In this paper it is introduced the notion of $\varepsilon$-homomorphism from a commutative semigroup $S_1$ into a commutative uniform semigroup $S_2$. The existence of homomorphism, which is $\varepsilon$-close to an $\varepsilon$-homomorphism, is proved. It is introduced the triangular bounded homomorphism and the relation to different type of boundedness is proved.