An algebraic characterization of C(L), L a locale;yet more on "closed under countable composition".
by
Anthony W. Hager
Wesleyan University
Coauthors: Richard N. Ball (University of Denver)

W is the category of archimedean l-groups with weak unit;any archimedean f-ring with identity is a W-object.For W-objects,there is the canonical Yosida representation (which for an f-ring is the Henriksen-Johnson representaion):A is isomorphic to an l-group (or f-ring) A* in D(YA) for a compact Hausdorff space YA,with the unit becoming the constant function 1 and A* separating points.

A is closed under countable composition (ccc) if g(a*(n)) is in A* for each sequence (a(n)) from A and g in C(P),P=the countable power of the reals R (Henriksen-Isbell-Johnson).

The propert ccc is undeniably defined in terms of data from A,but one would not accuse it of having a classicaly l-algebraic flavor. But ccc is very interesting:A is ccc iff A is isomorphic to some C(L),L a locale (Isbell)-and no similarly concise characterization of C(X),X a space,is known-and L can be taken to be the opposite of the frame of W-kernels of A (Madden);each W-object has a ccc-hull cccA,which is a monoreflection (Aron-Hager), and in fact,the largest essential monoreflection (Ball-Hager),thus is the functorial analogue of Conrad's essential closure. In fact,any essential and quotient-closed monoreflection has the form S-ccc for S a sub-l-group of C(P).

Here we algebraicize ccc:In terms of intuitively motivated equations and inequations,we define a convergence structure on a W-object A,called singly-indicated uniform convergence (siuc),and a corresponding completion,constructed intrinsically and algbraically,for which: A is ccc (isomorphic to a C(L)) iff A is siuc-complete,and cccA (=C(L),for L=the locale of W-kernels) is the siuc-completion.

Date received: January 30, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cain-07.