Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

The rational first-order Pavelka logic is an expansion of the infinite-valued first-order ?ukasiewicz logic ?? by truth constants. For this logic, we introduce a cumulative hypersequent calculus G¹?? and a noncumulative hypersequent calculus G²?? without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus G?? with structural rules. In particular, we show that every G??-provable sentence is G¹??-provable and a ??-sentence in the prenex form is G??-provable if and only if it is G²??-provable. For a tableau version of the calculus G²??, we describe a family of proof search algorithms that allow us to construct a proof of each G²??-provable sentence in the prenex form.     

Quasirecognition by prime graph of <Emphasis Type="Italic">L</Emphasis><Subscript>10</Subscript>(2)

Let G be a finite group. The prime graph of G is denoted by Γ(G). The main result we prove is as follows: If G is a finite group such that Γ(G) = Γ(L ₁₀(2)) then G/O ₂(G) is isomorphic to L ₁₀(2). In fact we obtain the first example of a finite group with the connected prime graph which is quasirecognizable by its prime graph. As a consequence of this result we can give a new proof for the fact that the simple group L ₁₀(2) is uniquely determined by the set of its element orders.     

On the composition factors of a group with the same prime graph as B n (5)

Let G be a finite group. The prime graph of G is a graph whose vertex set is the set of prime divisors of |G| and two distinct primes p and q are joined by an edge, whenever G contains an element of order pq. The prime graph of G is denoted by Γ(G). It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if G is a finite group such that Γ(G) = Γ(B _n (5)), where n ? 6, then G has a unique nonabelian composition factor isomorphic to B _n (5) or C _n (5).     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G ? B _p (3) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

The Fourier transform of Herz spaces on certain groups

LetG be a locally compact abelian topological group containing a suitable sequence of compact open subgroups and let Γ be its dual group. LetK (α,p, q; G) andK (α,p, q; Γ) denote the so-called Herz spaces onG and Γ, respectively. In this paper we shall prove that for 1<p≤2 and 0≤α<1/p′=1?1/p, the Fourier transform mapsK (α,p, p; G) continuously intoK (?α,p′, 2; Γ). The proof requires two results that are of independent interest: an extension of the Hausdorff-Young inequality to certain weightedL _p-spaces onG and a Littlewood-Paley theorem for certain weightedL _p-spaces onG.     

Two applications of semi-stable graphs

A Michigan graph G on a vertex set V is called semi-stable if for some υ?V, Γ(G_υ) = Γ(G)_υ. It can be shown that all regular graphs are semi-stable and this fact is used to show (i) that if Γ(G) is doubly transitive then G = K_n or K?_n, and (ii) that Γ(G) can be recovered from Γ(G_υ). The second result is extended to the case of stable graphs.     

Positive Frege and its Scott‐style semantics

We show that the untyped λ ‐calculus can be extended with Frege's interpretation of propositional notions, provided we restrict β ‐conversion to positive expressions. The system of illative λ ‐calculus so obtained admits a natural Scott‐style semantics. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Shared Framework for Consequence Operations and Abstract Model Theory

In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically.     

Mesures de Patterson-Sullivan dans les espaces symétriques

Let X = G/K be a symmetric space of noncompact type, Γ a Zariski-dense subgroup of G with critical exponent δ(Γ). We show that all Γ-invariant conformal densities of dimension δ(Γ) (e.g. Patterson-Sullivan densities) have their support contained in a same and single G-orbit on the geometric boundary of X. In the lattice case, we explicitly determine δ(Γ) and this G-orbit, and we establish the uniqueness of such densities.     

Distributive-Lattice Semantics of Sequent Calculi with Structural Rules

The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables (viz., metavariables for finite sets of formulas), upon the basis of the conception of model introduced in (Fuzzy Sets Syst 121(3):27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models (called its semantics) that a rule is derivable in the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant. Finally, we exemplify this application by studying sequent calculi for some of such logics.      

Quasirecognition by prime graph of 2 D n (3α) where n = 4m + 1 ≥ 21 and α is odd

Let G be a finite group. The prime graph of G is denoted by Γ(G). In this paper, as the main result, we show that if G is a finite group such that Γ(G) = Γ(² D _n (3^α)), where n = 4m+ 1 and α is odd, then G has a unique non-Abelian composition factor isomorphic to ² D _n (3^α). We also show that if G is a finite group satisfying |G| = |² D _n (3^α)|, and Γ(G) = Γ(² D _n (3^α)), then G ? ² D _n (3^α). As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for ² D _n (3^α). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that ² D _n (3^α) is quasirecognizable by the spectrum.     

