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

TL;DR

This paper extends the de Bruijn-Erd"os theorem within the framework of fuzzy linear spaces, exploring combinatorial properties, definitions, and formulas in a generalized, lattice-based setting with new concepts like fuzzy points and lines.

Contribution

It introduces a generalized theorem of de Bruijn and Erd"os in fuzzy linear spaces, along with new definitions, properties, and formulas for combinatorics in this context.

Findings

Generalizes de Bruijn-Erd"os theorem to fuzzy linear spaces

Defines fuzzy points and lines with combinatorial properties

Shows differences in formulas within fuzzy lattice structures

Abstract

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].