Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T06:58:03.359Z Has data issue: false hasContentIssue false
  • English
  • Français

Pseudocomplements in groupoids

Part of: Groupoids

Published online by Cambridge University Press:  09 April 2009

K. Nirmala Kumari Amma
K. Nirmala Kumari Amma
Affiliation:
Department of Mathematics University of Kerala Kariavattom Trivandrum
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.

This paper is devoted to a study of pseudocomplements in groupoids. A characterization of an intraregular groupoid is obtained in terms of prime ideals. It is proved that the set of dense elements of an intraregular groupoid S with 0 is the intersection of all the maximal filters of S and that the set of normal elements of an intraregular groupoid closed for pseudocomplements forms a Boolean algebra under natural operations. It is shown that the pseudocomplement of an ideal of an intraregular groupoid with 0 is the intersection of all the minimal prime ideas not containing it.

Keywords

Groupoidintraregularsemigrouppseudocomplementnormal elementcutcomplementprime idealfilterBoolean algebra.

MSC classification

Secondary: 20L99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 2 , September 1978 , pp. 209 - 219
Copyright
Copyright © Australian Mathematical Society 1978

References

Birkhoff, G. (1948), Lattice Theory, Colloq. Publ. 25 (Amer. Math. Soc, Providence, R.I.).Google Scholar
Clifford, A. H. and Preston, G. B. (1961), “The algebraic theory of semigroups, I”, Math. Surveys 7 (Amer. Math. Soc.).Google Scholar
Frink, O. (1941), “Representation of Boolean algebras”, Bull. Math.Soc. 47, 755–756.CrossRefGoogle Scholar
Frink, O. (1962), “Pseudocomplements in semilattices”, Duke Math. J. 29, 505–514.Google Scholar
Frink, O. and Smith, R. S. (1972), “On the distributivity of the lattice of filters of a groupoid”, Pacific J. Math. 42, 313–322.CrossRefGoogle Scholar
Venkatanarasimhan, P. V. (1970), “Semi-ideals in posets”, Math. Ann. 185, 338–348.CrossRefGoogle Scholar
Vankatanatasimhan, P. V. (1971), “Pseudocomplements in posets”, Proc. Amer. Math. Soc. 28, 9–17.CrossRefGoogle Scholar
Vankatanatasimhan, P. V. (1974a), “Semi-ideals in semilattices”, Colloq. Math. 30, 203–212.CrossRefGoogle Scholar
Vankatanatasimhan, P. V. (1974b), “Ideals in pseudocomplemented lattices and semilattices”, Acta Sci. Math. (Szeged) 36, 221–226.Google Scholar