Article in volume
Authors:
M. Sambasiva Rao
Department of Mathematics
MVGR College of Engineering, Chintalavalasa
Vizianagaram, Andhra Pradesh, India-535005
email: mssraomaths35@rediffmail.com
Title:
$\sigma$-filters of distributive lattices
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 44(2) (2024) 261-276
Accepted: 2023-02-21 , Available online: 2024-09-10 , https://doi.org/10.7151/dmgaa.1436
Abstract:
The concept of $\sigma $-filters is introduced in distributive lattices and
studied some properties of these classes of filters. Two sets of equivalent
conditions are derived one for every $\mu $-filter to become a $\sigma $-filter
and the other for every filter to become a $\sigma $-filter of a distributive
lattice. A one-to-one correspondence is established between the set of all prime
$\sigma $-filters of a distributive lattice and the set of all prime
$\sigma $-filters of its quotient lattice with respect to a congruence.
Keywords:
prime filter, co-annihilator, $\mu$-filter, O-filter, $\sigma$-filter, $pm$-lattice
References:
- G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. XXV (Providence, 1967), USA.
- S. Burris and H.P. Sankappanavar, A Cource in Universal Algebra (Springer Verlag, 1981).
- W.H. Cornish, Normal lattices, J. Austral. Math. Soc. 14 (1972) 200–215.
https://doi.org/10.1017/S1446788700010041 - W.H. Cornish, Annulets and $\alpha $-ideals in distributive lattices, J. Austral. Math. Soc. 15 (1973) 70–77.
https://doi.org/10.1017/S1446788700012775 - Y.Y. Dong and X.L. Xin, $\alpha $-filters and prime $\alpha $-filter spaces in residuated lattices, Soft Comput. 23 (10) (2018) 3207–3216.
https://doi.org/10.1007/s00500-018-3195-9 - O. Frink, Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962) 505–514.
https://doi.org/10.1215/S0012-7094-62-02951-4 - Y.S. Pawar and N.K. Thakare, $pm$-lattices, Algebra Univ. 7 (1977) 259–263.
https://doi.org/10.1007/BF02485435 - S. Rasouli, Generalized co-annihilators in residuated lattices An. Univ. Craiova Ser. Mat. Inform. 45 (2)(2018), 190–207.
https://doi.org/10.52846/ami.v45i2.827 - M. Sambasiva Rao, Normal filters of distributive lattices, Bull. Section of Logic 41 (3) (2012) 131–143.
- M. Sambasiva Rao, $O$-filters and minimal prime filters of lattices, Southeast Asian Bull. Math. 39 (2015) 113–121.
- M. Sambasiva Rao, $\mu $-filters of distributive lattices, Southeast Asian Bull. Math. 40 (2016) 251–264.
- H.P. Sankappanavar, Heyting algebras with dual pseudo-complementation, Pacific J. Math. 117 (2) (1985) 405–415.
https://doi.org/10.2140/pjm.1985.117.405 - T.P. Speed, Some remarks on a class of distributive lattices, J. Aust. Math. Soc. 9 (1969) 289–296.
https://doi.org/10.1017/S1446788700007205 - M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386.
https://doi.org/10.1215/S0012-7094-70-03748-8
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