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Almost distributive lattices

Part of: Distributive lattices Boolean algebras (Boolean rings)

Published online by Cambridge University Press:  09 April 2009

U. Maddana Swamy  and
G. C. Rao
U. Maddana Swamy
Affiliation:
Department of Mathematics, Andhra University, Waltair-530 003, India
G. C. Rao
Affiliation:
Department of Mathematics, Andhra University, Waltair-530 003, India
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Abstract

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The concept of ‘Almost Distributive Lattices’ (ADL) is introduced. This class of ADLs includes almost all the existing ring theoretic generalisations of a Boolean ring (algebra) like regular rings, P-rings, biregular rings, associate rings, P1-rings, triple systems, etc. This class also includes the class of Baer-Stone semigroups. A one-to-one correspondence is exhibited between the class of relatively complemented ADLs and the class of Almost Boolean Rings analogous to the well-known Stone's correspondence. Many concepts in distributive lattices can be extended to the class of ADLs through its principal ideals which from a distributive lattice with 0. Subdirect and Sheaf representations of an ADL are obtained.

MSC classification

Secondary: 06D99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 1 , June 1981 , pp. 77 - 91
Copyright
Copyright © Australian Mathematical Society 1981

References

Arens, R. F. and Kaplansky, I. (1948), ‘Topological representation of algebras’, Trans. Amer. Math. Soc. 63, 457–481.CrossRefGoogle Scholar
Boole, G. (1854), ‘An investigation into the laws of thought’, London (Reprinted by Open Court Publishing Co., Chelsea, 1940).Google Scholar
McCoy, N. H. and Montgomery, (1937), ‘A representation of generalised Boolean rings’, Duke Math. J. 3, 455–459.CrossRefGoogle Scholar
Swamy, U. Maddana (1974), Representation of algebras by sections of sheaves (Doctoral Thesis, Department of Mathematics, Andhra University, Waltair, India).CrossRefGoogle Scholar
Swamy, U. Maddana (1980), ‘Baer-Stone semigroups’, Semigroup Forum 19, 385–386.CrossRefGoogle Scholar
Swamy, U. Maddana and Manikymba, P., ‘The equivalence of Baer rings and associate rings’, to appear.Google Scholar
Saracino, D. and Weispfield, V. (1975), Model theory and algebra (Lecture Notes in Mathematics 498, pp. 307–383, Springer-Verlag, Berlin).CrossRefGoogle Scholar
Subrahmanyam, N. V. (1960a), ‘Lattice theory for certain classes of rings’, Math. Ann. 139, 275–286.CrossRefGoogle Scholar
Subrahmanyam, N. V. (1960a), ‘Structure theory for generalised Boolean rings’, Math. Ann. 141, 297–310.Google Scholar
Subrahmanyam, N. V. (1963), ‘An extension of Boolean lattice theory’, Math. Ann. 151, 332–345.Google Scholar
Sussman, I. (1958), ‘A generalisation for Boolean rings’, Math. Ann. 136, 326–338.Google Scholar
Von Neumann, J. (1936), ‘On regular rings’, Proc. Nat. Acad. Sci., USA 22, 707–713.CrossRefGoogle ScholarPubMed