We present some abstract theorems showing how domination properties equivalent to being or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang, and Yu [2009] that every degree above any in is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results.

We prove that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

In this paper it is shown that the meet-inaccessible degrees are dense in R. The construction uses an 0′-priority argument. As a consequence, the meet-inaccessible degrees and the meet-accessible degrees give a partition of R into two sets, either of which is a nontrivial dense subset of R and generates R − {0} under joins (thus an automorphism base of R).

Orthogonality between two stably embedded definable sets is preserved under the addition of constants.

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:

Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we have

The proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the following

Messy Fact 0.2. There are countable Borel equivalence relationssuch that:

(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;

(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;

(iii) if f: Xx → Xxis a measurable reduction ofExto itself then

(iv)

is a standard Borel space on which the projection function

is Borel and the equivalence relation Ê given by

if and only ifx = x′ andzExz′ is Borel;

(V)

is Borel.

We first prove the theorem granted this messy fact. We then prove the fact.

(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).

Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small stable abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.

We show that the lattice L20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (ℐ), We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π2 necessary and sufficient condition for embedding a finite pseudolattice into ℐ.

This paper presents a new correctness criterion for marked Danos-Reginer graphs (D-R graphs, for short) of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL.

As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.