B-algebra is a non-empty set X with a constant 0 and a binary operation ∗, satisfying the certain axioms. A special form of B-algebra is a BM -algebra. Their relationship are every BM -algebra is a B-algebra and every 0-commutative Balgebra is a BM -algebra. The concept of fq-derivation in B-algebra is discussed. In this thesis, the fq-derivation in BM -algebra and left fq-derivation in …