Article in volume
Authors:
Mukkamala Sambasiva Rao
Department of Mathematics
MVGR College of Engineering, Vizianagaram
Andhra Pradesh, India-535005
email: mssraomaths35@rediffmail.com
Sagiraju Venkata Siva Rama Raju
Academic Support Department
Abu Dhabi Polytechnic
Abu Dhabi, UAE
email: shivram2006@yahoo.co.in
Title:
Coherent lattices
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 44(2) (2024) 287-299
Received: 2023-02-15 , Revised: 2023-04-16 , Accepted: 2023-04-18 , Available online: 2024-09-10 , https://doi.org/10.7151/dmgaa.1463
Abstract:
The notion of coherent lattices is introduced and established relations
between a coherent lattice and that of a generalized Stone lattice, Boolean
algebra, quasi-complemented lattice, and normal lattice. A set of equivalent
conditions is given for every sublattice of a lattice to become a coherent
lattice. Some equivalent conditions are given for every interval of a lattice to
become a coherent sublattice. Coherent lattices are characterized with the
help of certain properties of lters and dense elements.
Keywords:
Coherent lattice, generalized Stone lattice, Boolean algebra
References:
- G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. XXV (Providence, USA, 1976).
- S. Burris and H.P. Sankappanavar, A Course in Univerasal Algebra (Springer Verlag, 1981).
- I. Chajda, R. Halaš and J. Kühr, Semilattice structures (Heldermann Verlag, Germany, 2007).
- W.H. Cornish, Normal lattices, J. Austral. Math. Soc. 14 (1973) 167–179.
https://doi.org/10.1017/S1446788700010041 - W.H. Cornish, Congruences on distributive pseudo-complemented lattices, Bull. Austral. Math. Soc. 8 (1973) 161–179.
https://doi.org/10.1017/S0004972700042404 - W.H. Cornish, Annulets and $\alpha$-ideals in distributive lattices, J. Austral. Math. Soc. 15 (1973) 70–77.
https://doi.org/10.1017/S1446788700012775 - W.H. Cornish, Quasi-complemented lattices, Comm. Math. Univ. Carolinae 15(3) (1974) 501–511.
- O. Frink, Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962) 505–514.
https://doi.org/10.1215/S0012-7094-62-02951-4 - G. Gratzer, General lattice theory (Academic Press, New York, San Francisco, USA, 1978).
- J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc., Sec. B 13 (1963) 31–50.
https://doi.org/10.1112/plms/s3-13.1.31 - M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377-386.
https://doi.org/10.1215/S0012-7094-70-03748-8 - A.P. Paneendra Kumar, M. Sambasiva Rao and K. Sobhan Babu, Filters of distributive lattices generated by dense elements, Palestine J. Math. 11(2) (2022) 45–54.
- A.P. Paneendra Kumar, M. Sambasiva Rao and K. Sobhan Babu, Generalized prime $D$-filters of distributive lattices, Archivum Mathematicum 57(3) (2021) 157–174.
https://doi.org/10.5817/AM2021-3-157 - M. Sambasiva Rao, A note on $\sigma $-ideals of distributive lattices, Alg. Struct. and their Appl. 9(2) (2022) 163–179.
https://doi.org/10.22034/AS.2022.2720 - T.P. Speed, Some remarks on a class of distributive lattices, Jour. Aust. Math. Soc. 9 (1969) 289–296.
https://doi.org/10.1017/S1446788700007205 - M.H. Stone, A theory of representations for Boolean algebras, Tran. Amer. Math. Soc. 40 (1936) 37–111.
https://doi.org/10.2307/1989664
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