  A directed graph is semi-transitive if and only if it is acyclic and for any
directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq
2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow
u_j$ 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 $n\times m$ matrices over $\{-1,0,1\}$
with a single natural condition.
