On α–field and β–field

Abstract The objective of this paper is, first, to introduce and study the concept of α–field as a generalization of field, σ–field and δ–field, and we discuss the properties of this concept. Furthermore, we study the relationships between σ–field and σ–field. As a first σ–field is α–field. second, to introduce the concept of β–field as a generalization of σ–field, β–σ–field and ring. So, we prove that every σ–field is β–field and we obtain some important results deals with this concept. Finally, we introduce and study the concept of restriction of β– field and we prove that, if ℘ is a β– field of a set ℵ and K is a non-empty subsets of ℵ. Then ℘|K is a β–field of a set K.