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Filter Monads, Continuous Lattices and Closure Systems

Published online by Cambridge University Press:  20 November 2018

Alan Day
Alan Day*
Affiliation:
Lakehead University, Thunder Bay, Ontario
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The notion of a monad (triple) has become increasingly important as an extension of the classical universal algebraic approach to “algebraic” categories. Indeed the categories of algebras arising from a monad seem to be the most natural generalization of Birkhoffs equational classes. Moreover in [2], Barr's concept of a relational model of a monad also coincides nicely with both the concepts of partial algebras (when suitably restricted) and (Moore) closure systems.

In this paper, we wish to examine two particular monads determined by filters. The first is the filter monad F = (F, η, 𝛍) over Sets where FX is the set of all (not necessarily proper) filters on X.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 50 - 59
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Banaschewski, B., Essential extensions of T-spaces (preprint).Google Scholar
2. Barr, M., Relational algebras, Reports of the Mid West Category Seminar IV, S.\MacLane, éd., Springer Lecture Notes 137, New York (1970).Google Scholar
3. Linton, F. E. J., Some aspects of equational categories, Proc. Conf. on Categorical Algebra (La Jolla 1965), pp. 84–94 (Springer, Berlin, 1966).Google Scholar
4. MacLane, S., Categories for the working mathematician (Springer-Verlag, New York, 1971).Google Scholar
5. Scott, D., Continuous lattices, Technical Monograph PRG7, Oxford University Computing Laboratory, 1971.Google Scholar