Let $(X,*)$ be a commutative semigroup. A root function is a function $\gamma_n:X\to X$ for a fixed $n$, such that $\gamma_n(x)*\gamma_n(y)=\gamma_n(x*y)$ and $n[\gamma_n(x)]=\underbrace{\gamma_n(x)*\ldots*\gamma_n(x)}_n=x$ for all $x,y\in X$. Let $(X_1,*)$ be a commutative semigroup with a root function $\gamma_n$, and $(X_2,\otimes)$ be a partially ordered commutative semigroup with a root function $\delta_n$. A function $f:A\to X_2$ is called an $n$-convex function on a set $A\subset X_1$ with property that, for all $x_1\in A$ and $i=1,\ldots,n$, $\gamma_n(x_1*\ldots*x_n)\in A$ if it satisfies either $f(\gamma_n(x_1*\ldots*x_n))\leq\delta_n(\otimes_{i=1}^nf(x_1))$ or if the left hand and the right hand sides of the above inequality are incomparable for all $x_1\in A$ $(i=1,\ldots,n)$. This paper gives three theorems with respect to the properties of $n$-convex functions.