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Duffus, Dwight; Rival, Ivan

A structure theory for ordered sets. (English) Zbl 0459.06002

Discrete Math. 35, 53-118 (1981).
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Cited in 3 Reviews
Cited in 62 Documents

MSC:

06A06 Partial orders, general
06B05 Structure theory of lattices

Keywords:

irreducible ordered sets; retracts; direct products; order varieties; prime intervals
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\textit{D. Duffus} and \textit{I. Rival}, Discrete Math. 35, 53--118 (1981; Zbl 0459.06002)
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References:

[1] Alvarez, L.R., Undirected graphs as graphs of modular lattices, Canad. J. math., 17, 923-932, (1965) · Zbl 0173.51502
[2] Birkhoff, G., On the structure of abstract algebras, Proc. camb. phil. soc., 31, 441-464, (1935)
[3] Birkhoff, G., Sobre los grupos de automorfismos, Revista de la union matemática Argentina, 11, 155-157, (1946)
[4] Bonnet, R.; Pouzet, M., Extension et stratification d’ensembles dispersés, C.R. acad. sci. Paris, 268, A1512-A1515, (1969) · Zbl 0188.04203
[5] Burris, S., Subdirect representation in axiomatic classes, Colloq. math., 34, 191-197, (1976) · Zbl 0326.08005
[6] Crawley, P.; Dilworth, R.P., Algebraic theory of lattices, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0494.06001
[7] Chien, M.; Martin, P., Sur le nombre de sants d’une forêt, C.R. acad. sci. Paris, 275, A159-A161, (1972)
[8] Davis, A.C., A characterization of complete lattices, Pacific J. math., 5, 311-319, (1955) · Zbl 0064.26101
[9] Dilworth, R.P., A decomposition theorem for partially ordered sets, Ann. of math., 51, 161-166, (1950) · Zbl 0038.02003
[10] Duffus, D.; Poguntke, W.; Rival, I., Retracts and the fixed point problem for finite partially ordered sets, Canad. math. bull., 23, 231-236, (1980) · Zbl 0441.06003
[11] Duffus, D.; Rival, I., Path length in the covering graph of a lattice, Discrete math., 19, 139-158, (1977) · Zbl 0372.06005
[12] Duffus, D.; Rival, I., A logarithmic property for exponents of partially ordered sets, Canad. J. math., 30, 797-807, (1978) · Zbl 0497.06004
[13] Duffus, D.; Rival, I., Retracts of partially ordered sets, J. austral. math. soc., 27, 495-506, (1979), (Ser. A) · Zbl 0421.06002
[14] Duffus, D.; Rival, I., Graphs orientable as distributive lattices, Preprint, (1980) · Zbl 0513.06004
[15] Duffus, D.; Rival, I.; Simonovits, M., Spanning retracts of a partially ordered set, Discrete math., 32, 1-7, (1980) · Zbl 0453.06003
[16] Dushnik, B.; Miller, E.W., Partially ordered sets, Amer. J. math, 63, 600-610, (1941) · Zbl 0025.31002
[17] Fawcett, B., Graphs and ultrapowers, () · Zbl 1342.05029
[18] Fournier, R., Irreducible subdirect representations of graphs, Proc. fifth british comb. conf., 203-210, (1975)
[19] Frucht, R., On the construction of partially ordered systems with a given group of automorphisms, Amer. J. math., 72, 195-199, (1950) · Zbl 0035.01501
[20] Ghouilà-Houri, A., Characterization of nonoriented graphs whose edges can be oriented in such a way as to obtain the graph of an order relation, C.R. acad. sci. Paris, 254, 1370-1371, (1962) · Zbl 0105.35503
[21] Gilmore, P.C.; Hoffman, A.J., A characterization of comparability graphs and of interval graphs, Canad. J. math., 16, 539-548, (1964) · Zbl 0121.26003
[22] R.L. Graham, A.C. Yao, and F.F. Yao, Some monotonicity properties of partial orders, SIAM J. Algebraic Discrete Methods. · Zbl 0496.68043
[23] Greene, C.; Kleitman, D.J., The structure of sperner k-families, J. combinatorial theory (A), 20, 41-68, (1976) · Zbl 0361.05016
