Chain conditions on \(\textit{BL}\)-algebras
The aim of this paper is to solve the open problem appeared in Motamed and Moghaderi (Soft Comput 2012), about the relation between Noetherian (Artinian) \(\textit{BL}\)-algebras in short exact sequences. Also, a better theorem to improve its results is suggested. The relation between Noetherian and Artinian \(\textit{BL}\)-algebras is found, the concept of length for a filter in \(\textit{BL}\)-algebras is introduced and properties of finite length \(\textit{BL}\)-algebras are developed. Finally, it is proved that any \(\textit{BL}\)-algebra has finite length if and only if be Noetherian and Artinian.
noetherian (artinian) \(\textit{bl}\)-algebra sequence length maximal filter prime filter