Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-03T11:50:27.369Z Has data issue: false hasContentIssue false
  • English
  • Français

Semi-Brouwerian algebras

Published online by Cambridge University Press:  09 April 2009

P. V. Ramana Murty
P. V. Ramana Murty
Affiliation:
Department of Mathematics, College of Arts, Andhra University Waltair, A.P., India
Rights & Permissions [Opens in a new window]

Extract

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.

Ever since David Ellis has shown that a Boolean algebra has a natural structure of an autometrized space, the interest in such spaces has led several authors to study various autometrized algebras like Brouwerian algebras [9], Newman algebras [4], Lattice ordered groups [6], Dually residuated lattice ordered semigroups [7] etc. However all these spaces are lattices (with the exception of Newman algebra which is not even a partially ordered set); and a natural question would be whether there are semilattices with a natural structure of an autometrized space. In the present paper we observe that the dual of an implicative semilattice [8] is a generalization of Brouwerian algebra and it has a natural structure of an autometrized space.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 3 , November 1974 , pp. 293 - 302
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Birkhoff, G., ‘Lattice theory’, (Am. Math. Colloquium publications, (25), (1948)).Google Scholar
[2]Ellis, D., ‘Autometrized Boolean Algebras I’, Canad. J. Math. 3 (1951), 83–87.Google Scholar
[3]Ellis, D., ‘Autometrizes Bollean algebras II’, Canad. J. Math. 3 (1951), 145–147.CrossRefGoogle Scholar
[4]Kamalaranjan, Roy, ‘Newmannian geometry I’, Bull. Calcutta Math. Soc. 52 (1960), 187–194.Google Scholar
[5]Swamy, K. L. Narasimha, ‘A general theory of autometrized algebras’, Math. Annalen 157 (1964), 65–74.CrossRefGoogle Scholar
[6]Swamy, K. L. Narasimha, ‘Autometrized lattice ordered groups I’, Math. Annalen 154 (1964), 406–412.CrossRefGoogle Scholar
[7]Swamy, K. L. Narasimha, ‘Dually residuated lattice ordered semigroups’, Math. Annalen 159 (1965), 105–114.CrossRefGoogle Scholar
[8]Nemitz, W. C., ‘Implicative semilattices’, Trans. Amer. Math. Soc. 117 (1965), 128–142.CrossRefGoogle Scholar
[9]Nordhaus, E. A. and Lapidus, Leo, ‘Brouwerian geometry’, Canad. J. Math. 117 (1965), 6 (1954), 217–229.Google Scholar