Archive for Mathematical Logic最新文献

Building on the work of Avraham, Rubin, and Shelah, we aim to build a variant of the Fraïssé theory for uncountable models built from finite submodels. With this aim, we generalize the notion of an increasing set of reals to other structures. As an application, we prove that the following is consistent: there exists an uncountable, separable metric space X with rational distances, such that every uncountable partial 1-1 function from X to X is an isometry on an uncountable subset. We aim for a general theory of structures with this kind of properties. This includes results about the automorphism groups, and partial classification results.

We investigate the degree spectra of computable relations on canonically ordered natural numbers ((omega ,<)) and integers ((zeta ,<)). As for ((omega ,<)), we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all (Delta _2) degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on ((omega ,<)), answering the question whether there are infinitely many such spectra. As for ((zeta ,<)), we prove the following dichotomy result: given an arbitrary computable relation R on ((zeta ,<)), its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for ((omega ,<)) obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).

We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the ({text {pcf}})-conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:

Theorem If ABSP holds for an ascending sequence ( langle aleph _{n_{m}} rangle _{m}) (( n_{m} in omega )) then there is an inner model with measurables (kappa < aleph _{omega }) of arbitrarily large Mitchell order below (aleph _{omega }), that is: (sup left{ alpha mid {exists }kappa < aleph _{omega } o ( kappa ) ge alpha right} = aleph _{omega }). A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales.

We investigate the class of m-hypergraphs whose substructures with l elements have more than s m-element subsets that do not form a hyperedge. The class will have the free amalgamation property if s is small, but it does not if s is large. We find the boundary of s. Suppose the class has the free amalgamation property. In the case (m ge 3), we demonstrate that the random structure for the class has continuum-many automorphisms with a single orbit. The situation differs from the case of Henson graphs. In the case of generic hypergraphs constructed by Hrushovski’s method using a predimension function, we also demonstrate that they have no automorphisms with a single orbit.

In this paper we introduce the class of weak Heyting–Brouwer algebras (WHB-algebras, for short). We extend the well known duality between distributive lattices and Priestley spaces, in order to exhibit a relational Priestley-like duality for WHB-algebras. Finally, as an application of the duality, we build the tense extension of a WHB-algebra and we employ it as a tool for proving structural properties of the variety such as the finite model property, the amalgamation property, the congruence extension property and the Maehara interpolation property.

In recent research, the prime and irreducible elements of strong finite factorization domains were studied. It was shown that strongly computable strong finite factorization domains (SCSFFD) have necessarily computable irreducible elements and a computable division algorithm. However, the question of how to best classify this class of structures is left unanswered. This work provides a classification for SCSFFDs by showing the existence of a computable norm where norm-form equations can be solved computably. This classification provides the intuition to extend further the notion of strong computability to finite factorization domains in general.

We study (omega )-categorical MS-measurable structures. Our main result is that a certain class of (omega )-categorical Hrushovski constructions, supersimple of finite SU-rank is not MS-measurable. These results complement the work of Evans on a conjecture of Macpherson and Elwes. In constrast to Evans’ work, our structures may satisfy independent n-amalgamation for all n. We also prove some general results in the context of (omega )-categorical MS-measurable structures. Firstly, in these structures, the dimension in the MS-dimension-measure can be chosen to be SU-rank. Secondly, non-forking independence implies a form of probabilistic independence in the measure. The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.

We investigate the covering numbers of some ideals on ({^{omega }}{2}{}) associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, ({mathfrak {C}_{2}}), remain small in the Sacks model.