×

zbMATH — the first resource for mathematics

On uniquely complemented posets. (English) Zbl 1091.06002
The authors extend some classical results of uniquely complemented lattices to uniquely complemented posets with \(0\) and \(1\). They prove that a uniquely complemented join-semilattice having the general disjointness property is Boolean. They extend the Birkhoff-von Neumann theorem to larger classes of lattices, such as the class of modular lattices with the general disjointness property.

MSC:
06A06 Partial orders, general
06A12 Semilattices
06C05 Modular lattices, Desarguesian lattices
06C15 Complemented lattices, orthocomplemented lattices and posets
06C20 Complemented modular lattices, continuous geometries
06D15 Pseudocomplemented lattices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams M. E., Sichler J.: Lattices with unique complementation, Pac. J. Math. 92 (1981), 1–13 · Zbl 0468.06005
[2] Chajda I.: Complemented ordered sets, Arch. Math. (Brno) 28 (1992), 25–34 · Zbl 0785.06002
[3] Chajda I., Halaš R.: Characterizing triplets for modular pseudocomplemented ordered sets, Math. Slovaca 50(No. 5) (2000), 513–524 · Zbl 0986.06001
[4] Chen C. C.: On uniquely complemented lattices, J. Nanyang Univ. 3 (1969), 380–384
[5] Chen C. C., Grätzer G.: On the construction of complemented lattices, J. Algebra 11 (1969), 56–63 · Zbl 0185.03701 · doi:10.1016/0021-8693(69)90101-X
[6] Dilworth R. P.: Lattices with unique complements, Trans. Am. Math. Soc. 57 (1945), 123–154 · Zbl 0060.06103 · doi:10.1090/S0002-9947-1945-0012263-6
[7] Frink O.: Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505–514. · Zbl 0114.01602 · doi:10.1215/S0012-7094-62-02951-4
[8] Grätzer G.: General Lattice Theory, Birkhäuser, New York, 1998.
[9] Grillet P. A., Varlet J. C.: Complementedness conditions in lattices, Bull. Soc. r. Sci. Liège 36 (1967), 628–642. · Zbl 0157.34202
[10] Halaš R.: Pseudocomplemented ordered sets, Arch. Math. (Brno) 29 (1993), 153–160. · Zbl 0801.06007
[11] Halaš R.: Some properties of Boolean ordered sets, Czech. Math. J. (Praha), 46(121) (1996), 93–98. · Zbl 0904.06002
[12] Janowitz M. F.: Section semicomplemented lattices, Math. Z. 63 (1968), 63–76. · Zbl 0167.01902 · doi:10.1007/BF01110457
[13] Jayaram C.: Complemented semilattices, Math. Sem. Notes 8 (1980), 259–267. · Zbl 0453.06005
[14] V. V. Joshi and Waphare B. N.: Characterizations of 0-distributive posets, Math. Bohem. 130(1) (2005), 73–80. · Zbl 1112.06001
[15] Larmerová J. and Rachunek J.: Translations of distributive and modular ordered sets, Acta. Univ. Palacki. Olomonc., 91 (1988), 13–23 · Zbl 0693.06003
[16] McLaughlin, J. E.: Atomic lattices with unique comparable complements, Proc. Am. Math. Soc. 7 (1956), 864–866. · Zbl 0071.25702 · doi:10.1090/S0002-9939-1956-0081262-3
[17] Niederle J.: Boolean and distributive ordered sets: Characterization and representation by sets, Order 12 (1995), 189–210. · Zbl 0838.06004 · doi:10.1007/BF01108627
[18] Pawar M. M. and Waphare B. N.: On Stone posets and strongly pseudocomplemented posets, J. Indian Math. Soc. (N.S.) 68(1–4) (2001), 91–95. · Zbl 1141.06300
[19] Saarimäki M.: Disjointness of lattice elements, Math. Nachr. 159 (1992), 169–174 · Zbl 0773.06014 · doi:10.1002/mana.19921590112
[20] Salii V.N.: Lattices with unique complements, Transl. Math. Monogr., Amer. Math. Soc. Providence, RI 69 (1988).
[21] Thakare N. K., Maeda S. and Waphare B.N.: Modular pairs and covering property in posets, J. Indian Math. Soc. (N.S.) (in press). · Zbl 1117.06300
[22] Thakare N.K., Pawar M.M. and Waphare B.N.: Modular pairs, standard elements, neutral elements and related results in posets, J. Indian Math. Soc. (N.S.)(to appear). · Zbl 1117.06301
[23] Varlet, J.: Contributions à l’étude des treillis pseudo-complémentés et des treillis de Stone, Mém. Soc. R. Sci. Liege, Collect 8 (1963), 1–71. · Zbl 0113.01803
[24] Venkatanarasimhan P.V.: Pseudo-complements in posets, Proc. Am. Math. Soc. 28(No.1) (1971), 9–17. · Zbl 0218.06002 · doi:10.1090/S0002-9939-1971-0272687-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.