Geodesic
Characterization of posets of intervals
Archivum mathematicum, Tome 36 (2000) no. 3, pp. 171-181 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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If $A$ is a class of partially ordered sets, let $P(A)$ denote the system of all posets which are isomorphic to the system of all intervals of $A$ for some $A\in A.$ We give an algebraic characterization of elements of $P(A)$ for $A$ being the class of all bounded posets and the class of all posets $A$ satisfying the condition that for each $a\in A$ there exist a minimal element $u$ and a maximal element $v$ with $u\le a\le v,$ respectively.
If $A$ is a class of partially ordered sets, let $P(A)$ denote the system of all posets which are isomorphic to the system of all intervals of $A$ for some $A\in A.$ We give an algebraic characterization of elements of $P(A)$ for $A$ being the class of all bounded posets and the class of all posets $A$ satisfying the condition that for each $a\in A$ there exist a minimal element $u$ and a maximal element $v$ with $u\le a\le v,$ respectively.
Classification : 06A06
Keywords: partially ordered set; interval 
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2000_36_3_a1/}
}
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Lihová, Judita. Characterization of posets of intervals. Archivum mathematicum, Tome 36 (2000) no. 3, pp. 171-181. http://geodesic.mathdoc.fr/item/ARM_2000_36_3_a1/

[1] Igošin, V. I.: Selfduality of lattices of intervals of finite lattices. Inst. matem. Sibir. Otdel. AN SSSR, Meždunarodnaja konferencija po algebre posvjaščennaja pamjati A. I. Maĺceva, Tezisyy dokladov po teoriji modelej i algebraičeskich sistem, Novosibirsk 1989, s. 48.

[2] Igošin, V. I.: Lattices of intervals and lattices of convex sublattices of lattices. Uporjadočennyje množestva i rešotki. Saratov 6 (1990), 69–76.

[3] Igošin, V. I.: Identities in interval lattices of lattices. Coll. Math. Soc. J. Bolyai 33 (Contributions to Lattice Theory), Szeged 1980 (1983), 491–501. | MR

[4] Igošin, V. I.: On lattices with restriction on their intervals. Coll. Math. Soc. J. Bolyai 43 (Lectures in Universal Algebra), Szeged 1983 (1986), 209–216.

[5] Igošin, V. I.: Algebraic characteristic of lattices of intervals. Uspechi matem. nauk 40 (1985), 205–206. | MR

[6] Igošin, V. I.: Semimodularity in lattices of intervals. Math. Slovaca 38 (1988), 305–308. | MR

[7] Jakubík, J.: Selfduality of the system of intervals of a partially ordered set. Czechoslov. Math. J. 41 (1991), 135–140. | MR

[8] Jakubík, J., Lihová, J.: Systems of intervals of partially ordered sets. Math. Slovaca 46 (1996 No. 4), 355–361. | MR

[9] Kolibiar, M.: Intervals, convex sublattices and subdirect representations of lattices. Universal Algebra and Applications, Banach Center Publications, Vol. 9, Warsaw 1982, 335–339. | MR | Zbl

[10] Lihová, J.: Posets having a selfdual interval poset. Czechoslov. Math. J. 44 (1994), 523–533. | MR

[11] Lihová, J.: On posets with isomorphic interval posets. Czechoslov. Math. J. 49 (1999), 67–80. | MR

[12] Slavík, V.: On lattices with isomorphic interval lattices. Czechoslov. Math. J. 35 (1985), 550–554. | MR