On semi maximal filters in BL-algebras

In this paper, first we study the semi maximal filters in linear $BL$-algebras and we prove that any semi maximal filter is a primary filter. Then, we investigate the radical of semi maximal filters in $BL$-algebras. Moreover, we determine the relationship between this filters and other types of filters in $BL$-algebras and G"{o} del algebra. Specially, we prove that in a G"{o}del algebra, any fantastic filter is a semi maximal filter and any semi maximal filter is an (n-fold) positive implicative filter. Also, in a $BL$-algebra, any semi maximal and implicative filter is a positive implicative filter.Finally, we give an answer to the open problem in [S. Motamed, L. Torkzadeh, A. Borumand Saeid and N. Mohtashamnia, Radical of filters in BL-algebras, Math. Log. Quart. 57, No. 2, (2011), 166-179 ].