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Common Fixed Points of Commuting Monotone Mappings

Published online by Cambridge University Press:  20 November 2018

James S. W. Wong
James S. W. Wong*
Affiliation:
University of Alberta, Edmonton, Alberta
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We are concerned here with the existence of fixed or common fixed points of commuting monotone self-mappings of a partially ordered set into itself. Let X be a partially ordered set. A self-mapping ƒ of X into itself is called an isotone mapping if x ⩾ y implies ƒ(x) ⩾ ƒ(y). Similarly, a self-mapping ƒ of X into itself is called an antitone mapping if x ⩾ y implies ƒ(x) ⩽ ƒ(y). An element X0 ∈ X is called well-ordered complete if every well-ordered subset with x0 as its first element has a supremum. An element x0 ∈ X is called chain-complete if every non-empty chain C ⊆ X such that x ⩾ x0 for all x ∈ C, has a supremum. X is called a well-ordered-complete semi-lattice if every non-empty well-ordered subset has a supremum. X is called a complete semi-lattice if every non-empty subset of X has a supremum.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 617 - 620
Copyright
Copyright © Canadian Mathematical Society 1967

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