On Dilworth's decomposition theorem for partially ordered sets

Tverberg, Helge

doi:10.1016/s0021-9800(67)80079-6

Cited by 33 publications

(7 citation statements)

References 4 publications

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“…Shortly aer Dilworth's seminal paper [1] a "Note" [2] was published containing an algorithmic proof, that is, a proof which also gives a method to �nd a combination of a maximal antichain and a minimal path cover. e other proofs [1,[3][4][5] are nonalgorithmic. e key issue in [2] is the relation between a minimal path cover and a maximal antichain in on the one hand and a maximal matching and a minimal vertex cover (in this order) in an associated bipartite graph on the other hand.…”

Section: Introductionmentioning

confidence: 99%

Another Note on Dilworth's Decomposition Theorem

Pijls

Potharst

2013

Journal of Discrete Mathematics

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Section: Introductionmentioning

confidence: 99%

Another Note on Dilworth's Decomposition Theorem

Pijls

Potharst

2013

Journal of Discrete Mathematics

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“…Many proofs of Dilworth's Theorem are known [5,9,10,14,17,18]. Among them, the argument provided by Fulkerson [10] is straightforward, by which a bipartite graph G S with bipartite (V 1 , V 2 ) for S = {a 1 , a 2 , ...,a n } is constructed, where V 1 = {x 1 , x 2 , ..., x n }, V 2 = {y 1 , y 2 , ..., y n } and an edge joining x i V 1 to y j V 2 whenever a i a j .…”

Section: Introductionmentioning

confidence: 99%

On the Decomposition of Posets into Minimum Set Node-Disjoint Chains

Chen¹,

Chen²

2013

Proceedings of the International Conference on Computer, Networks and Communication Engineering (ICCNCE 2013)

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One of the most famous results in the theory of partially ordered sets is due to Dilworth (1950) who showed that the size of a minimum decomposition (into chains) of a partially ordered set S is equal to the size of a maximum antichain, which is a subset of pairwise incomparable elements. However, up to now, the bestalgorithm to decompose S into a minimum set of chains needs O(n 3) time, where n is the number of the elements in S. In this paper, we address this problem and propose an algorithm which produces a minimum decomposition in O(n 2) time and O(m + n)space, where is the size of a maximum anti chain and m is the number of relations between elements (i.e., the number of pairs (a, c) such that ac).In general, is much smaller than n.

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“…In [6] , Dilworth discussed a partially ordered set (Poset) as a set with binary relation ≤ such that:…”

Section: Introductionmentioning

confidence: 99%

Characterization of Chain as a Regular Semi Group

Kehinde¹,

Makanjuola²

2008

J. of Mathematics and Statistics

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Problem Statement: There are some special classes of semi group namely: regular and eventually regular, abundant, orthodox, quasi-adequate. The objective of this study were to: (i) Define a new class of semi group on a Poset and give related examples (ii) Study and establish conditions that characterized Chain as a regular semi group. Approach: Tests of some of characteristics of semi group like associativity, commutativity, and regular semi group were carried out on this new class. Results: Conditions were obtained that showed it is associative and regular. Conclusion: Hence the results suggest that since Chain is regular, there are many other things we can still do this with class of semi group such as: (i) Whether one can characterize all the Green's equivalences and their starred analogues (ii) Whether one can characterize all the congruencies of the given semi group (iii) Whether one characterize all the subsemigroups of the given semi group.

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On Dilworth's decomposition theorem for partially ordered sets

Cited by 33 publications

References 4 publications

Another Note on Dilworth's Decomposition Theorem

Another Note on Dilworth's Decomposition Theorem

On the Decomposition of Posets into Minimum Set Node-Disjoint Chains

Characterization of Chain as a Regular Semi Group

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