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partially ordered set; interval
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.
[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 0724279
[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 0795195
[6] Igošin, V. I.: Semimodularity in lattices of intervals. Math. Slovaca 38 (1988), 305–308. MR 0978760
[7] Jakubík, J.: Selfduality of the system of intervals of a partially ordered set. Czechoslov. Math. J. 41 (1991), 135–140. MR 1087633
[8] Jakubík, J., Lihová, J.: Systems of intervals of partially ordered sets. Math. Slovaca 46 (1996 No. 4), 355–361. MR 1472629
[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 0738826 | Zbl 0506.06003
[10] Lihová, J.: Posets having a selfdual interval poset. Czechoslov. Math. J. 44 (1994), 523–533. MR 1288170
[11] Lihová, J.: On posets with isomorphic interval posets. Czechoslov. Math. J. 49 (1999), 67–80. MR 1676841
[12] Slavík, V.: On lattices with isomorphic interval lattices. Czechoslov. Math. J. 35 (1985), 550–554. MR 0809041
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