scholarly journals THE EXPECTED JAGGEDNESS OF ORDER IDEALS

2017 ◽  
Vol 5 ◽  
Author(s):  
MELODY CHAN ◽  
SHAHRZAD HADDADAN ◽  
SAM HOPKINS ◽  
LUCA MOCI
Keyword(s):  
Probability Distribution ◽  
Young Diagram ◽  
Partially Ordered Sets ◽  
Order Ideal ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Combinatorial Theorem ◽  
Order Ideals ◽  
Geometric Formula

Thejaggednessof an order ideal$I$in a poset$P$is the number of maximal elements in$I$plus the number of minimal elements of$P$not in$I$. A probability distribution on the set of order ideals of$P$istoggle-symmetricif for every$p\in P$, the probability that$p$is maximal in$I$equals the probability that$p$is minimal not in$I$. In this paper, we prove a formula for the expected jaggedness of an order ideal of $P$under any toggle-symmetric probability distribution when$P$is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015,arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS

2013 ◽  
Vol 89 (2) ◽  
pp. 279-292 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PATTANACHAI RAWIWAN
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered

AbstractFor a semigroup $S$, let ${S}^{1} $ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let ${S}^{1} = S$. Mitsch defined the natural partial order $\leqslant $ on a semigroup $S$ as follows: for $a, b\in S$, $a\leqslant b$ if and only if $a= xb= by$ and $a= ay$ for some $x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.

scholarly journals A Theorem on Partially Ordered Sets and its Application

1969 ◽  
Vol 9 (3-4) ◽  
pp. 361-362
Author(s):  
Vladimir Devidé
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered ◽  
Image Position

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that

scholarly journals On the endomorphism semigroup of an ordered set

1995 ◽  
Vol 37 (2) ◽  
pp. 173-178 ◽  
Author(s):  
T. S. Blyth
Keyword(s):  
Maximal Element ◽  
Ordered Set ◽  
Minimal Element ◽  
Hasse Diagram ◽  
Endomorphism Semigroup ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Image Position

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of(a) all antichains;(b) all quasi-complete chains;(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);(f) the six-element crown C6 with Hasse diagramA similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2].

To the spectral theory of partially ordered sets

2018 ◽  
Vol 60 (3) ◽  
pp. 578-598
Author(s):  
Yu. L. Ershov ◽  
M. V. Schwidefsky
Keyword(s):  
Spectral Theory ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

Modelling Nondeterministic Concurrent Processes with Event Structures

1991 ◽  
Vol 14 (1) ◽  
pp. 39-73
Author(s):  
Rita Loogen ◽  
Ursula Goltz
Keyword(s):  
Dynamic Behaviour ◽  
Partially Ordered Sets ◽  
Parallel Composition ◽  
Ordered Sets ◽  
Communicating Sequential Processes ◽  
Partially Ordered ◽  
Event Structures ◽  
Transition Relation ◽  
Concurrent Processes ◽  
Sequential Processes

We present a non-interleaving model for non deterministic concurrent processes that is based on labelled event structures. We define operators on labelled event structures like parallel composition, nondeterministic combination, choice, prefixing and hiding. These operators correspond to the operations of the “Theory of Communicating Sequential Processes” (TCSP). Infinite processes are defined using the metric approach. The dynamic behaviour of event structures is defined by a transition relation which describes the execution of partially ordered sets of actions, abstracting from internal events.

Estimating the area under a receiver operating characteristic curve using partially ordered sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ehsan Zamanzade ◽  
Xinlei Wang
Keyword(s):  
Roc Curve ◽  
Characteristic Curve ◽  
Partially Ordered Sets ◽  
Sampling Technique ◽  
Real Data ◽  
Cost Effective ◽  
Ordered Sets ◽  
Operating Characteristics ◽  
Partially Ordered ◽  
Receiver Operating

AbstractRanked set sampling (RSS), known as a cost-effective sampling technique, requires that the ranker gives a complete ranking of the units in each set. Frey (2012) proposed a modification of RSS based on partially ordered sets, referred to as RSS-t in this paper, to allow the ranker to declare ties as much as he/she wishes. We consider the problem of estimating the area under a receiver operating characteristics (ROC) curve using RSS-t samples. The area under the ROC curve (AUC) is commonly used as a measure for the effectiveness of diagnostic markers. We develop six nonparametric estimators of the AUC with/without utilizing tie information based on different approaches. We then compare the estimators using a Monte Carlo simulation and an empirical study with real data from the National Health and Nutrition Examination Survey. The results show that utilizing tie information increases the efficiency of estimating the AUC. Suggestions about when to choose which estimator are also made available to practitioners.

scholarly journals Model Checking Existential Logic on Partially Ordered Sets

2016 ◽  
Vol 17 (2) ◽  
pp. 1-35 ◽  
Author(s):  
Simone Bova ◽  
Robert Ganian ◽  
Stefan Szeider
Keyword(s):  
Model Checking ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

Planar zero divisor graphs of partially ordered sets

2012 ◽  
Vol 137 (1-2) ◽  
pp. 27-35 ◽  
Author(s):  
M. Afkhami ◽  
Z. Barati ◽  
K. Khashyarmanesh
Keyword(s):  
Zero Divisor ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered
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