# $\alpha $-ideals in $0$-distributive posets

Mathematica Bohemica (2015)

- Volume: 140, Issue: 3, page 319-328
- ISSN: 0862-7959

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topMokbel, Khalid A.. "$\alpha $-ideals in $0$-distributive posets." Mathematica Bohemica 140.3 (2015): 319-328. <http://eudml.org/doc/271604>.

@article{Mokbel2015,

abstract = {The concept of $\alpha $-ideals in posets is introduced. Several properties of $\alpha $-ideals in $0$-distributive posets are studied. Characterization of prime ideals to be $\alpha $-ideals in $0$-distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal $I$ of a $0$-distributive poset is non-dense, then $I$ is an $\alpha $-ideal. Moreover, it is shown that the set of all $\alpha $-ideals $\alpha \mathop \{\rm Id\}(P)$ of a poset $P$ with $0$ forms a complete lattice. A result analogous to separation theorem for finite $0$-distributive posets is obtained with respect to prime $\alpha $-ideals. Some counterexamples are also given.},

author = {Mokbel, Khalid A.},

journal = {Mathematica Bohemica},

keywords = {$0$-distributive poset; ideal; $\alpha $-ideal; prime ideal; non-dense ideal; minimal ideal; annihilator ideal},

language = {eng},

number = {3},

pages = {319-328},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {$\alpha $-ideals in $0$-distributive posets},

url = {http://eudml.org/doc/271604},

volume = {140},

year = {2015},

}

TY - JOUR

AU - Mokbel, Khalid A.

TI - $\alpha $-ideals in $0$-distributive posets

JO - Mathematica Bohemica

PY - 2015

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 140

IS - 3

SP - 319

EP - 328

AB - The concept of $\alpha $-ideals in posets is introduced. Several properties of $\alpha $-ideals in $0$-distributive posets are studied. Characterization of prime ideals to be $\alpha $-ideals in $0$-distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal $I$ of a $0$-distributive poset is non-dense, then $I$ is an $\alpha $-ideal. Moreover, it is shown that the set of all $\alpha $-ideals $\alpha \mathop {\rm Id}(P)$ of a poset $P$ with $0$ forms a complete lattice. A result analogous to separation theorem for finite $0$-distributive posets is obtained with respect to prime $\alpha $-ideals. Some counterexamples are also given.

LA - eng

KW - $0$-distributive poset; ideal; $\alpha $-ideal; prime ideal; non-dense ideal; minimal ideal; annihilator ideal

UR - http://eudml.org/doc/271604

ER -

## References

top- Balasubramani, P., Venkatanarasimhan, P. V., Characterizations of the $0$-distributive lattice, Indian J. Pure Appl. Math. 32 (2001), 315-324. (2001) Zbl0984.06007MR1826759
- Cornish, W. H., 10.1017/S1446788700012775, J. Aust. Math. Soc. 15 (1973), 70-77. (1973) Zbl0274.06008MR0344170DOI10.1017/S1446788700012775
- Grätzer, G., General Lattice Theory. New appendices by the author with B. A. Davey et al, Birkhäuser Basel (1998). (1998) MR1670580
- Grillet, P. A., Varlet, J. C., Complementedness conditions in lattices, Bull. Soc. R. Sci. Liège (electronic only) 36 (1967), 628-642. (1967) Zbl0157.34202MR0228389
- Halaš, R., Characterization of distributive sets by generalized annihilators, Arch. Math., Brno 30 (1994), 25-27. (1994) MR1282110
- Halaš, R., Rachůnek, J., Polars and prime ideals in ordered sets, Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-59. (1995) MR1369627
- Jayaram, C., Prime $\alpha $-ideals in an $0$-distributive lattice, Indian J. Pure Appl. Math. 17 (1986), 331-337. (1986) MR0835346
- Joshi, V. V., Mundlik, N., Prime ideals in $0$-distributive posets, Cent. Eur. J. Math. 11 (2013), 940-955. (2013) Zbl1288.06002MR3032342
- Joshi, V. V., Waphare, B. N., Characterizations of $0$-distributive posets, Math. Bohem. 130 (2005), 73-80. (2005) Zbl1112.06001MR2128360
- 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
- Pawar, Y. S., Khopade, S. S., $\alpha $-ideals and annihilator ideals in $0$-distributive lattices, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 49 (2010), 63-74. (2010) Zbl1245.06023MR2797524
- Pawar, Y. S., Mane, D. N., $\alpha $-ideals in $0$-distributive semilattices and $0$-distributive lattices, Indian J. Pure Appl. Math. 24 (1993), 435-443. (1993) Zbl0789.06005MR1234802

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