Cycle-free cuts of mutual rank probability relations

Abstract

It is well known that the linear extension majority (LEM) relation of a poset of size n ≥ 9 can contain cycles. In this paper we are interested in obtaining minimum cutting levels αm such that the crisp relation obtained from the mutual rank probability relation by setting to 0 its elements smaller than or equal to αm, and to 1 its other elements, is free from cycles of length m. In a first part, theoretical upper bounds for αm are derived using known transitivity properties of the mutual rank probability relation. Next, we experimentally obtain minimum cutting levels for posets of size n ≤ 13. We study the posets requiring these cutting levels in order to have a cycle-free strict cut of their mutual rank probability relation. Finally, a lower bound for the minimum cutting level α4 is computed. To accomplish this, a family of posets is used that is inspired by the experimentally obtained 12-element poset requiring the highest cutting level to avoid cycles of length 4.

Estimating Octanol / Water Partition Coefficients by Order Preserving Mappings

“…A computational and mathematical attractive method to estimate h av is the lattice theoretical one, 16 which avoids storing and managing all L i in LT (Appendix). The idea may be sketched as follows:…”

“…(b) A linear extension is then a path from the bottom (empty set) to the top object (the poset itself). (c) By an appropriate organisation of the computational algorithm, important quantities of the poset such as average height 16 can be obtained. To manually find all down sets of a poset (X, IB), we perform the following steps: (4), and here we show how we apply each one of the five steps mentioned before:…”

“…Following, 16,18,19 these four items are the basis of a computational approach to calculate average heights. In the software PyHasse this concept is available too (avrank5.py), where the Python-code by Wienand (2003) 19 was correspondingly adjusted.…”

“…It has been proven that LPOM computational complexity is O(n 2 ). 26 There are two LPOM variants, namely LPOM0 and LPOMext. The first one assumes all x-incomparables to be isolated objects; then both down set and up set of x are considered as chains and a simple counting of the probability for each y  U(x) localized above or below x, allows estimating h av (LPOM0)(x).…”

“…According to h av exact and to h av (LPOMext), we found the following equivalence classes of chlorophenols: [2,3], [5,7], [6,9], [8,10], [11,14], [12,13], [15,16], [17,18]; and according to h av (LPOM0), these ones: [2,3], [5,7,8,10], [6,9], [11,14,15,16], [12,13], [17,18]. The fusion of [5,7] and [8,10] as well as that of [11,14] and [15,16] in LPOM0 arises from the additional symmetries because the incomparable objects are considered as isolated.…”