Dedekind completion
Connections between conjoint structures and
concatenation structures (p. 77)
 Recall that conjoint measurements can be used to quantify
attributes where it is not possible concatenate
 Formal definition: S’pose A and P are nonempty sets and ≥ is
a binary relation on A x P. Then C = < A x P, ≥ > is a conjoint
structure if f for each a, b
A and p, q
P the following
three conditions are satisfied:
 Weak ordering
 Independence
 ap ≥ bq iff aq ≥bq
 ap ≥ aq iff bp ≥bq
 ≥ A and ≥ P are total orders
Can you say Thomsen condition?
I knew you could!
 C is said to satisfy the Thomsen condition if f for all a,b,c
A and
p,q,r
P, ar~cq and cp~br imply ap~bq
 For a 0
A and p 0
P, C
is said to solvable relative to
a 0 p 0 if f
 For each a
 For each ap
ap
A there exists a π (a)
P such that ap 0 ~ a 0 π(a)
AxP, there exists ξ(a,p)
A such that ξ(a,p) p 0 ~
 C is said to be unrestrictedly A -solvable if f for each a
A
and p,q
P, there exists a b
A such that ap~bq. The
definition of P-solvable is similar. If C is unrestictedly A and
P-solvable it is solvable
 Let J be an (infinite or finite) interval of integers. Then a
sequence {a j } is said to boundediff for some c,d
A, c ≥ aj
≥d for all j
I am not making this up! (p. 78)
 The Holman operation - S’pose C = < A x P, ≥ > is a conjoint
structure that is solvable relative to a 0 p 0 . The Holman
induced operation on A relative to a 0 p 0 denoted ○ a , is
defined by: for each a,b
A
a○ A b = ξ[a, π (b)]
 The Holman operation recodes information in a conjoint
structure as operations on one of its components
Definition 10 (p. 78)
 Let A be a nonempty set ≥ a binary relation on A , ○ a binary
relation on A and a 0 an element of A . Then A = < A , ≥ , ○, a 0
> is said to be a total concatenation structure if f the
following five conditions hold
 ≥ is a total order and ○ is monotonic
 The restriction of A to A + = {a|a
A and a> a 0 } is a PCS
 The restriction of A to A - = {a|a
A and a< a 0 } but with the
converse order is a PCS
 Acts as a 0 element, i.e. is the lowest element and yields the identity
upon concatenation
 Archimedean property
Now it’s time for my operation to do some
work!
 Theorem 11: Given a solvable conjoint structure, the Holman
operation is closed, monotonic and positive over A + (and
negative over A - ). If the conjoint structure is Archimedean the
positive and negative substructures are Archimedean in
standard sequences. Further if the larger conjoint structure is
both solvable and Archimedean then the union of
substructures is a closed, solvable, total concatenation
structure
 ii. S’pose A = < A x P, ≥ > is a closed total concatenation
structure. Then there is a conjoint structure C = < A x A , ≥’
> that is solvable relative a 0 a 0 and that induces A . If A is
Archimedean in dif ferences, then C is Archimedean. If A is
solvable, Archimedean in standard sequences, and
associative, then C is solvable and Archimedean \
 Punch line: The induced operations are basically two positive
concatenation structures separated by a 0
The relation of automorphisms in terms of
induced operations (p.80)
 We need to know whether the order automorphisms of C are
factorizable in the following sense
 C = < A x P, ≥ > is a conjoint structure and α is an order
automorphism of C . Then α is factorizable if f there exist
functions θ and η where θ is a 1:1 mapping of A onto A and η
is a 1:1 mapping of P onto P s.t.
α = < θ, η> i.e. α(ap) = θ(a) η(p)
 In a conjoint structure the identity of independent structures
should be preserved by automorphisms
 Theorem 12 shows that if the conjoint structure has a
factorizable automorphism the induced operations are
basically the same
Left and right multiplication
 S’pose A = < A , ≥ , ○> is a concatenation structure, then for
each a
A , define left multiplication a L by a L (b) = a ○ b,
for all b
A for which the right hand side is defined.
Define right multiplication analogously by b ○ a
 In a conjoint structure any pair multiplications in the Holman
structures induced on each factor generates a factorizable
transformation. But this is not typically an automorphism.
