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Ch 4: Difference Measurement

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Difference Measurement In Ch 3 we saw the kind of representation you can get with a concatenation operation on an ordered set A “The question arises whether similarly tight representations ever exist when there is no concatenation operation.” 136

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Difference Measurement Extensive measurement: consider a set of movable rods Difference measurement: consider fixed points on a line. Consider a set of intervals between points We can construct standard sequences in A with an auxiliary, uncalibrated rod to lay off equal intervals 136

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Difference Measurement Denoting elements of A by a, b, e, d, we denote intervals in A by ab, cd, etc. We distinguish between ab and ba. Comparison with a set of movable rods generates an ordering on the intervals in A. ab ≿ cd if some rod does not exceed ab but exceeds or matches cd. 137

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Axiomatization of Difference Measurement Holder (1901) showed how the measurement of intervals between points on a line can be reduced to extensive measurement. Standard sequences of equally spaced elements a 1, a 2, a 3,..., where the intervals a 1 a 2 ∼ a 2 a 3 ∼... Equivalent intervals are identified with a single element, their equivalence class 143

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Otto Ludwig Hölder

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Positive Difference Structures 145

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145

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147 Interpret A as the set of endpoints of intervals. A* is the set of positive intervals, and is a subset of A x A.

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Positive Difference Structures 147 Transitivity

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Positive Difference Structures 147 Axiom 3 guarantees that there are no null intervals. Note it also follows that A* is not reflexive or symmetric.

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Positive Difference Structures 147 Weak monotonicity: this is needed to guarantee that concatenation of non-adjacent intervals gets the right results

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Positive Difference Structures 147 Archimedean axiom: a n a 1 = (n-1)a 2 a 1

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Positive Difference Structures 147 Archimedean axiom: a n a 1 = (n-1)a 2 a 1

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Positive Difference Structures 147

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147

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Algebraic Difference Structures 151 We now allow for negative and null intervals, so we don’t need A*.

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Algebraic Difference Structures 151 Axioms 2 and 3 of Definition 1 are here replaced by Axiom 2. It is a pretty intuitive axiom

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Algebraic Difference Structures 151 Axioms 3-5 are correspond to axioms 4-6 of Definition 1

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Algebraic Difference Structures 151

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Cross Modality Difference Structures 165 Solvability axiom: The first part says that any element in A i x A i can be matched with an element in A 1 x A 1. The second part is just the normal solvability property for A 1. But because of the first part, it follows that all the A i have the solvability property. This is also why the Archimedean axiom is formulated for A 1.

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Finite, Equally Spaced Difference Structures 167

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Absolute-Difference Structures 172 Axiom 3: Betweenness is well behaved i) If b is between a and c, and if c is between b and d, then c and b are between a and d. ii) If b is between a and c and c is between a and d, then ad exceeds bd

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Absolute-Difference Structures 172 Weak Monotonicity: If b is between a and c and b’ is between a’ and c’, and ab ∼ a’b’, then bc ≿ b’c’ iff ac ≿ a’c’

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Absolute-Difference Structures 172 Solvability: if ab ≿ cd, then there is some d’ that is between a and b such that ad’ ∼ cd

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Absolute-Difference Structures 172 Archimedean: a i is between a 1 and a i+1 for all i, and successive intervals are non-null. a i a 1 is strictly bounded.

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Absolute-Difference Structures 173

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