Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T20:06:01.786Z Has data issue: false hasContentIssue false
  • English
  • Français

The Lattice of Equational Classes of Commutative Semigroups

Published online by Cambridge University Press:  20 November 2018

Evelyn Nelson
Evelyn Nelson*
Affiliation:
McMaster University, Hamilton, Ontario
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.

There has been some interest lately in equational classes of commutative semigroups (see, for example, [2; 4; 7; 8]). The atoms of the lattice of equational classes of commutative semigroups have been known for some time [5]. Perkins [6] has shown that each equational class of commutative semigroups is finitely based. Recently, Schwabauer [7; 8] proved that the lattice is not modular, and described a distributive sublattice of the lattice.

The present paper describes a “skeleton” sublattice of the lattice, which is isomorphic to A × N+ with a unit adjoined, where A is the lattice of pairs (r, s) of non-negative integers with r ≦ s and s ≧ 1, ordered component-wise, and N+ is the natural numbers with division.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 875 - 895
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

This paper is a revision of the author's doctoral dissertation, written under the supervision of Dr. G. Bruns at McMaster University, Hamilton, Ontario.

References

1. Burris, S. and Nelson, E., Embedding Hm in the lattice of equational classes of commutative semigroups, Proc. Amer. Math. Soc. 80 (1971), 37–39.Google Scholar
2. Dean, R. A. and Evans, T., A remark on varieties of lattices and semigroups, Proc. Amer. Math. Soc. 21 (1969), 394–396.Google Scholar
3. Trevor, Evans, The number of semigroup varieties, Quart. J. Math. Oxford Ser. 19 (1968), 335–336.Google Scholar
4. Trevor, Evans, The lattice of semigroup varieties, Semigroup Forum 2 (1971), 1–43.Google Scholar
5. Kalicki, J. and Scott, D., Equational completeness of abstract algebras, Nederl. Akad. Wetensch. Proc. Ser. A 58 (1955), 650–658.Google Scholar
6. Peter, Perkins, Bases for equational theories of semi-groups, J. Algebra 11 (1969), 298–314.Google Scholar
7. Schwabauer, R., A note on commutative semigroups, Proc. Amer. Math. Soc. 20 (1969), 503–504.Google Scholar
8. Schwabauer, R., Commutative semigroup laws, Proc. Amer. Math. Soc. 22 (1969), 591–595.Google Scholar