However, solvability at a point and pair multiplications that
yield automorphism are suf ficient to imply the Thomson
condition (Theorem 13)
Total concatenation structures induced by
closed Idempotent concatenation structures
(p. 81-82)
 In chapter 6 we recoded each bisymmetric structure as an
additive conjoint structure, we will use a similar tactic here
 S’pose A = < A , ≥, ○ > is a closed concatenation structure.
The the conjoint structure induced by A is C =< A x A , ≥’ >
Where for all a,b,c,d in A , ab ≥’ cd if f a ○ b ≥ c ○ d
 Where A is a closed concatenation structure and C is the
conjoint structure induced by A the following hold:
 If A is solvable, C is restrictedly solvable
 C is Archimedean iff A is Archimedean in differences
 S’pose A is idempotent and α is a mapping of A onto A. Then α is
an automorphism A of iff (α, α) is a factorizable automorphism of C
(Theorem 14)
More pandering to Jenny
Dilation (p. 82)
 S’pose A = < A , ≥, ○ > is a concatenation structure and α is
an automorphism of A . Then α is said to be a dilation of a if f
α(a) = a and it is said to be a translation of if f it is either the
identity of it is not a dilation for any a
 In other words, it is a translation if a has all the same points
as α(a) or none of the same points
 Consider linear transformations, the transformation x  rx +s
has a fixed point if r =1 and s =0 or if r ≠ 1
Dilation (p. 82)
 So… S’pose A is a closed, idempotent, solvable
concatenation structure J is the total concatenation
structure induced at a via Definitions 9 and 13 and α is an
automorphism of A .
 α is a dilation at a iff α automorphism of J a
 α is a translation iff α is an isomorphism of J a ontoJ α(a) where α(a) ≠
a
 This decomposes an idempotent structure into a family of
induced total concatenation structures that are all isomorphic
under the translation of the idempotent structure
 The dilations are the automorphisms of the induced total
concatenation structures
Intensive structures and
the Doubling function
 Let * denote the intensive operation.
 Let there be a 0 element that can be sensibly be joined to
each element of the set and let it play the role of the 0 in
mathematical average
 b is double a if f b*0 ~ a
 If we can do this we may think of a the halvable element of b
that we could introduce ○, s.t. a ○ a = b
 But it is not clear how to adjoin 0 to the function and so there
is the less direct definition 15
A less clear and direct way of characterizing a
doubling function (p. 84)
 Let A be a nonempty set, ≥ be a binar y relation on A , and * be a
par tial intensive operation on A (Definition 2). Suppose B is a
subset of A and φis a function from B onto A . Then φ is said to be
a doubling function of A = < A , ≥ , ○> if f for all a,b in A





Φis strictly increasing
If a is in B and a > b then b is in B
If a > b, then c in A such that b*c is defined and in B and a > φ (b*c)
* is a positive function
Suppose that a n is in A, n = 1,2,…., are such that if a n-1 are in B then
a n ~ φ (a n-1 )* a 1. For any b, either there exists an integer n s.t. a n is not an
element of B or a n ≥ b. Such is a sequence is called the standard sequence
of b.
I don’t see how φis a doubling function, unless we aren’t supposed to get the
doubling function until theorem 16, theorem 17 states the doubling function
is unique or there is one and only one other doubling function with a domain
that differs by just one point and the double of b is the maximal point in A
General representation and uniqueness of of
conjoint structures
 The existence of a representation of a conjoint structure that
is unrestrictedly solvable and Archimedean follows almost
immediately from the facts that it’s induced structure is a
total concatenation structure (theorem 11), that such a
structure is made up of two PCSs and that each PCS has a
representation (Theorem 3)
 Theorem 19: S’pose A = < A x P, ≥ > is a conjoint structure
that is Archimedean and solvable. Then there exists a
numerical operation ◙ and a function φfrom a and a function
ψ from P into Re such that
 Φ(a 0 )=0, ψ(a 0 )=0
 0 acts as the identity for ◙ whether it is on the right or left
 φ ψmaintain the ordering of ap ≥ bq, i.e. Φ(a) ◙ ψ(p) ≥ φ(b) ψ(q)
 Theorem 20: gives us one -point uniqueness (after a 0 p 0 has
been mapped to (0,0))
More generally
More generally we may be interested in the representation
and uniqueness of concatenation structures that are distinct
from PCSs
 Theorem 21 gives us that a concatenation structure that is
closed solvable and Archimedean in dif ferenceis wither 1 or 2
point unique