On the complexity of axiomatizations of the class of representable quasi‐polyadic equality algebras

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omegaUsing games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omega$ and $l< n,$ $k < n$, k′ < ω are natural numbers, then Σ contains infinitely equations in which ? occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Thin equivalence relations in scaled pointclasses

For ordinals α beginning a Σ₁ gap in $\mathrm{L}(\mathbb {R})$, where $\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ is closed under number quantification, we give an inner model‐theoretic proof that every thin $\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ equivalence relation is $\Delta _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ in a real parameter from the (optimal) hypothesis $\mathsf {AD}^{\mathrm{J}_{\alpha }(\mathbb {R})}$.     

On the rank of certain matrices

This note is an appendix to the paper “Osculating spaces and diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier)” by M. Bolognesi and G. Pirola.     

Powers of positive elements in C*‐algebras

In this paper, we show that Ogasawa’s theorem has a proof in Bishop style constructive mathematics (BISH). In 25 , we introduced the elementary constructive theory of C*‐algebras in BISH, but we did not discuss the powers of positive elements there. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Chain homogeneous Souslin algebras

Assuming Jensen's principle ?⁺ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todor?evi?.     

On definability of types of finite Cantor‐Bendixson rank

We prove that every type of finite Cantor‐Bendixson rank over a model of a first‐order theory without the strict order property is definable and has a unique nonforking extension to a global type. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Subrecursive functions on partial sequences

The paper studies a domain theoretical notion of primitive recursion over partial sequences in the context of Scott domains. Based on a non-monotone coding of partial sequences, this notion supports a rich concept of parallelism in the sense of Plotkin. The complexity of these functions is analysed by a hierarchy of classes similar to the Grzegorczyk classes. The functions considered are characterised by a function algebra generated by continuity preserving operations starting from computable initial functions. Its layers are related to those above by showing , thus generalising results of Schwichtenberg/Müller and Niggl. Received: 18 November 1996     

Coding into HOD via normal measures with some applications

We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the mutually non isomorphic ℓp(ℓq) spaces

We extend a result of Pe?czyński showing that {?_p(?_q): 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of ?_p(?_q) are also given.     

Simple monadic theories and partition width

We study tree‐like decompositions of models of a theory and a related complexity measure called partition width. We prove a dichotomy concerning partition width and definable pairing functions: either the partition width of models is bounded, or the theory admits definable pairing functions. Our proof rests on structure results concerning indiscernible sequences and finitely satisfiable types for theories without definable pairing functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Kolmogorov complexity and characteristic constants of formal theories of arithmetic

We investigate two constants c _T and r _T , introduced by Chaitin and Raatikainen respectively, defined for each recursively axiomatizable consistent theory T and universal Turing machine used to determine Kolmogorov complexity. Raatikainen argued that c _T does not represent the complexity of T and found that for two theories S and T , one can always find a universal Turing machine such that $c_\mathbf {S}= c_\mathbf {T}$. We prove the following are equivalent: $c_\mathbf {S}\ne c_\mathbf {T}$ for some universal Turing machine, $r_\mathbf {S}\ne r_\mathbf {T}$ for some universal Turing machine, and T proves some Π₁‐sentence which S cannnot prove. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

De Morgan Heyting algebras satisfying the identity xn(′*) ≈ x(n+1)(′*)

In this paper we investigate the sequence of subvarieties $ {\mathcal {SDH}_n} $of De Morgan Heyting algebras characterized by the identity x^n(′*) ≈ x^(n+1)(′*). We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in $ {\mathcal {SDH}_1} $ by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in $ {\mathcal {SDH}_1} $. We extend these results for finite algebras in the general case $ {\mathcal {SDH}_n} $. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Failure of n‐uniqueness: a family of examples

In this paper, the connections between model theory and the theory of infinite permutation groups (see 11 ) are used to study the n‐existence and the n‐uniqueness for n‐amalgamation problems of stable theories. We show that, for any n ? 2, there exists a stable theory having (k + 1)‐existence and k‐uniqueness, for every k ? n, but has neither (n + 2)‐existence nor (n + 1)‐uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3 . © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Radical of filters in BL‐algebras

In this paper, the notion of the radical of a filter in BL‐algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV‐algebra if and only if D_s(A) ? F. After that we define the notion of semi maximal filter in BL‐algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL‐algebra. Moreover, we prove that A/F is a semi simple BL‐algebra if and only if F is a semi maximal filter of A. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.     

Low sets without subsets of higher many‐one degree

Given a reducibility ?_r, we say that an infinite set A is r‐introimmune if A is not r‐reducible to any of its subsets B with |A\B| = ∞. We consider the many‐one reducibility ?_m and we prove the existence of a low₁ m‐introimmune set in Π⁰₁ and the existence of a low₁ bi‐m‐introimmune set.     

A set of axioms for nonstandard extensions

We give an axiomatic characterization for complete elementary extensions, that is, elementary extensions of the first‐order structure consisting of all finitary relations and functions on the underlying set. Such axiom systems have been studied using various types of primitive notions (e.g., [1, 3, 6]). Our system uses the notion of partial functions as primitive. Properties of nonstandard extensions are derived from five axioms in a rather algebraic way, without the use of metamathematical notions such as formulas or satisfaction. For example, when applied to the real number system, it provides a complete framework for developing nonstandard analysis based on hyperreals without having to construct them and without any use of logic. This has possible pedagogical and expository applications as presented in, e.g., [5], [6]. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Recollements of extension algebras

Let A be a finite-dimensional algebra over arbitrary base field k. We prove: if the unbounded derived module category D-(Mod-A) admits symmetric recollement relative to unbounded derived module categories of two finite-dimensional k-algebras B and C:D- (Mod - B) D-(Mod - A) D-(Mod - C),then the unbounded derived module category D-(Mod - T(A)) admits symmetric recollement relative to the unbounded derived module categories of T(B) and T(C):D-(Mod - T(B)) D-(Mod - T(A)) D-(Mod -T(C)).