Generic representations of abelian groups and extreme amenability

If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group $\overline {\pi (\Gamma )} $ for the generic π ∈ Hom(Γ, G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $\overline {\pi (\Gamma )} $ ; in the other two, we show that the generic $\overline {\pi (\Gamma )} $ is extremely amenable. We also show that if Γ is torsionfree, the centralizer of the generic π is as small as possible, extending a result of Chacon and Schwartzbauer from ergodic theory.     

Note on group distance magic complete bipartite graphs

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ? from V to an Abelian group Γ of order n such that the weight $w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ? _p -distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K _m,n is a group distance magic graph if and only if n + m ? 2 (mod 4).     

A novel approach towards fuzzy Γ-ideals in ordered Γ-semigroups

In many applied disciplines like computer science, coding theory and formal languages, the use of fuzzified algebraic structures especially ordered semigroups play a remarkable role. In this paper, we introduce a new concept of fuzzy Γ-ideal of an ordered Γ-semigroup G called an (∈, ∈ ?q _k )-fuzzy Γ-ideal of G. Fuzzy Γ-ideal of type (∈, ∈ ∨q _k ) are the generalization of ordinary fuzzy Γ-ideals of an ordered Γ-semigroup G. A new characterization of ordered Γ-semigroups in terms of an (∈, ∈ ∨q _k )-fuzzy Γ-ideal is given. We show that a fuzzy subset λ of an ordered Γ-semigroup G is an (∈, ∈ ∨q _k )-fuzzy Γ-ideal of G if and only if U (λ; t) is a Γ-ideal of G for all \(t \in \left( {0,\frac{{1 - k}} {2}} \right]\) . We also investigate some important characterization theorems in terms of this notion. Finally, regular ordered Γ-semigroups are characterized by the properties of their (∈, ∈ ∨q _k )-fuzzy Γ-ideals.     

Γ-amenability of locally compact groups

Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner＇s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.     

Radius two trees specify χ-bounded classes

A class of graphs χ is said to be χ-bounded, with χ-binding function f, if for all G ? Γ, χ (G) ≦ f (ω(G)), where χ(G) is the chromatic number of G and ω(G) is the clique number of G. It has been conjectured that for every tree T, the class of graphs that do not induce T is χ-bounded. We show that this is true in the case where T is a tree of radius two.     

Strong liftings with application to measurable cross sections in locally compact groups

There are two principal theorems. The adjustment theorem asserts that a lifting may be changed on a set of measure zero so as to become slightly stronger. In conjunction with the standard lifting theorem, it yields generalizations (with shorter proofs) of a number of known results in the theory of strong liftings. It also inspires a characterization of strong liftings, when the measure is regular, by the fact that they induce upon every open set an artificial “closure” of that set which differs from it by a set of measure zero. The projection theorem asserts that, in the presence of a strict disintegration, a strong lifting may be transferred or “projected” from one topological measure space onto another. In conjunction with Losert's example, it yields regular Borel, measures, carried on compact Hausdorff spaces of arbitrarily large weight, which everywhere fail to have the strong lifting property. It also provides the final link needed to obtained, with no separability assumptions, a measurable cross section (or right inverse) for the canonical map Ω:G→G/H, whereG is an arbitrary locally compact group, and whereH is an arbitrary closed subgroup ofG.     

On standard models of fuzzy region connection calculus

The Region Connection Calculus (RCC) is perhaps the most influential topological relation calculus. Based on the first-order logic, the RCC, however, does not fully meet the needs of applications where the vagueness of entities or relations is important and not ignorable. This paper introduces standard models for the fuzzy region connection calculus (RCC) proposed by Schockaert et al. (2008) [18]. Each of such a standard fuzzy RCC model is induced by a standard RCC model in a natural way. We prove that each standard fuzzy RCC model is canonical in the sense that any satisfiable set of fuzzy RCC8 constraints have a solution in it. A polynomial realization algorithm is also provided. As a side product, we show similar sets of fuzzy constraints have similar solutions if both are satisfiable. This allows us to approximate fuzzy RCC constraints that have arbitrary bounds by those have bounds with finite precision.     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.