Higher-dimensional models of networks

Networks are often studied as graphs, where the vertices stand for entities in the world and the edges stand for connections between them. While relatively easy to study, graphs are often inadequate for modeling real-world situations, especially those that include contexts of more than two entities. For these situations, one typically uses hypergraphs or simplicial complexes. In this paper, we provide a precise framework in which graphs, hypergraphs, simplicial complexes, and many other categories, all of which model higher graphs, can be studied side-by-side. We show how to transform a hypergraph into its nearest simplicial analogue, for example. Our framework includes many new categories as well, such as one that models broadcasting networks. We give several examples and applications of these ideas

First-order Nilpotent Minimum Logics: first steps

Following the lines of the analysis done in [BPZ07, BCF07] for first-order G\"odel logics, we present an analogous investigation for Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra. We establish a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, we analyze axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The introduction section has been rewritten, and many modifications have been done to improve the readability; moreover, numerous references have been added. Concerning the technical side, some proofs has been shortened or made more clear, but the mathematical content is substantially the same of the previous versio

On Integer Additive Set-Indexers of Graphs

A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \rightarrow2^{X}$ such that the function $f^{\oplus}:E(G)\rightarrow2^{X}-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus} f(v)$ for every $uv{\in} E(G)$ is also injective, where $2^{X}$ is the set of all subsets of $X$ and $\oplus$ is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph. An IASI $f$ is said to be a {\em weak IASI} if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ and an IASI $f$ is said to be a {\em strong IASI} if $|g_f(uv)|=|f(u)| |f(v)|$ for all $u,v\in V(G)$. In this paper, we study about certain characteristics of inter additive set-indexers.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1312.7674 To Appear in Int. J. Math. Sci.& Engg. Appl. in March 201

Scalar cardinalities for divisors of a fuzzy cardinality

The cardinality of a finite fuzzy set can be defined as a scalar or a fuzzy quantity. The fuzzy cardinalities are represented by means the generalized natural numbers, where it is possible to define arithmetical operations, in particular the division by a natural number. The main result obtained in this paper is that, if determined conditions are assured, the scalar cardinality of a finite fuzzy set, B, whose fuzzy cardinality is a rational part of the fuzzy cardinality of another fuzzy set, A, is obtained by the same division of the scalar cardinality of A