Paul Taylor asks:
f(a) & a = f(true) & a
I wonder whether anyone has noticed this formula before?
There is a section of my thesis devoted to operators on Heyting algebras that satisfy this formula. More generally, I considered "extensional operators" l : A -> A (A is a H.A.), which are characterized by any of the following equivalent conditions: (a) x C y implies l(x) C l(y) (x,y in A, C a Heyting-congruence on A) (b) a & l(x) <= l(a * x) <= a -> l(x) (a,x in A, * in {&,->}) (c) a * l(x) = a * l(a o x) (a,x in A, *,o in {&,->}) (d) a * x = a * y implies a o l(x) = a o l(y) (a,x,y in A, *,o in {&,->}) (e) x <-> y <= l(x) <-> l(y) (x,y in A) The name "extensional operator" comes from (e). If A is complete, extensional operators on A are in 1-1 correspondence with morphisms Omega -> Omega in the topos Sh(A). "Logical operators" are what I called extensional operators satisfying l(true) = true -- or, equivalently, x <= l(x) (x in A). The quasinuclei of Banaschewski are then precisely the monotone logical operators, and nuclei are (therefore) precisely the extensional closure operators. Arbitrary extensional operators can be "built" from "below" or from "above" by specific extensional operators, generalizing the way nuclei can be built from open, closed, and quasiclosed nuclei, and several "structure theorems" of this kind exist. One can also characterize the subsets of A that can appear as the fixedpoint sets of logical operators (equivalently the prefixedpoint sets of extensional operators). My thesis goes on to study "regular operators" -- the regular elements in the lattice of logical operators, or, equivalently, those operators r satisfying r(a->b) = a->r(b) (a,b in A) -- which have an amazing number of properties (e.g., they are idempotent, they preserve "regular" joins and "stable" meets, their fixedpoint sets are the same as the fixedpoint sets of logical operators, any two regular operators commute, and the join of regular operators is function composition, to name several). The set of regular operators on a frame A forms a complete Boolean algebra isomorphic to N^2(A)/--, i.e., the double-negation quotient of the *second assembly* of A (where N(A) = the frame of nuclei on A = the frame of frame congruences on A; see Johnstone, Stone Spaces, pp. 51--57, for more information on the functor N and its iterates). This provides a concrete embedding of any frame into a complete Boolean algebra. I have used the theory regular operators to derive an explicit formula for joins of nuclei, characterize "free meets" in frames, and to answer some questions about fibrewise-Boolean locales. My thesis also characterizes universal monos (and obtains some results about limits and colimits) in the categories of frames and kappa-frames, and generalizes some results of Banaschewki on finitely generated frame extensions. It appeared as CMU tech report CMU-CS-94-186. Todd Wilson Computer Science Department Cornell University