The notion of (right) left mapping on equality algebras is introduced, and related properties are investigated. In order for the kernel of (right) left mapping to be filter, we investigate what conditions are required. Relations between left mapping and $\rightarrow$-endomorphism are investigated. Using left mapping and $\rightarrow$-endomorphism, a characterization of positive implicative equality algebra is established. By using the notion of left mapping, we define $\rightarrow$-endomorphism and prove that the set of all $\rightarrow$-endomorphisms on equality algebra is a commutative semigroup with zero element. Also, we show that the set of all right mappings on positive implicative equality algebra makes a dual BCK-algebra.