Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ {\cdot m}}\longrightarrow M\overset{ {\cdot m}}\longrightarrow \cdots $$ We denote the space of this mapping telescope by $M_\infty$. The sequence of the above mapping telescope induces a sequence of homomorphisms of Pontrjagin rings on homology $$ H_*(M)\overset{ {(\cdot m)_*}}\longrightarrow H_*(M)\overset{ {(\cdot m)_*}}\longrightarrow H_*(M)\overset{ {(\cdot m)_*}}\longrightarrow \cdots $$ By Homology of mapping telescope, the direct limit of the above sequence on homology is $H_*(M_\infty)$.
In the paper Homology fibrations and group completion theorem, by McDuff-Segal, page 281, line 13-14: If $H_*(M)[\pi_0(M)^{-1}]$ can be formed by "right fractions" (cf. page 279, bottom line 4-7, Homology fibrations and group completion theorem) then the direct limit of the above sequence on homology is precisely $H_*(M)[\pi_0(M)^{-1}]$.
My question: (1). how to use "right fractions" to prove that the direct limit of the above sequence on homology is precisely $H_*(M)[\pi_0(M)^{-1}]$?
(2). given a specific topological monoid $M$, is there a direct way to check whether $M$ satisfies the condition: $H_*(M)[\pi_0(M)^{-1}]$ can be formed by "right fractions"?