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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
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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

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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