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### The interval inclusion number of a partially ordered set

1991 Discrete Mathematics
West, The interval inclusion number of a partially ordered set, Discrete Mathematics 88 (1991) 259-277.  ...  We introduce the interval inclusion number (or interval number) i(P) as the smallest t such that P has a containment representation fin which each f(x) is the union of at most t intervals.  ...  By a poset (partially ordered set) we mean a pair (P, CP), where P is a set of elements and cP is a partial order on P, i.e. an antisymmetric, irreflexive, transitive binary relation.  ...

### Page 232 of The American Mathematical Monthly Vol. 61, Issue 4 [page]

1954 The American Mathematical Monthly
If the partially ordered set is a chain, it has a “natural” topology defined by taking the open intervals, consisting of elements x such that a<x 3b), to be neighborhoods of each of their elements.  ...  In a partially ordered set which is not a chain, this definition is not a useful one.  ...

### Interval sets and interval-set algebras

Yiyu Yao
2009 2009 8th IEEE International Conference on Cognitive Informatics
An interval set is an interval in the power set lattice based on a universal set and is a family of subsets of the universal set.  ...  Two types of interval-set algebras are examined based on an inclusion ordering and a knowledge ordering, respectively. Related studies are summarized.  ...  The relative sets consider both inclusion (truth) ordering and knowledge ordering of a bilattice.  ...

### Page 3180 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews
C. (1-BELL) Interval orders and circle orders. Order 5 (1988), no. 3, 225-234. There are two natural (partial) orderings on a set of intervals of real numbers.  ...  As applications, the lattice of subsets of a finite set and the lattice of divisors of a natural number are considered.” 90f:06002 06A10 11A05 11D09 11F06 20HOS Beck, Istvan (N-OSLO-I) Partial orders and  ...

### FL-GrCCA: A granular computing classification algorithm based on fuzzy lattices

Hongbing Liu, Shengwu Xiong, Zhixiang Fang
2011 Computers and Mathematics with Applications
In view of this, this work proposes a partial order relation and lattice computing, respectively, for dealing with the aforementioned issues.  ...  We compare the performance of FL-GrCCA with the performance of popular classification algorithms, including support vector machines (SVMs) and the fuzzy lattice reasoning (FLR) classifier, for a number  ...  This work was supported in part by the National Natural Science Foundation of China (Grant No. 40701153, 40971233), Natural Science Foundation of Henan Province (102300410178, 2009B520025) and self-determined  ...

### Page 654 of Mathematical Reviews Vol. , Issue 94b [page]

1994 Mathematical Reviews
A partially ordered set is called a k-sphere order if it is isomorphic to some set of balls in R‘, ordered by inclusion.  ...  A partially ordered set is called a k-sphere order if it is isomorphic to some set of balls in R*, ordered by inclusion.  ...

### Ideals In Partially Ordered Sets

Orrin Frink
1954 The American mathematical monthly
to the order relation of set inclusion.  ...  This corresponds to the definition of order for the real numbers. This is a rela- tion of simple ordering, and determines a topology, using open intervals for neighborhoods.  ...

### Page 724 of Mathematical Reviews Vol. 54, Issue 3 [page]

1977 Mathematical Reviews
The authors study the partially ordered set (J(0, n), <) consisting 5059 06 ORDER, LATTICES, ORDERED ALGEBRAIC STRUCTURES 724 of all closed intervals of real numbers (including degenerate intervals) with  ...  The set of all subgraphs of a graph ordered by inclusion forms a distributive lattice L(G) so that a result of R. P.  ...

### Fuzzy lattice neural network (FLNN): a hybrid model for learning

V. Petridis, V.G. Kaburlasos
1998 IEEE Transactions on Neural Networks
Sufficient conditions for the existence of an inclusion measure in a mathematical lattice are shown.  ...  Hence a novel theoretical foundation is introduced in this paper, that is the framework of fuzzy lattices or FL-framework, based on the concepts fuzzy lattice and inclusion measure.  ...  In conclusion is a partially ordered set. Consider the following lemma.  ...

### An upper bound on the "dimension of interval orders"

I Rabinovitch
1978 Journal of combinatorial theory. Series A
The main results of this paper are two distinct characterizations of interval orders and an upper bound on the dimension of an interval order as a function of its height.  ...  In particular, interval orders of height 1 have dimension of at most 2.  ...  (X, P) is an interval order if and only if H,(X) is linearly ordered by set inclusion.  ...

### Inclusion within Continuous Belief Functions [article]

Dorra Attiaoui , Boutheina Ben Yaghlane
2015 arXiv   pre-print
Within this paper we will propose and present two forms of inclusion: The strict and the partial one. In order to develop this relation, we will study the case of consonant belief function.  ...  Defining and modeling the relation of inclusion between continuous belief function may be considered as an important operation in order to study their behaviors.  ...  Partial inclusion between belief densities induced by normal distributions The partial inclusion is defined in order to give us the proportion of the intersection between two pdf s. 1) Partial inclusion  ...

### An Introduction to Some Spaces of Interval Functions [article]

Roumen Anguelov
2004 arXiv   pre-print
The paper gives a brief account of the spaces of interval functions defined through the concepts of H-continuity, D-continuity and S-continuity.  ...  Various ways in which the spaces of H-continuous, D-continuous and S-continuous interval functions complement the spaces of continuous real functions are discussed.  ...  A partial order naturally related to interval spaces is the relation inclusion.  ...

### Page 2417 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews
{For the entire collection see MR 92b:05001.} 92e:06012 06A07 Madej, Tom (1-IL-C); West, Douglas B. (1-IL) The interval inclusion number of a partially ordered set.  ...  We introduce the interval inclusion number (or interval number) i(P) as the smallest ¢ such that P has a containment representation f in which each f(x) is the union of at most ¢ intervals.  ...

### Fuzzy Alexandrov Topologies Associated to Fuzzy Interval Orders

Gianni Bosi, Chiaramaria Panozzo, Magal`ı Ernestine Zuanon
2020 Asian Research Journal of Mathematics
In this way, we generalize a well known result by Rabinovitch (1978), according to which a crisp partial order is a crisp interval order if and only if the family of all the strict upper sections of the  ...  We characterize the fuzzy T0 - Alexandrov topologies on a crisp set X, which are associated to fuzzy interval orders R on X.  ...  In this way, we generalize a well known result by [4] , according to which a crisp partial order is a crisp interval order if and only if the family of all the strict upper sections of the partial order  ...

### Page 5126 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews
An ordered set P is called a bounded bitolerance order if for each x € P there exist a closed interval J, = [a(x),d(x)] (probably of the set of all real numbers, although this is not explicitly stated)  ...  (English summary ) Order 11 (1994), no.2, 127-134. Let &,(s,t) be the partially ordered set consisting of all subsets with size s or ¢ of an n-element set, ordered by inclusion.  ...
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