[24] Grötzch, H., Ein dreifarbensatz für dreikreisfreie netze auf der kugel, Wiss. Z. martin-luther univ. halle-Wittenberg. math. naturwiss. reihe, 8, 109-120, (1958)
[25] Hashimoto, J., On the product decomposition of partially ordered sets, Math. japonicae, 1, 120-123, (1948) · Zbl 0041.37801
[26] Hell, P.; Hell, P., Subdirect products of bipartite graphs, (), 852-866 · Zbl 0308.05125
[27] Huntington, E.V., Sets of independent postulates for the algebra of logic, Trans. amer. math. soc., 5, 288-309, (1904) · JFM 35.0087.02
[28] Jakubik, J., On lattices whose graphs are isomorphic, Czechoslovak math. J., 4, 131-140, (1954), (Russian) · Zbl 0059.02602
[29] Jakubik, J., Unoriented graphs of modular lattices, Czechoslovak math. J., 25, 240-246, (1975) · Zbl 0314.06006
[30] Kelly, D., The 3-irreducible partially ordered sets, Canad. J. math, 29, 367-383, (1977) · Zbl 0357.06004
[31] Kelly, D.; Rival, I., Crowns, fences, and dismantlable lattices, Canad. J. math., 26, 1257-1271, (1974) · Zbl 0271.06003
[32] Kelly, D.; Rival, I., Certain partially ordered sets of dimension three, J. combinatorial theory (A), 18, 239-242, (1975) · Zbl 0308.06003
[33] Kelly, D.; Rival, I., Planar lattices, Canad. J. math., 27, 636-665, (1975) · Zbl 0312.06003
[34] H.A. Kierstead, An effective version of Dilworth’s theorem, Trans. Amer. Math. Soc. (to appear). · Zbl 0485.03019
[35] Knaster, B., Un théorème sur LES fonctions d’ensembles, Ann. soc. polon. math, 6, 133-134, (1928) · JFM 54.0091.04
[36] Knuth, D., The art of computer programming, Vol. 3, (1973), Addison-Wesley Reading, MA
[37] Lawler, E.G., Sequencing jobs to minimize total weighted completion time subject to precedence constraints, Ann. discrete math., 2, 75-90, (1978) · Zbl 0374.68033
[38] MacNeille, H.M., Partially ordered sets, Trans. amer. math. soc, 42, 416-460, (1937) · Zbl 0017.33904
[39] Mosesjan, K.M., Strongly basable graphs, Akad. nauk armjan. SSR dokl., 54, 134-138, (1972), (Russian)
[40] Mosesjan, K.M., Certain theorems on strongly basable graphs, Akad. nauk armjan. SSR dokl., 54, 241-245, (1972), (Russian)
[41] Mosesjan, K.M., Basable and strongly basable graphs, Akad. nauk armjan. SSR dokl., 55, 83-86, (1972), (Russian)
[42] Nowakowski, R.; Rival, I., Fixed-edge theorem for graphs with loops, J. graph theory, 3, 339-350, (1979) · Zbl 0432.05030
[43] Ore, O., Theory of graphs, (1967), Amer. Math. Soc Providence, RI
[44] Pouzet, M., Private communication, (1980)
[45] Rival, I., Lattices with doubly irreducible elements, Canad. math. bull., 17, 91-95, (1974) · Zbl 0293.06003
[46] Rival, I., A fixed point theorem for finite partially ordered sets, J. combinatorial theory (A), 21, 309-318, (1976) · Zbl 0357.06003
[47] Rival, I.; Willie, R., The smallest order variety containing all chains, Discrete math., 35, (1981), in this volume. · Zbl 0467.06002
[48] Sabidussi, G.; Sabidussi, G., Subdirect representations of graphs, (), 1199-1226 · Zbl 0308.05124
[49] L.A. Schepp, The FKG inequality and some monotonicity properties of partial orders, SIAM J. Algebraic Discrete Methods.
[50] Szpilrajn, E., Sur l’extension de l’ordre partiel, Fund. math., 16, 386-389, (1930) · JFM 56.0843.02
[51] Tarski, A., A lattice-theoretical fixpoint theorem and its applications, Pacific J. math., 5, 285-309, (1955) · Zbl 0064.26004
[52] Trotter, W.T.; Moore, J.I., Characterization problems for graphs, partially ordered sets, lattices, and families of sets, Discrete math., 16, 361-381, (1976) · Zbl 0356.06007
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