Mathematics

In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions F and G under consideration are close to two nicely behaved functions A and B, such that the average value of A(n − h)B(n) over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of NK (N)/φ(N), where NK (N)/φ(N) log N is the expected number of primes such that a random elliptic curve over ra-tionals has N points when reduced over those primes.

We show that Intuitionistic Open Induction iop is not closed under the rule DN S(∃ − 1). This is established by constructing a Kripke model of iop+¬L y (2y > x), where L y (2y > x) is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by P A − plus the scheme of weak ¬¬LN P for open formulas, where universal quantification on the parameters precedes double negation. We also show that for any open formula ϕ(y) having only y free, (P A −) i L y ϕ(y). We observe that the theories iop, i∀ 1 and iΠ 1 are closed under Friedman's translation by negated formulas and so under V R and IP. We include some remarks on the classical worlds in Kripke models of iop. 1.1 Let DOR (resp. P A −) be the finite set of usual axioms (including Trichotomy) for discretely ordered commutative rings with 1 (resp. their nonnegative parts) in the language L = {+, ·, < , 0, 1} of arithmetic. Peano Arithmetic P A (resp. Heyting Arithmetic HA) is the classical (resp. intuitionistic, obtained by dropping the principle P EM of excluded middle whose instance P EM ϕ on a formula ϕ is ϕ ∨ ¬ϕ) first order theory axiomatized by P A − together with the induction scheme whose instance with respect to a distinguished free variable x on a formula ϕ(x, y) is I x ϕ = I x ϕ(x, y) : ∀y(ϕ(0, y) ∧ ∀x(ϕ(x, y) → ϕ(x + 1, y)) → ∀xϕ(x, y)).

Assembling a localic map f : L → M from localic maps fi : Si → M , i ∈ J, defined on closed respectively open sublocales (J finite in the closed case) follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead, we present simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.

The concept of λ-super socle of C(X), denoted by S λ (X) (that is, the set of elements of C(X) such that the cardinality of their cozerosets are less than λ, where λ is a regular cardinal number with λ ≤ |X|) is introduced and studied. Using this concept we extend some of the basic results concerning SCF (X), the super socle of C(X) to S λ (X), where λ ≥ ℵ0. In particular, we determine spaces X for which SCF (X) and S λ (X) coincide. The one-point λ-compactification of a discrete space is algebraically characterized via the concept of λ-super socle. In fact we show that X is the one-point λ-compactification of a discrete space Y if and only if S λ (X) is a regular ideal and S λ (X) = Ox, for some x ∈ X.

We discuss the congruences θ that are connected as elements of the (totally disconnected) congruence frame CL, and show that they are in a one-to-one correspondence with the completely prime elements of L, giving an explicit formula. Then we investigate those frames L with enough connected congruences to cover the whole of CL. They are, among others, shown to be TD-spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.

We show that in ZF set theory without choice, the Ultrafilter Principle (UP) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasicontinuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from UP but also from DC, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets, which is not provable in ZF set theory.

The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical 'tangle modality' connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.

Locally compact Hausdorff spaces and their one-point com-pactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where lattice-theoretic methods can be used to obtain topological results. Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such. Partial frames are meet-semilattices in which not all subsets need have joins. A distinguishing feature of their study is that a small collection of axioms of an elementary nature allows one to do much that is traditional for frames or locales. The axioms are sufficiently general to include as examples σ-frames, κ-frames and frames. In this paper, we present the notion of a continuous partial frame by means of a suitable " way-below " relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main

The much-studied projectable hull of an ℓ-group G ≤ pG is an essential extension, so that, in the case that G is archimedean with weak unit, " G ∈ W " , we have for the Yosida representation spaces a " covering map " Y G ← Y pG. We have earlier [8] shown that (1) this cover has a characteristic minimality property, and that (2) knowing Y pG, one can write down pG. We now show directly that for A , the boolean algebra in the power set of the minimal prime spectrum Min(G), generated by the sets U (g) = {P ∈ Min(G) : g / ∈ P } (g ∈ G), the Stone space SA is a cover of Y G with the minimal property of (1); this extends the result from [1] for the strong unit case. Then, applying (2) gives the pre-existing description of pG, which includes the strong unit description of [1]. The present methods are largely topological, involving details of covering maps and Stone duality.

In this paper, we consider the forgetful functor from the category LDcpo of local dcpos (respectively, Dcpo of dcpos) to the category Pos of posets (respectively, LDcpo of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree S-ldcpos over a local dcpo, where S is a local dcpo monoid. The main results are: (1) The forgetful functor U : LDcpo −→ Pos has a left adjoint, but does not have a right adjoint; (2) The inclusion functor I : Dcpo −→ LDcpo has a left adjoint, but does not have a right adjoint; (3) The forgetful functor U : LDcpo-S −→ LDcpo has both left and right adjoints; (4) If (S, ·, 1) is a good ldcpo-monoid, then the forgetful functor U : LDcpo-S −→ Pos-S has a left adjoint.

We find a criterion for a morphism of coalgebras over a Barr-exact category to be effective descent and determine (effective) descent mor-phisms for coalgebras over toposes in some cases. Also, we study some ex-actness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between corresponding categories of coalgebras. As a result, we find conditions under which the induced functors preserve natural number objects as well as a criterion for them to be exact. Also this enable us to give a criterion for split epis in a category of coalgebras to be effective descent.