  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.
