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Keywords:
convex ordered sets; convex isomorphism
convex ordered sets; convex isomorphism
Summary:
V. I. Marmazejev introduced in [5] the following concept: two lattices are convex isomorphic if their lattices of all convex sublattices are isomorphic. He also gave a necessary and sufficient condition under which lattices are convex isomorphic, in particular for modular, distributive and complemented lattices. The aim of this paper is to generalize this concept to ordered sets and to characterize convex isomorphic ordered sets in the general case of modular, distributive or complemented ordered sets. These concepts were defined in [1], [2], [4].
References:
[1] Chajda I.: Complemented ordered sets. Arch. Math. (Brno), to appear. MR 1201863 | Zbl 0983.06002
[2] Chajda I., Rachůnek J.: Forbided configuration for distributive and modular ordered sets. Order 5 (1989), 407-423. DOI 10.1007/BF00353659 | MR 1010389
[3] Igosin V. I.: Lattices of intervals and lattices of convex sublattices of lattices. (Russian), Mežvuzovskij naučnyj sbornik 6-Uporjadočennyje množestva i rešetky, Saratov (1980), 69-76.
[4] Larmerová J., Rachůnek J.: Translations of Distributive and modular ordered sets. Acta UPO, Fac. rer. nat., 91 (Mathematica XXVII, 1988), 13-23. MR 1039879 | Zbl 0693.06003
[5] Marmazejev V. I.: The lattice of convex sublattices of a lattice. Mežvuzovskij naučnyj sbornik 6-Uporjadočennyje množestva i rešetky, Saratov (1986), 50-58. (In Russian.) MR 0957970
[6] Szász G.: Théorie des trellis. Akadémiai Kaidó, Budapest, 1971.
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