No abstract

No abstract

No abstract

The theory of finitely supported algebraic structures is related to Pitts theory of nominal sets (by equipping finitely supported sets with finitely supported internal algebraic laws). It represents a reformulation of Zermelo Fraenkel set theory obtained by requiring every set theoretical construction to be finitely supported according to a certain action of a group of permutations of some basic elements named atoms. Its main purpose is to let us characterize infinite algebraic structures, defined involving atoms, only by analyzing their finite supports. The first goal of this paper is to define and study different kinds of infinities and the notion of 'cardinality' in the framework of finitely supported structures. We present several properties of infinite cardinalities. Some of these properties are extended from the non-atomic Zermelo Fraenkel set theory into the world of atomic objects with finite support, while other properties are specific to finitely supported structures. We also compare alternative definitions of 'infinite finitely supported set', and we finally provide a characterization of finitely supported countable sets.Proposition 2.2) is called the support of x and is denoted by supp(x). An empty supported element is called equivariant; this means that x ∈ X is equivariant if and only if π • x = x, ∀π ∈ S A .3. Let (X, •) be an S A -set. We say that X is an invariant set if for each x ∈ X there exists a finite set S x ⊂ A which supports x.Proposition 2.2 [1] Let X be an S A -set and let x ∈ X. If there exists a finite set supporting x (particularly, if X is an invariant set), then there exists a least finite set supporting x which is constructed as the intersection of all finite sets supporting x.

No abstract

No abstract

We propose and support the possibility that the shape of topological density 2-point function in pure-glue QCD is crucially, and possibly entirely, determined by the space-time folding (geometry) of the double-sheet sign-coherent structure of Ref. [1], while the distribution of topological density within individual sheets only determines the overall magnitude of the correlator at finite physical distances. A specific manifestation of this, discussed here, is that the shape of the correlation function (encoding e.g. the masses of pseudoscalar glueballs) is reproduced upon the replacement q(x) → sgn(q(x)), i.e. by considering the double sheet of the same space-time geometry but with constant magnitude of topological density. Combined with previous results on the fundamental topological structure, this suggests that a collective degree of freedom describing topological fluctuations of QCD vacuum can be viewed as a global space-filling homogeneous double membrane. Selected possibilities for practical uses of this are discussed.

In the new framework of the extended Fraenkel-Mostowski set theory, we define an extended interchange function as an action on a permutation group, and a new notion of permutative renaming by generalizing an existing notion of finitary permutative renaming. Some algebraic and combinatorial properties of permutative renamings expressed by using the Fraenkel-Mostowski axioms remain valid in the new framework even if we replace one axiom with a weaker axiom.Mathematics Subject Classification 2010: 03E25, 03E30, 03B70.