Some properties of Higman-Thompson monoids and digital circuits

We define various monoid versions of the R. Thompson group $V$, and prove connections with monoids of acyclic digital circuits. We show that the monoid $M_{2,1}$ (based on partial functions) is not embeddable into Thompson's monoid ${\sf tot}M_{2,1}$, but that ${\sf tot}M_{2,1}$ has a submonoid that maps homomorphically onto $M_{2,1}$. This leads to an efficient completion algorithm for partial functions and partial circuits. We show that the union of partial circuits with disjoint domains is an element of $M_{2,1}$, and conversely, every element of $M_{2,1}$ can be decomposed efficiently into a union of partial circuits with disjoint domains.