This thesis discusses a develompment of BN1-algebra with addition of companion and homomorphism concepts. An algebra statifies of companion if there is an operation ⊙, then ((x ⊙ y)x) ∗ y = 0 and (z ∗ x) ∗ y = 0 ⇒ z ∗ (x ⊙ y) = 0. In this thesis, we develop the formula definition of companion BN1-algebra. Then a mapping ' : X → Y is said to be a homomorphism, if it statis…