By a *digraph,* let us mean an ordered pair $(X,r)$ with $r : X \times X \rightarrow B,$ where $X$ is a set and $B = \{\mathrm{False}, \mathrm{True}\}.$

Then supposing $\mathbb{X} =(X,r)$ is a digraph, we may ask: under what circumstances is $r$ the forgetful image of a homomorphism $\mathbb{X}^\mathrm{op} \times \mathbb{X} \rightarrow \mathbb{B},$ where $\mathbb{B} = \{\mathrm{False} \leq \mathrm{True}\}$ is the Boolean domain? Note that $\mathbb{B} = (B,\leq)$ is also a digraph.

Unpacking the definitions, we observe that $\mathbb{X} = (X,\rightarrow)$ has the above property iff it is *semitransitive,* by which I mean that for all $x,y,a,b \in X,$ we have the following, where the fraction line denotes entailment.

$$\dfrac{x \rightarrow a \rightarrow b \rightarrow y}{x \rightarrow y}$$

Now observe that transitivity implies semitransitivity, and that in the presence of reflexivity, these properties are equivalent. However, not every semitransitive digraph $\mathbb{X} = (X,\rightarrow)$ is transitive. For a minimal counterexample, consider the digraph $\mathbb{X} = \{x \rightarrow a \rightarrow y\}$ with three distinct elements and two arrows. $\mathbb{X}$ is vacuously semitransitive because there is no sequence of three arrows with respect to which the conclusion of the semitransitivity condition needs to be checked. However, $\mathbb{X}$ fails to be transitive because $x \rightarrow y$ is false.

**Question.** I'm looking for interesting or "natural" examples of semitransitive digraphs that fail to be transitive. Ideas, anyone?