Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-21T13:41:20.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime ideal characterization of chain based lattices

Part of: Distributive lattices Boolean algebras (Boolean rings) Categories and geometry

Published online by Cambridge University Press:  09 April 2009

U. Maddna Swamy ,
G. C. Rao  and
P. Manikyamba
U. Maddna Swamy
Affiliation:
Department of Mathematics, Andhra University,, Waltair-530003, India
G. C. Rao
Affiliation:
Department of Mathematics, Andhra University,, Waltair-530003, India
P. Manikyamba
Affiliation:
Department of Mathematics, Andhra University,, Waltair-530003, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Epstein and Horn, in their paper ‘Chain based lattices’, characterized P1-lattices, and P2-lattices in terms of their prime ideals. But no such prime ideal characterization for P0-lattices was given. Our main aim in this paper is to characterize P0-lattices in terms of their prime ideals. We also give a necessary and sufficient condition for a P-algebra to be a P0-lattice (and hence a P2-lattice).

Keywords

B-normal latticesP-algebrachain based latticessheaf of bounded distributive latticescontinuity axiomP(L) is determined by a subset of L.

MSC classification

Secondary: 06D99: None of the above, but in this section 18F20: Presheaves and sheaves
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 1 , September 1980 , pp. 98 - 106
Copyright
Copyright © Australian Mathematical Society 1980

References

Birkhoff, G. (1967), Lattice theory (3rd ed., A.M.S. Colloquium Publications, Amer. Math. Soc., Proidence, RI.).Google Scholar
Cignoli, R. (1971), ‘Stone filters and ideals in distributive lattices’, Bull. Math. Soc. Sci. Math. R.S. Roumanie 15, 131–137.Google Scholar
Cignoli, R. (1978), ‘The lattice of global sections of sheaves of chains’, Algebra Universalis 8, 357–373.CrossRefGoogle Scholar
Cornish, W. H. (1972), ‘Normal lattices’, J. Austral. Math. Soc. 14, 200–215.CrossRefGoogle Scholar
Epstein, G. (1960), ‘Lattice theory of Post algebras’, Trans. Amer. Math. Soc. 95, 300–317.CrossRefGoogle Scholar
Epstein, G. and Horn, A. (1974), ‘P-algebras, an abstraction from Post algebras’, Algebra Universalis 4, 195–206.CrossRefGoogle Scholar
Epstein, G. and Horn, A. (1975), ‘Chain based lattices’, Pacific J. Math. 55, 65–84.CrossRefGoogle Scholar
Hoffman, K. H. (1972), ‘Representation of algebras by continuous sections’, Bull. Amer. Math. Soc. 73, 291–373.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 and Manikyamba, P. (1979), ‘Prime ideal characterization of Stone lattices’, Math. Sem. Notes, Kobe Univ. 7, 25–31.Google Scholar
Swamy, U. Maddana and Manikyamba, P. ‘Representation of certain classes of distributive lattices by sections of sheaves’, Internat. J. Math. and Mathematical Phys, to appear.Google Scholar
Subrahmanyam, N. V. (1978), Lectures on sheaf theory (Lecture Notes, Department of Mathematics, Andhra University, Waltair, India).Google Scholar
Traczyk, T. (1963), ‘Axioms and some properties of Post algebras’ Colloq. Math. 10, 193–209.CrossRefGoogle Scholar