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.

翻译：我们定义了R. Thompson群$V$的多种幺半群版本，并证明了它们与无环数字电路幺半群之间的联系。我们证明了基于部分函数的幺半群$M_{2,1}$不能嵌入Thompson幺半群${\sf tot}M_{2,1}$，但${\sf tot}M_{2,1}$存在一个子幺半群，该子幺半群可同态映射到$M_{2,1}$上。这为部分函数与部分电路提供了一种高效的完备化算法。我们证明了定义域互不相交的部分电路的并集属于$M_{2,1}$；反之，$M_{2,1}$中的每个元素均可高效分解为定义域互不相交的部分电路的并集。