0-distributive posets

Khalid A. Mokbel; Vilas S. Kharat

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 3, page 325-335
  • ISSN: 0862-7959

Abstract

top
Several characterizations of 0-distributive posets are obtained by using the prime ideals as well as the semiprime ideals. It is also proved that if every proper l -filter of a poset is contained in a proper semiprime filter, then it is 0 -distributive. Further, the concept of a semiatom in 0-distributive posets is introduced and characterized in terms of dual atoms and also in terms of maximal annihilator. Moreover, semiatomic 0-distributive posets are defined and characterized. It is shown that a 0 -distributive poset P is semiatomic if and only if the intersection of all non dense prime ideals of P equals ( 0 ] . Some counterexamples are also given.

How to cite

top

Mokbel, Khalid A., and Kharat, Vilas S.. "0-distributive posets." Mathematica Bohemica 138.3 (2013): 325-335. <http://eudml.org/doc/260577>.

@article{Mokbel2013,
abstract = {Several characterizations of 0-distributive posets are obtained by using the prime ideals as well as the semiprime ideals. It is also proved that if every proper $l$-filter of a poset is contained in a proper semiprime filter, then it is $0$-distributive. Further, the concept of a semiatom in 0-distributive posets is introduced and characterized in terms of dual atoms and also in terms of maximal annihilator. Moreover, semiatomic 0-distributive posets are defined and characterized. It is shown that a $0$-distributive poset $P$ is semiatomic if and only if the intersection of all non dense prime ideals of $P$ equals $(0]$. Some counterexamples are also given.},
author = {Mokbel, Khalid A., Kharat, Vilas S.},
journal = {Mathematica Bohemica},
keywords = {0-distributive poset; ideal; semiprime ideal; prime ideal; semiatom; semiatomic 0-distributive poset; 0-distributive poset; prime ideal; semiprime ideal; semi-atom; semi-atomic poset},
language = {eng},
number = {3},
pages = {325-335},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {0-distributive posets},
url = {http://eudml.org/doc/260577},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Mokbel, Khalid A.
AU - Kharat, Vilas S.
TI - 0-distributive posets
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 3
SP - 325
EP - 335
AB - Several characterizations of 0-distributive posets are obtained by using the prime ideals as well as the semiprime ideals. It is also proved that if every proper $l$-filter of a poset is contained in a proper semiprime filter, then it is $0$-distributive. Further, the concept of a semiatom in 0-distributive posets is introduced and characterized in terms of dual atoms and also in terms of maximal annihilator. Moreover, semiatomic 0-distributive posets are defined and characterized. It is shown that a $0$-distributive poset $P$ is semiatomic if and only if the intersection of all non dense prime ideals of $P$ equals $(0]$. Some counterexamples are also given.
LA - eng
KW - 0-distributive poset; ideal; semiprime ideal; prime ideal; semiatom; semiatomic 0-distributive poset; 0-distributive poset; prime ideal; semiprime ideal; semi-atom; semi-atomic poset
UR - http://eudml.org/doc/260577
ER -

References

top
  1. Beran, L., Length of ideals in lattices, Collect. Math. 46 (1995), 21-33. (1995) Zbl0842.06006MR1366126
  2. Grätzer, G., General Lattice Theory. New appendices by the author with B. A. Davey, et al., (Second ed.), Birkhäuser, Basel (1998). (1998) MR1670580
  3. Grillet, P. A., Varlet, J. C., Complementedness conditions in lattices, Bull. Soc. R. Sci. Liège 36 (1967), 628-642. (1967) Zbl0157.34202MR0228389
  4. Halaš, R., Characterization of distributive sets by generalized annihilators, Arch. Math., Brno 30 (1994), 25-27. (1994) MR1282110
  5. Halaš, R., Rachůnek, J., Polars and prime ideals in ordered sets, Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-59. (1995) MR1369627
  6. Jayaram, C., Semiatoms in semilattices, Math. Semin. Notes, Kobe Univ. 10 (1982), 351-366. (1982) Zbl0522.06003MR0704918
  7. Joshi, V. V., Waphare, B. N., 10.21136/MB.2005.134222, Math. Bohem. 130 (2005), 73-80. (2005) Zbl1112.06001MR2128360DOI10.21136/MB.2005.134222
  8. Kharat, V. S., Mokbel, K. A., 10.21136/MB.2009.140636, Math. Bohem. 134 (2009), 19-30. (2009) Zbl1212.06001MR2504684DOI10.21136/MB.2009.140636
  9. Kharat, V. S., Mokbel, K. A., 10.1007/s11083-008-9087-3, Order 25 (2008), 195-210. (2008) Zbl1155.06003MR2448404DOI10.1007/s11083-008-9087-3
  10. Pawar, Y. S., 0-1 distributive lattices, Indian J. Pure Appl. Math. 24 (1993), 173-179. (1993) Zbl0765.06015MR1210389
  11. Pawar, Y. S., Dhamke, V. B., 0 -distributive posets, Indian J. Pure Appl. Math. 20 (1989), 804-811. (1989) Zbl0676.06009MR1012883
  12. Pawar, Y. S., Thakare, N. K., 10.4153/CMB-1978-080-6, Can. Math. Bull. 21 (1978), 469-475. (1978) MR0523589DOI10.4153/CMB-1978-080-6
  13. Rachůnek, J., On 0 -modular and 0 -distributive semilattices, Math. Slovaca 42 (1992), 3-13. (1992) MR1159487
  14. Varlet, J. C., A generalization of the notion of pseudo-complementedness, Bull. Soc. R. Sci. Liège 37 (1968), 149-158. (1968) Zbl0162.03501MR0228390
  15. Varlet, J. C., Distributive semilattices and Boolean lattices, Bull. Soc. R. Sci. Liège 41 (1972), 5-10. (1972) Zbl0237.06011MR0307991

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.