Chapter 19:Non-additive

representations

Bennett HolmanFoundations of Measurement

What is essential nonadditivity

The fact that a representation is nonadditive is not sufficient 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 effects (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 = RAWhere R is the probability of recognizing the presence of side effectsand A is the conditional probability of attributing the side effect 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 nonadditiveQ = 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) ◙ zE.g. averaging: Let x ◙ y = (x + y)/2Using 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 = ½One consequence of considering arbitrary

binary operations is that finding a representation can be seen as a process of discovery

Whereas the uniqueness problem is best conceptualized as a process ofscale construction

Def 1: Concatenation Structures (p. 26)

A = < A, ≥ , ○ > is a concatenation structure iff 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 preserves:

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 variables, using the order topology on its range and the relative product order topology on its domain

Let’s use our new words

S’pose x ◙ y = x + y, if x or y is less than 3, and x ◙ y = xy otherwiseThe 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 7Nonassociative 4 ◙ (2 ◙ 2) ≠ (4 ◙ 2) ◙ 2

4 ◙ (4) ≠ (6) ◙ 2, (16 ≠ 8)

Closed (x ◙ y) is always definedPositive (x ◙ y) is strictly greater than x or yRestrictively 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 ≠

160It is not Autodistributive (a ○ b) ○ c ~ (a ○ c) ○ (b ○ c)

(2 ○ 3) ○ 4 ≠ (2 ○ 4) ○ (3 ○ 4) 5 ○ 4 ≠ 8 ○ 12, 20 ≠ 96

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 very 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. theaverage, 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 = 1a ◙ 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))

= 1.875unbounded 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 difference sequence if there exists a b,c in A (s.t. b and c are distinct) such that FOR ALL j, j + 1, aj+1 ◙ b ~ aj ◙ c

This captures the notion of equivalent spacingHere any b, s.t. b = c + 1, will give us the correct spacing.

Solution 2: Regular sequences

While a difference sequence will be sufficient 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, aj+1 ◙ b > aj ◙ c and b ◙ aj+1 > c ◙ aj

Theorem 1 (p. 37)

S’pose A is a Dedekind complete concatenation structurei. 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 sequenceii. If A is solvable, then it is Archimedean in difference 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’ iff 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 = (x2 + y2 + cxy)1/2

The two structures are related by the homomorphism x x1/2

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 PCS2. 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 preserve order. Further 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 difficult 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’) > cUpper 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 iff it satisfies upper and lower semicontinuity

19.4 completions of total orders and PCSs

Prior literature pursued different 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 intervals often 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 start from the toplogical postulates and show the structural and the Archimedean properties (see theorem 1) but not conversely

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 intervals 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 Re2: 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 bsuch that a>b>c

So the integers have gaps but no holesThe rationals have holes but no gapsLexigraphic ordering of a plane has neither,

but has no countable order-dense subsetTheorem 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 r2 < 2The 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 iff 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 iff for all a,b,c A and p,q,r P, ar~cq and cp~br imply ap~bq

For a0 A and p0 P, C is said to solvable relative to a0 p0 iff For each a A there exists a π (a) P such that ap0 ~ a0π(a)

For each ap AxP, there exists ξ(a,p) A such that ξ(a,p) p0 ~ ap

C is said to be unrestrictedly A-solvable iff 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 {aj} is said to boundediff for some c,d A, c ≥ aj ≥d for all j

C is said to be Archimedean iff every bounded standard sequence on A is finite

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 a0 p0. The Holman induced operation on A relative to a0 p0 denoted ○a, is defined by: for each a,b A

a○Ab = 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 a0 an element of A. Then A = < A, ≥ , ○, a0 > is said to be a total concatenation structure iff the following five conditions hold ≥ is a total order and ○ is monotonic The restriction of A to A+ = {a|a A and a> a0} is a PCS

The restriction of A to A- = {a|a A and a< a0} 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 a0 a0 and that induces A . If A is Archimedean in differences, 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 a0

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 iff 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 automorphismsTheorem 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 aL by aL (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 sufficient 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 iff 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 iff α(a) = a and it is said to be a translation of iff 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 iff 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 = bBut 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 binary relation on A, and * be a partial 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, ≥ , ○> iff 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 an is in A, n = 1,2,…., are such that if an-1 are in B then

an ~ φ (an-1)* a1. For any b, either there exists an integer n s.t. an is not an element of B or an ≥ 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(a0)=0, ψ(a0)=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 a0p0has 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 differenceis wither 1 or 2 point unique