Intersection of two monoids generated by two element codes

This article provides a formalization of the classification of intersection \( \{x,y\}^* \cap \{u,v\}^*\) of two monoids generated by two element codes. Namely, the intersection has one of the following forms \( \{\beta,\gamma\}^* \quad \text{ or } \quad \left(\beta_0 + \beta(\gamma(1+\delta+ \cdots + \delta^t))^*\epsilon\right)^*.\) Note that it can be infinitely generated. The result is due to [Karhumäki, 84]. Our proof uses the terminology of morphisms which allows us to formulate the result in a shorter and more transparent way.