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Weak chain-completeness and fixed point property for pseudo-ordered sets. (English) Zbl 1081.06004

Summary: In this paper the notion of weak chain-completeness is introduced for pseudo-ordered sets as an extension of the notion of chain-completeness of posets (see G. Markowsky [Algebra Univers. 6, 53–68 (1976; Zbl 0332.06001)]), and it is shown that every isotone map of a weakly chain-complete pseudo-ordered set into itself has a least fixed point.

MSC:

06B05 Structure theory of lattices
06A75 Generalizations of ordered sets

Citations:

Zbl 0332.06001
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References:

[1] P. Crawley and R. P. Dilworth: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, 1973. · Zbl 0494.06001
[2] J. Lewin: A simple proof of Zorn’s lemma. Amer. Math. Monthly 98 (1991), 353–354. · Zbl 0749.04002
[3] G. Markowski: Chain-complete posets and directed sets with applications. Algebra Universalis 6 (1976), 54–69.
[4] H. L. Skala: Trellis theory. Algebra Universalis 1 (1971), 218–233. · Zbl 0242.06003
[5] H. Skala: Trellis Theory. Mem. Amer. Math. Soc. 121, Providence, 1972. · Zbl 0242.06004
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