Semi-transitivity of directed split graphs generated by morphisms

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.