Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T07:43:05.613Z Has data issue: false hasContentIssue false
  • English
  • Français

Double MS -algebras

Published online by Cambridge University Press:  14 November 2011

T. S. Blyth  and
J. C. Varlet
T. S. Blyth
Affiliation:
Mathematical Institute, University of St Andrews
J. C. Varlet
Affiliation:
Institut de Mathématique, Université de Liége, B-4000 Liège, Belgium
Get access Rights & Permissions [Opens in a new window]

Synopsis

We introduce the notion of a double MS-algebra (L, 0, +) as an MS-algebra (L, 0) whose dual isan MS-algebra (Ld, +), with certain linking conditions concerning the operations x↦x0 and x↦x+. We determine necessary and sufficient conditions whereby an MS-algebra can be made into a double MS-algebra and show that this, when possible, can be done in one and only one way. We also consider the notion of a bistable subvariety of MS-algebras, namely a subvariety R with the property that, for every double MS-algebra (L, 0, +), whenever L, 0 ↦ R, we have (Ld,+)↦ R. Finally, we determine those subvarieties R of MS that are dense (in the sense that every MS-algebra L ↦ R can be made into a double MS-algebra), and those that are sparse (in the sense that if L ↦ R can be made into a double MS-algebra then it belongs to a proper subclass of R).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 1-2 , 1984 , pp. 37 - 47
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Blyth, T. S. and Janowitz, M. F.. Residuation Theory (Oxford: Pergamon Press, 1972).Google Scholar
2Blyth, T. S. and Varlet, J. C.. On a common abstraction of de Morgan algebras and stone algebras. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 301–308.CrossRefGoogle Scholar
3Blyth, T. S. and Varlet, J. C.. Subvarieties of the class of MS-algebras. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 157–169.CrossRefGoogle Scholar