On a generalized theorem of de Bruijn and Erd\"os in d-dimensional Fuzzy Linear Spaces

In this study we follow a new framework for the theory that offers us, other than traditional, a new angle to observe and investigate some relations between finite sets, F-lattice L and their elements. The theory is based on the Fuzzy Linear Spaces (FLS) S=(N,D). In this case, to operate on these spaces the necessary preliminaries, concepts and operations in lattices relative to FLS are introduced. Some definitions, such that k-fuzzy point, k-fuzzy line are given. Then we correspond these definitions to the definitions in usually linear spaces. We investigate some combinatorics properties of FLS. In some examples in the case where ILI=3*. We see some differences. In general, taking an ordered lattice Ln={0,a1,a2,...,an,1} we observe how some combinatorics formulas and properties are changed. In FLS the dimension concept is a set. We produce some general formulas by using some trivial examples. Furthermore, we generalize de Bruijn-Erd\"os Theorem in [2].Comment: 5 page

Pairwise balanced designs with prescribed minimum dimension

The dimension of a linear space is the maximum positive integer $d$ such that any $d$ of its points generate a proper subspace. For a set $K$ of integers at least two, recall that a pairwise balanced design PBD$(v,K)$ is a linear space on $v$ points whose lines (or blocks) have sizes belonging to $K$. We show that, for any prescribed set of sizes $K$ and lower bound $d$ on the dimension, there exists a PBD$(v,K)$ of dimension at least $d$ for all sufficiently large and numerically admissible $v$

On the Wilson Monoid of a Pairwise Balanced Design

We give a new perspective of the relationship between simple matroids of rank 3 and pairwise balanced designs, connecting Wilson's theorems and tools with the theory of truncated boolean representable simplicial complexes. We also introduce the concept of Wilson monoid W(X) of a pairwise balanced design X. We present some general algebraic properties and study in detail the cases of Steiner triple systems up to 19 points, as well as the case where a single block has more than 2 element

Cographs

Cographs--defined most simply as complete graphs with colored lines--both dualize and generalize ordinary graphs, and promise a comparably wide range of applications. This article introduces them by examples, catalogues, and elementary properties. Any finite cograph may be realized in several ways, including inner products, polynomials, geometrically, or by "fat intersections." Particular classes then considered include sum cographs (points in Z or Zn; the line C(P,Q) joining points P and Q defined C(P,Q) = P+Q); difference cographs (C(P,Q) = |P-Q|; and intersection cographs (points are sets; C(P,Q) = P intersect Q). Intersection cographs, especially, promise many applications; described here are some to aesthetics. Point-line cographs turn out equivalent to linear spaces. Finally solved here is an interesting group-theoretic problem arising from group cographs (points in a group; C(P,Q) = {PQ,QP})