### Abstract

Let F be a monotone operator on the complete lattice L into itself. Tarski’s lattice theoretical fixed point theorem states that the set of fixed points of F is a nonempty complete lattice for the ordering of L. We give a constructive proof of this theorem showing that the set of fixed points of F is the image of L by a lower and an upper preclosure operator. These preclosure operators are the composition of lower and upper closure operators which are defined by means of limits of stationary transfinite iteration sequences for F. In the same way we give a constructive characterization of the set of common fixed points of a family of commuting operators. Finally we examine some consequences of additional semicontinuity hypotheses.

Original language | English (US) |
---|---|

Pages (from-to) | 43-57 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 82 |

Issue number | 1 |

State | Published - 1979 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*82*(1), 43-57.

**Constructive versions of tarski's fixed point theorems.** / Cousot, Patrick; Cousot, Radhia.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 82, no. 1, pp. 43-57.

}

TY - JOUR

T1 - Constructive versions of tarski's fixed point theorems

AU - Cousot, Patrick

AU - Cousot, Radhia

PY - 1979

Y1 - 1979

N2 - Let F be a monotone operator on the complete lattice L into itself. Tarski’s lattice theoretical fixed point theorem states that the set of fixed points of F is a nonempty complete lattice for the ordering of L. We give a constructive proof of this theorem showing that the set of fixed points of F is the image of L by a lower and an upper preclosure operator. These preclosure operators are the composition of lower and upper closure operators which are defined by means of limits of stationary transfinite iteration sequences for F. In the same way we give a constructive characterization of the set of common fixed points of a family of commuting operators. Finally we examine some consequences of additional semicontinuity hypotheses.

AB - Let F be a monotone operator on the complete lattice L into itself. Tarski’s lattice theoretical fixed point theorem states that the set of fixed points of F is a nonempty complete lattice for the ordering of L. We give a constructive proof of this theorem showing that the set of fixed points of F is the image of L by a lower and an upper preclosure operator. These preclosure operators are the composition of lower and upper closure operators which are defined by means of limits of stationary transfinite iteration sequences for F. In the same way we give a constructive characterization of the set of common fixed points of a family of commuting operators. Finally we examine some consequences of additional semicontinuity hypotheses.

UR - http://www.scopus.com/inward/record.url?scp=84972546036&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972546036&partnerID=8YFLogxK

M3 - Article

VL - 82

SP - 43

EP - 57

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -