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Retracts of partially ordered sets

Part of: Combinatorics

Published online by Cambridge University Press:  09 April 2009

Dwight Duffus  and
Ivan Rival
Dwight Duffus
Affiliation:
The University of Calgary Calgary, Canada
Ivan Rival
Affiliation:
The University of Calgary Calgary, Canada
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Abstract

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Let P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.

Keywords

partially ordered setretractcrowndismantlable.

MSC classification

Secondary: 05A05: Permutations, words, matrices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 4 , June 1979 , pp. 495 - 506
Copyright
Copyright © Australian Mathematical Society 1979

References

Birkhoff, G. (1937), ‘Generalized arithmetic’, Duke Math. J. 3, 443–454.Google Scholar
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Colloq. Math. János Bolyai. 18 (1976), 271–292.Google Scholar
Duffus, D., Poguntke, W. and Rival, I. (1977), ‘Retracts and the fixed point problem for finite partially ordered sets’, Canad. Math. Bull. (to appear).Google Scholar
Kelly, D. and Rival, I. (1974), ‘Crowns, fences, and dismantlable lattices’, Canad. J. Math. 26, 1257–1271.CrossRefGoogle Scholar
Rival, I. (1976), ‘A fixed point theorem for finite partially ordered sets’, J. Combinatorial Theory Ser. A, 21, 309–318.CrossRefGoogle Scholar