Archive for Mathematical Logic最新文献

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 The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.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.

In this article we investigate the complexity of isomorphisms between scattered linear orders of constructive ranks. We give the general upper bound and prove that this bound is sharp. Also, we construct examples showing that the categoricity level of a given scattered linear order can be an arbitrary ordinal from 3 to the upper bound, except for the case when the ordinal is the successor of a limit ordinal. The existence question of the scattered linear orders whose categoricity level equals the successor of a limit ordinal is still open.

We study Basic Arithmetic, (textsf{BA}) introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). (textsf{BA}) is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of (textsf{BA}) is a proper sub-class of the primitive recursive functions. Three extensions of (textsf{BA}), called (textsf{BA}+mathsf U), (mathsf {BA_{mathrm c}}) and (textsf{EBA}) are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of (textsf{BA}) are exactly the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in (textsf{BA}), (textsf{BA}+mathsf U), (mathsf {BA_{mathrm c}}), but holds in (textsf{EBA}).

By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic and a uniform proof of undecidability of their full theory, both in the language of rings without parameters. This vastly extends the scope of a method due to Raphael Robinson, which deals with a restricted class of polynomial integral domains.

We investigate the problem of punctual (fully primitive recursive) presentability of algebraic structures up to primitive recursive and computable isomorphism. We show that for mono-unary structures and undirected graphs, if a structure is not punctually categorical then it has infinitely many punctually non-isomorphic punctual presentations. We also show that the punctual degrees of any computably almost rigid structure as well as the order ((mathbb {Z},<)) are dense. Finally we characterise the Boolean algebras which have a punctually 1-decidable presentation that is computably isomorphic to a 1-decidable presentation.

We formalize various counting principles and compare their strengths over (V^{0}). In particular, we conjecture the following mutual independence between:

• a uniform version of modular counting principles and the pigeonhole principle for injections,

• a version of the oddtown theorem and modular counting principles of modulus p, where p is any natural number which is not a power of 2,

• and a version of Fisher’s inequality and modular counting principles.

Then, we give sufficient conditions to prove them. We give a variation of the notion of PHP-tree and k-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have o(n)-evaluations. As for the remaining two, we utilize well-known notions of p-tree and k-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.

On the set of all first-order complete theories (T(sigma )) of a language (sigma ) we define a binary operation ({cdot }) by the rule: (Tcdot S= {{,textrm{Th},}}({Atimes Bmid Amodels T ,,text {and},, Bmodels S})) for any complete theories (T, Sin T(sigma )). The structure (langle T(sigma );cdot rangle ) forms a commutative semigroup. A subsemigroup S of (langle T(sigma );cdot rangle ) is called an absorption’s formula definable semigroup if there is a complete theory (Tin T(sigma )) such that (S=langle {Xin T(sigma )mid Xcdot T=T};cdot rangle ). In this event we say that a theory T absorbs S. In the article we show that for any absorption’s formula definable semigroup S the class ({{,textrm{Mod},}}(S)={Ain {{,textrm{Mod},}}(sigma )mid Amodels T_0,,text {for some},, T_0in S}) is axiomatizable, and there is an idempotent element (Tin S) that absorbs S. Moreover, ({{,textrm{Mod},}}(S)) is finitely axiomatizable provided T is finitely axiomatizable. We also prove that ({{,textrm{Mod},}}(S)) is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided.