NETS AND TRANSLATION NETS OF HIGHER DIMENSIONS

Abstract

Thesis starts with the introduction to ?Nets (finite)? in two dimension (2- nets) as given by Bruck and then proceed to three dimension (3-nets), which is due to Laskar. We also address the issues found in the basic properties of 2-nets and 3-nets by listing out the resolutions. Some of the drawbacks in Laskar?s definition for nets of dimension three led to the redefinition of 3-nets by fixing the parameters involved. We also prove that Laskar?s definition can be recovered from our new definition (except for a particular case). Translation Nets and partial congruence partition (PCP) are then surveyed in two dimension. Later, we introduce the concept of ?PCP? in three dimension denoted by PCP^(3) and prove the equivalence of PCP^(3) and translation nets of dimension three. We conclude with the definitions of Nets and PCP in n-dimension. Examples and properties are also stated.