The present study aimed to investigate the similarity monadic $BL$-algebras and prove the completeness of the similarity monadic basic logic. To this end, the similarity monadic $BL$-algebras were defined as: a variety of monadic $BL$-algebras equipped with unary operators $\forall$ and $\exists$, and the vital properties of the algebras were provided. In addition, the S-monadic filters in the similarity monadic $BL$-algebras were introduced and investigated. Correspondingly, the representable similarity monadic $BL$-algebras were characterized, and the notion of the logic of the similarity monadic $BL$-algebras were introduced to prove their soundness and completeness.
"Similarity monadic basic logic." Bull. Belg. Math. Soc. Simon Stevin 27 (3) 321 - 336, august 2020. https://doi.org/10.36045/bbms/1599616814