Additive Representability of Finite Measurement Structures

Slinko, Arkadii

doi:10.1007/978-3-540-79128-7_7

Cited by 9 publications

(4 citation statements)

References 39 publications

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“…The reference list in this paper is kept short. The survey by Slinko [7] is extensive and provides an excellent source of references.…”

Section: C23 Additivementioning

confidence: 99%

A Basic Set of Cancellation Violating Sequences for Finite Two-Dimensional Non-Additive Measurement

2023

Annales Mathematicae Silesianae

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Cancellation conditions play a central role in the representation theory of measurement for a weak order on a finite two-dimensional Cartesian product set X. A weak order has an additive representation if and only if it violates no cancellation conditions. Given X, a longstanding open problem is to determine the simplest set of cancellation conditions that is violated by every linear order that is not additively representable. Here, we report that the simplest set of cancellation conditions on a 5 by 5 product X is obtained.

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“…The reference list in this paper is kept short. The survey by Slinko [7] is extensive and provides an excellent source of references.…”

Section: C23 Additivementioning

confidence: 99%

A Basic Set of Cancellation Violating Sequences for Finite Two-Dimensional Non-Additive Measurement

2023

Annales Mathematicae Silesianae

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“…These total orders appear in a variety of settings with names that reflect the application at hand [14,23,6,11]. In probability theory, the total orders in F n are known as comparative probability orders, and they enjoy applications in decision theory and economics [21,14,16,28]. A comparative probability order is additively representable when there is a probability measure p :…”

Section: De Finetti Total Ordersmentioning

confidence: 99%

De Finetti Lattices and Magog Triangles

Beveridge

Calaway

Heysse

2021

Electron. J. Combin.

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The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

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“…In probability theory, the orderings in F n are known as comparative probability orders, and they enjoy applications in decision theory and economics [20,13,15,26]. A comparatively probability order is additively representable when there is a probability measure p : [n] → [0, 1] that induces the order, namely p(X) ≤ p(Y ) if and only if X Y .…”

Section: Introductionmentioning

confidence: 99%

de Finetti Lattices and Magog Triangles

Beveridge¹,

Calaway²,

Heysse³

2019

Preprint

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Let B n,2 denote the order ideal of the boolean lattice B n consisting of all subsets of size at most 2. Let F n,2 denote the poset extension of B n,2 induced by the rule: i < j implies {i} ≺ {j} and {i, k} ≺ {j, k}. We give an elementary bijection from the set F n,2 of linear extensions of F n,2 to the set of shifted standard Young tableau of shape (n, n − 1, . . . , 1), which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set F(1) n,2 of poset extensions so that each singleton is comparable with all of the doubletons. We show that F(1) n,2 is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

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Additive Representability of Finite Measurement Structures

Cited by 9 publications

References 39 publications

A Basic Set of Cancellation Violating Sequences for Finite Two-Dimensional Non-Additive Measurement

A Basic Set of Cancellation Violating Sequences for Finite Two-Dimensional Non-Additive Measurement

De Finetti Lattices and Magog Triangles

de Finetti Lattices and Magog Triangles

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