Geodesic
Going down in (semi)lattices of finite Moore families and convex geometries
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 249-271 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper we first study what changes occur in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then we study the set of all convex geometries which have the same poset of join-irreducible elements. We show that this set---ordered by set inclusion---is a ranked join-semilattice and we characterize its cover relation. We prove that the lattice of all ideals of a given poset $P$ is the only convex geometry having a poset of join-irreducible elements isomorphic to $P$ if and only if the width of $P$ is less than 3. Finally, we give an algorithm for computing all convex geometries having the same poset of join-irreducible elements.
In this paper we first study what changes occur in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then we study the set of all convex geometries which have the same poset of join-irreducible elements. We show that this set---ordered by set inclusion---is a ranked join-semilattice and we characterize its cover relation. We prove that the lattice of all ideals of a given poset $P$ is the only convex geometry having a poset of join-irreducible elements isomorphic to $P$ if and only if the width of $P$ is less than 3. Finally, we give an algorithm for computing all convex geometries having the same poset of join-irreducible elements.
Classification : 06A12
Keywords: closure system; Moore family; convex geometry; (semi)lattice; algorithm 
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     title = {Going down in (semi)lattices of finite {Moore} families and convex geometries},
     journal = {Czechoslovak Mathematical Journal},
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}
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Gabriela, Bordalo; Nathalie, Caspard; Bernard, Monjardet. Going down in (semi)lattices of finite Moore families and convex geometries. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 249-271. http://geodesic.mathdoc.fr/item/CMJ_2009_59_1_a17/

[1] Barbut, M., Monjardet, B.: Ordre et Classification, Algèbre et Combinatoire, tomes I--II. Hachette, Paris (1970). | MR

[2] Berman, J., Bordalo, G.: Finite distributive lattices and doubly irreducible elements. Disc. Math. 178 (1998), 237-243. | DOI | MR | Zbl

[3] Bordalo, G., Monjardet, B.: Reducible classes of finite lattices. Order 13 (1996), 379-390. | DOI | MR | Zbl

[4] Bordalo, G., Monjardet, B.: The lattice of strict completions of a finite poset. Alg. Univ. 47 (2002), 183-200. | DOI | MR | Zbl

[5] Bordalo, G., Monjardet, B.: Finite orders and their minimal strict completion lattices. Discuss. Math. Gen. Algebra Appl. 23 (2003), 85-100. | DOI | MR | Zbl

[6] Caspard, N.: A characterization theorem for the canonical basis of a closure operator. Order 16 (1999), 227-230. | DOI | MR | Zbl

[7] Caspard, N., Monjardet, B.: The lattice of closure systems, closure operators and implicational systems on a finite set: a survey. Disc. Appl. Math. 127 (2003), 241-269. | DOI | MR

[8] Caspard, N., Monjardet, B.: Some lattices of closure systems. Disc. Math. Theor. Comput. Sci. 6 (2004), 163-190. | MR | Zbl

[9] Chacron, J.: Nouvelles correspondances de Galois. Bull. Soc. Math. Belgique 23 (1971), 167-178. | MR | Zbl

[10] Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990). | MR | Zbl

[11] Dilworth, R. P.: Lattices with unique irreducible representations. Ann. of Math. 41 (1940), 771-777. | DOI | MR

[12] Edelman, P. H., Jamison, R. E.: The theory of convex geometries. Geom. Dedicata 19 (1985), 247-270. | DOI | MR | Zbl

[13] Erné, M.: Bigeneration in complete lattices and principal separation in ordered sets. Order 8 (1991), 197-221. | DOI | MR

[14] Lorrain, F.: Notes on topological spaces with minimum neighborhoods. Amer. Math. Monthly 76 (1969), 616-627. | DOI | MR | Zbl

[15] Monjardet, B.: The consequences of Dilworth's work on lattices with unique irreducible decompositions. Bogart, K. P., Freese, R., Kung, J. The Dilworth theorems. Selected papers of Robert P. Dilworth. Birkhaüser, Boston (1990), 192-201. | MR

[16] Monjardet, B., Raderanirina, V.: The duality between the anti-exchange closure operators and the path independent choice operators on a finite set. Math. Social Sci. 41 (2001), 131-150. | DOI | MR | Zbl

[17] Nation, J. B., Pogel, A.: The lattice of completions of an ordered set. Order 14 (1997), 1-7. | DOI | MR | Zbl

[18] Niederle, J.: Boolean and distributive ordered sets: characterization and representation by sets. Order 12 (1995), 189-210. | DOI | MR | Zbl

[19] Nourine, L.: Private communication. (2003).

[20] "Ore, O.: Some studies on closure relations. Duke Math. J. 10 (1943), 761-785. | DOI | MR

[21] Rabinovitch, I., Rival, I.: The rank of a distributive lattice. Disc. Math. 25 (1979), 275-279. | DOI | MR | Zbl

[22] Reading, N.: Order dimension, strong Bruhat order and lattice properties for posets. Order 19 (2002), 73-100. | DOI | MR | Zbl

[23] Schmid, J.: Quasiorders and sublattices of distributive lattices. Order 19 (2002), 11-34. | DOI | MR | Zbl

[24] Wild, M.: A theory of finite closure spaces based on implications. Adv. Math. 108 (1994), 118-139. | DOI | MR | Zbl