Ideals, congruences and annihilators on nearlattices

Ivan Chajda; Miroslav Kolařík

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2007)

  • Volume: 46, Issue: 1, page 25-33
  • ISSN: 0231-9721

Abstract

top
By a nearlattice is meant a join-semilattice having the property that every principal filter is a lattice with respect to the semilattice order. We introduce the concept of (relative) annihilator of a nearlattice and characterize some properties like distributivity, modularity or 0 -distributivity of nearlattices by means of certain properties of annihilators.

How to cite

top

Chajda, Ivan, and Kolařík, Miroslav. "Ideals, congruences and annihilators on nearlattices." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 46.1 (2007): 25-33. <http://eudml.org/doc/32460>.

@article{Chajda2007,
abstract = {By a nearlattice is meant a join-semilattice having the property that every principal filter is a lattice with respect to the semilattice order. We introduce the concept of (relative) annihilator of a nearlattice and characterize some properties like distributivity, modularity or $0$-distributivity of nearlattices by means of certain properties of annihilators.},
author = {Chajda, Ivan, Kolařík, Miroslav},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {nearlattice; semilattice; ideal; congruence; distributivity; modularity; $0$-distributivity; annihilator; nearlattice; semilattice; ideal; congruence; distributivity; modularity; 0-distributivity; annihilator},
language = {eng},
number = {1},
pages = {25-33},
publisher = {Palacký University Olomouc},
title = {Ideals, congruences and annihilators on nearlattices},
url = {http://eudml.org/doc/32460},
volume = {46},
year = {2007},
}

TY - JOUR
AU - Chajda, Ivan
AU - Kolařík, Miroslav
TI - Ideals, congruences and annihilators on nearlattices
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2007
PB - Palacký University Olomouc
VL - 46
IS - 1
SP - 25
EP - 33
AB - By a nearlattice is meant a join-semilattice having the property that every principal filter is a lattice with respect to the semilattice order. We introduce the concept of (relative) annihilator of a nearlattice and characterize some properties like distributivity, modularity or $0$-distributivity of nearlattices by means of certain properties of annihilators.
LA - eng
KW - nearlattice; semilattice; ideal; congruence; distributivity; modularity; $0$-distributivity; annihilator; nearlattice; semilattice; ideal; congruence; distributivity; modularity; 0-distributivity; annihilator
UR - http://eudml.org/doc/32460
ER -

References

top
  1. Abbott J. C., Semi-boolean algebra, Mat. Vestnik 4 (1967), 177–198. (1967) Zbl0153.02704MR0239957
  2. Beazer R., Hierarchies of distributive lattices satisfying annihilator conditions, J. London Math. Soc. 11 (1975), 216–222. (1975) Zbl0335.06008MR0387141
  3. Chajda I., Kolařík M., A decomposition of homomorphic images of nearlattices, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 45 (2006), 43–52. Zbl1123.06002MR2321296
  4. Chajda I., Kolařík M., Nearlattices, Discrete Math., to appear. Zbl1151.06004MR2446101
  5. Cornish W. H., The free implicative BCK-extension of a distributive nearlattice, Math. Japonica 27, 3 (1982), 279–286. (1982) Zbl0496.03046MR0663155
  6. Cornish W. H., Noor A. S. A., Standard elements in a nearlattice, Bull. Austral. Math. Soc. 26, 2 (1982), 185–213. (1982) Zbl0523.06006MR0683652
  7. Davey B., Some annihilator conditions on distributive lattices, Algebra Universalis 4 (1974), 316–322. (1974) Zbl0299.06007MR0357261
  8. Davey B., Nieminen J., Annihilators in modular lattices, Algebra Universalis 22 (1986), 154–158. (1986) Zbl0613.06004MR0870463
  9. Grätzer G.: General Lattice Theory., Birkhäuser Verlag, , Basel, 1978. (1978) MR0504338
  10. Halaš R., Subdirectly irreducible distributive nearlattices, Math. Notes 7, 2 (2006), 141–146. Zbl1120.06003MR2310273
  11. Hickman R., Join algebras, Communications in Algebra 8 (1980), 1653–1685. (1980) Zbl0436.06003MR0585925
  12. Mandelker M., Relative annihilators in lattices, Duke Math. J. 40 (1970), 377–386. (1970) Zbl0206.29701MR0256951
  13. Nieminen J., The Jordan–Hölder chain condition and annihilators in finite lattices, Tsukuba J. Math. 14 (1990), 405–411. (1990) Zbl0721.06008MR1085207
  14. Noor A. S. A., Cornish W. H., Multipliers on a nearlattices, Commentationes Mathematicae Universitatis Carolinae (1986), 815–827. (1986) MR0874675
  15. Scholander M., Trees, lattices, order and betweenness, Proc. Amer. Math. Soc. 3 (1952), 369–381. (1952) MR0048405
  16. Scholander M., Medians and betweenness, Proc. Amer. Math. Soc. 5 (1954), 801–807. (1954) MR0064749
  17. Scholander M., Medians, lattices and trees, Proc. Amer. Math. Soc. 5 (1954), 808–812. (1954) MR0064750

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.