A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

A simplified proof of arithmetical completeness theorem for provability logic GLP

We present a simplified proof of Japaridze’s arithmetical completeness theorem for the well-known polymodal provability logic GLP. The simplification is achieved by employing a fragment J of GLP that enjoys a more convenient Kripke-style semantics than the logic considered in the papers by Ignatiev and Boolos. In particular, this allows us to simplify the arithmetical fixed point construction and to bring it closer to the standard construction due to Solovay.     

Fuzzy equational logic

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established. Received: 15 March 2000 / Revised version: 20 September 2000 Published online: 12 December 2001     

A relative completeness theorem

Archiv der Mathematik - We prove a relative n-completeness theorem asserting that a complex space Y that fibers over a complex space X of dimension less than n is n-complete provided that Y admits...     

A topological completeness theorem

We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected. Received: 27 September 1996 / Revised version: 15 January 1998     

Gentzen-style axiomatizations in equational logic

The notion of a Gentzen-style axiomatization of equational theories is presented. In the standard deductive systems for equational logic axioms take the form of equations and the inference rules can be viewed as quasi-equations. In the deductive systems for quasi-equational logic the axioms, which are quasi-equations, can be viewed as sequents and the inference rules as Gentzen-style rules. It is conjectured that every finite algebra has a finite Gentzen-style axiomatization for its quasi-identities. We verify this conjecture for a class of algebras that includes all finite algebras without proper subalgebras and all finite simple algebras that are embeddable into the free algebra of their variety.Dedicated to the memory of Alan DayPresented by J. Sichler.Supported by an Iowa State University Research Assistantship.Supported by National Science Foundation Grant #DMS 8005870.     

A criterion for positive completeness in ternary logic

The operator of positive closure is considered on the set P _k of functions of k-valued logic. Some positive complete systems of functions are defined. It is proved that every positive complete class of functions from P _k is positive generated by the set of all functions depending on at most k variables. For each k ? 3, the three families of positive precomplete classes are defined. It is shown that, for k = 3, the 10 classes of these families constitute a criterion system.     

A characterization theorem for geometric logic

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.     

A characterization theorem for weak varieties

In this paper I define the notion of mixed product of partial algebras. It is a common generalization of two operators: weak subalgebras and products of partial algebras. I show that a class of similar partial algebras is a weak variety if and only if it is closed under the operators of mixed products and closed homomorphic images.     

A simple algebraic proof of the equational interpolation theorem

An algebraic proof is given of a natural version of Craig's interpolation lemma for equational logic.Presented by Bjarni Jónsson.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

On a weak form of equational compactness

In this paper, we discuss a property of large systems of equations over universal algebras which does not appear to be generally known, but which coincides with equational compactness for abelian groups. It is shown, for example, that if is a cardinal number with uncountable cofinality, then every finitely solvable system of equations over any countable algebra has a solvable subsystem consisting also of equations. As an application, this property is used to generalize some results of Jensen and Lenzing on the non-compactness of ultrapowers of modules. Received November 11, 1997; accepted in final form March 2, 1998.     

A Sahlqvist theorem for distributive modal logic

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.     

A stationary point theorem characterizing metric completeness

We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.     

Intuitionistic completeness of first-order logic

We constructively prove completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable in iFOL if and only if it is uniformly valid in intuitionistic evidence semantics as defined in intuitionistic type theory extended with an intersection operator.     

Analytic completeness theorem for singular biprobability models

The aim of the paper is to prove tha analytic completeness theorem for a logic L(∫₁, ∫₂)_A^s with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.