Hahn-Banach theorem for operators on the lattice normed $f$-algebras

Let $X$ and $E$ be $f$-algebras and $p:X\to E_{+}$ be a monotone vector norm. Then the triple $(X,p,E)$ is called a lattice-normed $f$-algebraic space. In this paper, we show a generalization of the extension of the Hahn-Banach theorem for operators on the lattice-normed $f$-algebras, in which the extension of one step of that is not similar to the other Hahn-Banach theorems. Also, we give some applications and results.