A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1rightarrow u_2rightarrow cdots rightarrow u_t$, $t geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_irightarrow u_j$ exist for $1 leq i < j leq t$. In this paper, we study semi-transitivity of families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving in the limit infinite directed split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. We fully classify semi-transitive infinite directed split graphs when a morphism in question can involve any $ntimes m$ matrices over ${-1,0,1}$ with a single natural condition.
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