Geodesic
Characterizations of totally ordered sets by their various endomorphisms
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 1, pp. 23-32.

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We characterize totally ordered sets within the class of all ordered sets containing at least three-element chains using a simple relationship between their isotone transformations and the so called 2-, 3-, 4-endomorphisms which are introduced in the paper. Another characterization of totally ordered sets within the class of ordered sets of a locally finite height with at least four-element chains in terms of the regular semigroup theory is also given.
Classification : 06A05, 20M17, 20M20
Keywords: endomorphisms; totally ordered sets—chains; isotone mappings; regular semigroups 
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     title = {Characterizations of totally ordered sets by their various endomorphisms},
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Hort, Daniel; Chvalina, Jan; Moučka, Jiří. Characterizations of totally ordered sets by their various endomorphisms. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 1, pp. 23-32. http://geodesic.mathdoc.fr/item/CMJ_2002__52_1_a3/