Chapter 19: Non-additive representations Bennett Holman Foundations 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 = RA Where R is the probability of recognizing the presence of side effects and 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 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 arent 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 = One 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 of scale 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 Lets use our new words Spose x y = x + y,

if x or y is less than 3, and x y = xy otherwise 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

Lets 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) (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 doesnt 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, aa

(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. Spose

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 Spose 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, 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 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) Spose 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 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) Spose 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 be a function from A into A. is a homomorphism of A into A iff the following hold 1. preserves the order of 2. preserves the results of So (a) (b) = (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 and 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 (a) > (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) Spose 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 (A) onto (A) such that for all a A (a) = h[(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 and are two isomorphisms from a totally ordered PCS onto the same numerical PCS and if h is an increasing function from into (theorem 3) then and are two homomorphisms such that -1 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

(a) > (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 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, orderdense 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 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 such that

The total order is a topologically connected simple-order is an isomorphism from A onto A (A) is order dense in A 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 didnt I take it The construction of a completion is exactly the same as Dedekinds 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 didnt 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 maps each element to the corresponding cut The ordering of the cuts is just set inclusion since for a > b cut (a) includes the cut (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: Spose 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 For each ap

A there exists a (a) AxP, there exists (a,p) P such that ap0 ~ a0(a) A such that (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 Psolvable 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- Spose 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 aAb = [a, (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 its 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. Spose 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 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 is an order automorphism of C . Then is factorizable iff there exist functions and where is a 1:1 mapping of A onto A and is a 1:1 mapping of P onto P s.t. = < , > i.e. (ap) = (a) (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 Spose 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 Spose 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 Spose A is idempotent and is a mapping of A onto A. Then is an automorphism A of iff (, ) is a factorizable automorphism of C (Theorem 14) More pandering to Jenny Dilation (p. 82) Spose A = < A , , > is a concatenation structure and is an automorphism of A . Then is said to be a dilation of a iff (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 (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 Spose A is a closed, idempotent, solvable concatenation structure J is the total concatenation structure induced at a via Definitions 9 and 13 and is an automorphism of A . is a dilation at a iff automorphism of J a is a translation iff is an isomorphism of J a

a ontoJ (a) where (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 = 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 binary relation on A, and * be a partial intensive operation on A (Definition 2). Suppose B is a subset

of A and is a function from B onto A. Then is said to be a doubling function of A = < A, , > iff for all a,b in A is 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 > (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 ~ (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 dont see how is a doubling function, unless we arent 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 its 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: Spose A = < A x P, > is a conjoint structure that is

Archimedean and solvable. Then there exists a numerical operation and a function from a and a function from P into Re such that (a0)=0, (a0)=0 0 acts as the identity for whether it is on the right or left maintain the ordering of ap bq, i.e. (a) (p) (b) (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