PREFRAME PRESENTATIONS PRESENT

Preframes (directed complete posets with finite meets that distribute over the directed joins) are the algebras for an infinitary essentially algebraic theory, and can be presented by generators and relations. This result is combined with a general argument concerning categories of commutative monoids to give a very short proof of the localic Tychonoff Theorem. It is also shown how frames can be presented as preframes, a result analogous to Johnstone’s construction of frames from sites, and an application is given.

[1]  E. Michael.  Topologies on spaces of subsets , 1951 .

[2]  J. Słomiński,et al.  The theory of abstract algebras with infinitary operations , 1959 .

[3]  S. Papert.  An abstract theory of topological subspaces , 1964 .

[4]  P. Gabriel,et al.  Lokal α-präsentierbare Kategorien , 1971 .

[5]  Peter Freyd,et al.  Aspects of topoi , 1972, Bulletin of the Australian Mathematical Society.

[6]  C. H. Dowker,et al.  Products and sums in the category of frames , 1976 .

[7]  Michael B. Smyth.  Power Domains , 1978, J. Comput. Syst. Sci..

[8]  M. Coste.  Localisation, spectra and sheaf representation , 1979 .

[9]  D. Wigner,et al.  Two notes on frames , 1979, Journal of the Australian Mathematical Society.

[10]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[11]  P. T. Johnstone,et al.  Tychonoff's theorem without the axiom of choice , 1981 .

[12]  Michael W. Mislove,et al.  Local compactness and continuous lattices , 1981 .

[13]  M. Smyth.  Power Domains and Predicate Transformers: A Topological View , 1983, ICALP.

[14]  A. Joyal,et al.  An extension of the Galois theory of Grothendieck , 1984 .

[15]  Igor Kříž,et al.  A constructive proof of the Tychonoff's theorem for locales , 1985 .

[16]  Colin McLarty.  Left exact logic , 1986 .

[17]  Bernhard Banaschewski,et al.  Another look at the localic Tychonoff theorem , 1988 .

[18]  Peter T. Johnstone,et al.  Weak products and Hausdorff locales , 1988 .

[19]  Thierry Coquand,et al.  An intuitionistic proof of Tychonoff's theorem , 1992, Journal of Symbolic Logic.