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Partially Ordered Sets
, 2000
"... this article we define the choice function of a nonempty set family that does not contain ; as introduced in [6, pages 8889]. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure nonempty set and order of the set, cha ..."
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Cited by 165 (4 self)
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this article we define the choice function of a nonempty set family that does not contain ; as introduced in [6, pages 8889]. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure nonempty set and order of the set
Ordered Sets
"... “And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The C ..."
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. The Cayley representation theorem then shows that this model is “correct”: every group is isomorphic to a group of permutations. In the same way, we want to define a partial order to be an abstract model of set containment ⊆, and then we should prove a representation theorem for partially ordered sets
ORDERED SETS
"... Abstract. Ordered sets are used as a computational model for motion planning problems. Every ordered set has a twodirectional point representation using subdivisions. These subdivision points correspond to direction changes along the path of motion. AMS subject classifications (1980). 06A10, 52A37, ..."
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Abstract. Ordered sets are used as a computational model for motion planning problems. Every ordered set has a twodirectional point representation using subdivisions. These subdivision points correspond to direction changes along the path of motion. AMS subject classifications (1980). 06A10, 52A37
Pseudocomplemented Ordered Sets
, 1993
"... The aim of this paper is to transfer the concept of pseudocomplement from lattices to ordered sets and to prove some basic results holding for pseudocomplemented ordered sets. ..."
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The aim of this paper is to transfer the concept of pseudocomplement from lattices to ordered sets and to prove some basic results holding for pseudocomplemented ordered sets.
Complemented Ordered Sets
 ARCH. MATH. (BRNO
, 1992
"... We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets repr ..."
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Cited by 3 (0 self)
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We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets
DENSE SUBSETS OF ORDERED SETS
, 1999
"... Abstract. Some modifications of the definition of density of subsets in ordered ( = partially ordered) sets are given and the corresponding concepts are compared. ..."
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Abstract. Some modifications of the definition of density of subsets in ordered ( = partially ordered) sets are given and the corresponding concepts are compared.
Tripled partially ordered sets
"... In this paper, we introduce tripled partially ordered sets and monotone functions on tripled partially ordered sets. Some basic properties on these new defined sets are studied and some examples for clarifying are given. ..."
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In this paper, we introduce tripled partially ordered sets and monotone functions on tripled partially ordered sets. Some basic properties on these new defined sets are studied and some examples for clarifying are given.
1. Ordered Sets
"... “And just how far would you like to go in? ” he asked.... “Not too far but just far enough so’s we can say that we’ve been there, ” said the first chief. “All right, ” said Frank, “I’ll see what I can do.” –Bob Dylan In group theory, groups are defined algebraically as a model of permutations. The C ..."
Abstract
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. The Cayley representation theorem then shows that this model is “correct”: every group is isomorphic to a group of permutations. In the same way, we want to define a partial order to be an abstract model of set containment ⊆, and then we should prove a representation theorem for partially ordered sets
OPTICS: Ordering Points To Identify the Clustering Structure
, 1999
"... Cluster analysis is a primary method for database mining. It is either used as a standalone tool to get insight into the distribution of a data set, e.g. to focus further analysis and data processing, or as a preprocessing step for other algorithms operating on the detected clusters. Almost all of ..."
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Cited by 511 (49 self)
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the intrinsic clustering structure accurately. We introduce a new algorithm for the purpose of cluster analysis which does not produce a clustering of a data set explicitly; but instead creates an augmented ordering of the database representing its densitybased clustering structure. This clusterordering
Compact representations of ordered sets
 In Proc. 15th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... We consider the problem of efficiently representing sets S of size n from an ordered universe U = {0,...,m−1}. Given any ordered dictionary structure (or comparisonbased ordered set structure) D that uses O(n) pointers, we demonstrate a simple blocking technique that produces an ordered set struct ..."
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Cited by 22 (3 self)
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We consider the problem of efficiently representing sets S of size n from an ordered universe U = {0,...,m−1}. Given any ordered dictionary structure (or comparisonbased ordered set structure) D that uses O(n) pointers, we demonstrate a simple blocking technique that produces an ordered set
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