Orders on Multisets and Discrete Cones

Abstract

Abstract. We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of Independence of Equal Submultisets (IES) introduced by Slinko (2002, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (1996) and Conder and Slinko (2004), we define functions f (n, k) and g(n, k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n, k) = n − 1 whenever n ≥ 3 and (n, k) = (5, 2). In the exceptional case, g(5, 2) = 3. We also prove that g(n, k) ≤ f (n, k) ≤ n and establish that for small n and k the functions g(n, k) and f (n, k) coincide.

An Axiomatic Development of Multi-Natural Numbers

“…Although the root of the studies in multiset is in combinatorics from ancient times ( [25], [26], [27]), the modern research in this field about the structural development in multiset context is a relatively new concept. Some research works on the relations and functions in multiset context ( [13], [14], [22]), multiset topology ( [1], [11], [12], [18]), multiset ordering ( [13]), multi group theory ( [10], [23], [24]) etc. have been done by some researchers.…”

Additive Representability of Finite Measurement Structures

“…We call it the power multiset structure. The power multiset model has numerous useful interpretations (see e.g., [40,5]). We report results on the power multiset structure in Section 5.…”

“…Here we follow Conder et al [5] who give a slightly different (but equivalent) definition of this concept, which makes it a close relative to the concept of the Independence of Equal Subalternatives and de Finetti's axiom.…”

“…Both independently prove that all orders on P[n] satisfying this axiom are additively representable. For the set P ≤k [n] of all multisets on [n] of cardinality ≤ k, Danilov gives an example of nonrepresentable orders on P ≤k [5] satisfying the modified de Finetti axiom.…”

“…we obtain functions f (n, k) and g(n, k). It is rather surprising that in this case better progress can be achieved in describing these functions than in the case of comparative probability orders [5]. The function g(n, k) is determined exactly: we have g(n, k) = n − 1 for (n, k) = (5, 2) and g(5, 2) = 3.…”