John Isbell

General function spaces, products and continuous lattices

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Foxs paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Foxs paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

Perfect addition sets

A subset S of Zn is called a perfect addition set if the form x+y, for x? y in S, does not represent 0 and represents all non-zero elements the same number of times. The known perfect addition sets with 1?2 elements are (up to equivalence) one for each prime n=4a+3>3 and one each for n=2,4. It is shown that there are no others with 21>n?9, or with l divisible by 5; and there are some other restrictions.

Functorial implicit operations

Two forms of Keislers characterization of functorial predicates are established for implicity definable infinitary operations. In particular, functorial and implicity definable ⇒ explicitly definable.

Some structure of Borel locales

All Borel classes of sublocales of the real line after the first ambiguous class (in particular, the limit ambiguous classes) have proper (=irreducible) representatives.

A frame with no admissible topology

Every finitary algebra is, of course, a topological algebra in the discrete topology. On the other hand, a topological Boolean σ-algebra is indiscrete; sequences (0, 0, …, x, x ,…) with join x converge to (0, 0,…) with join 0, so 0 is in { x } − , and similarly x is in {0} − . For lattices with infinite joins but only finite meets, such as topologies, the matter is more interesting.

Zero sets of polynomials in finite rings

In a field, of course, ann-th degree polynomial has at mostn zeros. Is there anything like this in rings? We find: ann-th degree polynomial in a finite ringA, ofr elements, that is not zero everywhere inA is non-zero at leastr/2n times. This is sharp for alln, even in commutative rings; perhaps also in unitary rings, though examples are lacking beyond the cubic case. However, the best such result for commutative unitary rings (not determined pastn=3) is better; (2/2n)r is proved, and the best coefficient is between that and its square root.

Notes on semigroup dominions

For the background of the present results see [i] or Scheiblich-Moores note [3]. i. The semigroup of all endomorphisms of a left vector space over a division ring i__ss absolutely closed. Proof. Let V be such a vector space~ A the semigroup of endomorphisms of V, E any extension of A. Observe that we have three representations of A as a semigroup of functions : on V, on A itself (by left multiplication) and on E (by left multiplication). If we wish, then, we have three representations of A as a semigroup of single-valued relations. We shall prove that A is left-isolated. For this, given equations in E of the form a 0 = xla I ,