Extended Curry‐Howard terms for second‐order logic

In order to allow the use of axioms in a second‐order system of extracting programs from proofs, we define constant terms, a form of Curry‐Howard terms, whose types are intended to correspond to those axioms. We also define new reduction rules for these new terms so that all consequences of the axioms can be represented. We finally show that the extended Curry‐Howard terms are strongly normalizable.     

Multivariate measures of concordance

In 1984 Scarsini introduced a set of axioms for measures of concordance of ordered pairs of continuous random variables. We exhibit an extension of these axioms to ordered n-tuples of continuous random variables, n ≥ 2. We derive simple properties of such measures, give examples, and discuss the relation of the extended axioms to multivariate measures of concordance previously discussed in the literature.     

A method for finding new sets of axioms for classes of semigroups

We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.     

Axioms of spheres in lightlike geometry of submanifolds

We prove that if an indefinite Kaehler manifold \(\bar {M}\) with lightlike submanifolds satisfies the axioms of holomorphic 2r-spheres, axioms of holomorphic 2r-planes, axioms of transversal r-spheres and axioms of transversal r-planes, then it is an indefinite complex space form.     

Resurrection axioms and uplifting cardinals

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.     

Set-valued nonlinear analogues of the shapley value

One of Shapley's classic axioms deals with the addition of the games. We consider the family of axiom systems depending on various operations over the set of cooperative games. The existence and the uniqueness of the correspondinga priori set-valued solutions are studied. The family of solutions (satisfying the proposed system of axioms) contains, as particular cases, the Shapley value and the -core. The modification of the proposed system of axioms is studied, prekernel being the maximal solution satisfying this modification.We would like to thank two anonymous referees for helpful comments.     

Quadratic homology

We describe axioms for a `quadratic homology theory' which generalize the classical axioms of homology. As examples we consider quadratic homology theories induced by 2-excisive homotopy functors in the sense of Goodwillie and the homology of a space with coefficients in a square group which generalizes the homology of a space with coefficients in an abelian group.

     

Algebraic structures on Grothendieck groups of a tower of algebras

The Grothendieck group of the tower of symmetric group algebras has a self-dual graded Hopf algebra structure. Inspired by this, we introduce by way of axioms, a general notion of a tower of algebras and study two Grothendieck groups on this tower linked by a natural paring. Using representation theory, we show that our axioms give a structure of graded Hopf algebras on each Grothendieck groups and these structures are dual to each other. We give some examples to indicate why these axioms are necessary. We also give auxiliary results that are helpful to verify the axioms. We conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras (in the sense of Loday) on their Grothendieck groups.     

Combinatorial Models for Weyl Characters

We present a set of axioms for combinatorial objects closely related to those for Kashiwara's crystals. We show that any model for the axioms, such as Littelmann's path model, has a character—a nonnegative sum of irreducible characters for a semisimple Lie group or algebra, or more generally, a symmetrizable Kac-Moody algebra. Moreover, there are simple explicit restriction rules and rules for decomposing the product of any such character by an irreducible character.     

The relation between GTi-spaces and fuzzy proximity spaces, G-compact spaces, fuzzy uniform spaces

In this paper, we study the relation between the fuzzy separation axioms, which had been introduced by the authors in 2001, and the fuzzy proximity defined by Katsaras in 1980. We study also the relation between our fuzzy separation axioms and the G-compactness defined by Gähler in 1995. Moreover, we show here the relation between these fuzzy separation axioms and the fuzzy uniform structures introduced and studied by Gähler and the first author in 1998.     

On replacement axioms for the Jacobi identity for vertex algebras and their modules

We discuss the axioms for vertex algebras and their modules, using formal calculus. Following certain standard treatments, we take the Jacobi identity as our main axiom and we recall weak commutativity and weak associativity. We derive a third companion property that we call “weak skew-associativity”. This third property in some sense completes an S₃-symmetry of the axioms, which is related to the known S₃-symmetry of the Jacobi identity. We do not initially require a vacuum vector, which is analogous to not requiring an identity element in ring theory. In this more general setting, one still has a property, occasionally used in standard treatments, which is closely related to skew-symmetry, which we call “vacuum-free skew-symmetry”. We show how certain combinations of these properties are equivalent to the Jacobi identity for both vacuum-free vertex algebras and their modules. We then specialize to the case with a vacuum vector and obtain further replacement axioms. In particular, in the final section we derive our main result, which says that, in the presence of certain minor axioms, the Jacobi identity for a module is equivalent to either weak associativity or weak skew-associativity. The first part of this result has appeared previously and has been used to show the (nontrivial) equivalence of representations of and modules for a vertex algebra. Many but not all of our results appear in standard treatments; some of our arguments are different from the usual ones.     

A parametrised choice principle and Martin's conjecture on Blackwell determinacy

We define a parametrised choice principle PCP which (under the assumption of the Axiom of Blackwell Determinacy) is equivalent to the Axiom of Determinacy. PCP describes the difference between these two axioms and could serve as a means of proving Martin's conjecture on the equivalence of these axioms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Ramsey bound on stable sets in Jordan pillage games

Jordan (J Econ Theory 131(1):26–44, 2006) defined ‘pillage games’, a class of cooperative games whose dominance operator is represented by a ‘power function’ satisfying coalitional and resource monotonicity axioms. In this environment, he proved that stable sets must be finite. We provide a graph theoretical interpretation of the problem which tightens the finite bound to a Ramsey number. We also prove that the Jordan pillage axioms are independent.     

Axiomatic characterizations of the weighted solidarity values

We define and characterize the class of all weighted solidarity values. Our first characterization employs the classical axioms determining the solidarity value (except symmetry), that is, efficiency, additivity and the A-null player axiom, and two new axioms called proportionality and strong individual rationality. In our second axiomatization, the additivity and the A-null player axioms are replaced by a new axiom called average marginality.     

On the axioms of planes in quaternionic geometry

Summary The axioms of planes in Riemannian geometry and Kaehlerian geometry have been largely studied. In this paper we study axioms for three kinds of planes in Quaternionic geometry: the axiom of quaternionic 4-planes, the axiom of half-quaternionic planes and the axiom of totally real planes. We also give a characterization of quaternionspaceforms in terms of the constancy of the totally real sectional curvatures.     

Bernays-Gödel type theory

We study the type-theoretical analogue of Bernays-Gödel set-theory and its models in categories. We introduce the notion of small structure on a category, and if small structure satisfies certain axioms we can think of the underlying category as a category of classes. Our axioms imply the existence of a co-variant powerset monad on the underlying category of classes, which sends a class to the class of its small subclasses. Simple fixed points of this and related monads are shown to be models of intuitionistic Zermelo-Fraenkel set-theory (IZF).     

Axiomatizing subcategories of Abelian categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any d-abelian category is equivalent to a d-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.     

Generalized Rigid Groups: Definitions,Basic Properties,and Problems

We find a natural generalization of the concept of rigid group. The generalized rigid groups are also called r-groups. The terms of the corresponding rigid series of every r-group can be characterized by both ?-formulas and ?-formulas. We find a recursive system of axioms for the class of r-groups of fixed solubility length. We define divisible r-groups and give an appropriate system of axioms. Several fundamental problems are stated.     

Incomplete Relative Semi-Abelian Categories

We propose relative (to a distinguished class E of epimorphisms) versions of the so-called old and new style axioms for semi-abelian categories, and prove the equivalence of these two sets of axioms. The same results were obtained before under much stronger completeness/cocompleteness assumptions. One of obvious purposes of such a generalization is to include the trivial case of E being the class of all isomorphisms when the ground category is an arbitrary pointed category.