Tag: semigroup

In the paper https://arxiv.org/pdf/1209.2578.pdf, Krieger is simply defining his notion of $mathcal{R}$-graph. It is an object $mathcal{G}_{mathcal{R}}(mathfrak{P},mathcal{E}^-,mathcal{E}^+)$, where $mathfrak{P}$ is the vertex set of a finite directed graph with edge set $mathcal{E}$, ${mathcal{E}^-,mathcal{E}^+}$ is a partition of $mathcal{E}$, the graph satisfies certain conditions, and $mathcal{R}$ is a relation on the edges that also satisfies certain conditions.

Specifically, for $mathfrak{q},mathfrak{r}inmathfrak{P}$ we define $mathcal{E}^-(mathfrak{q},mathfrak{r})$ to be the set of edges in $mathcal{E}^-$ from $mathfrak{q}$ to $mathfrak{r}$ and $mathcal{E}^+(mathfrak{q},mathfrak{r})$ to be the set of edges in $mathcal{E}^+$ from $mathfrak{r}$ to $mathfrak{q}$. For each $mathfrak{q},mathfrak{r}inmathfrak{P}$ we require that either both of these sets be empty or both be non-empty, and we further require that the directed graph $langlemathfrak{P},mathcal{E}^-rangle$ (and hence also ($langlemathfrak{P},mathcal{E}^+rangle$) be strongly connected.

The relation $mathcal{R}$ is the union of relations $mathcal{R}(mathfrak{q},mathfrak{r})subseteqmathcal{E}^-(mathfrak{q},mathfrak{r})timesmathcal{E}^+(mathfrak{q},mathfrak{r})$ for $mathfrak{q},mathfrak{r}inmathfrak{P}$.

Further, Kreiger construct a semigroup (with zero) $mathcal S(mathcal{G}_{mathcal{R}}(mathfrak{P},mathcal{E}^-,mathcal{E}^+))$ from an $R$-graph $mathcal{G}_{mathcal{R}}(mathfrak{P},mathcal{E}^-,mathcal{E}^+)$. The semigroup $mathcal S(mathcal{G}_{mathcal{R}}(mathfrak{P},mathcal{E}^-,mathcal{E}^+))$ contains an idempotent $1_{mathfrak p}, mathfrak p in mathfrak{P}$ and $mathcal E$ as a generating set. Besides $1_{mathfrak p}^2 = 1_{mathfrak p}, mathfrak p in mathfrak{P}$, the defining relations are
$$f^-g^+ = left{ begin{array}{cl}
1_{mathfrak{q}} & text{if} ; f^{-} in mathcal{E^{-}}(mathfrak{q}, mathfrak{r}), ; g^{+} in mathcal{E^{+}}(mathfrak{q}, mathfrak{r}), ; (f^{-}, g^{+}) in mathcal{R}(mathfrak{q}, mathfrak{r}), ; mathfrak q, mathfrak{r} in mathfrak{P},\
0 & text{if} ; f^{-} in mathcal{E^{-}}(mathfrak{q}, mathfrak{r}), ; g^{+} in mathcal{E^{+}}(mathfrak{q}, mathfrak{r}), ; (f^{-}, g^{+}) notin mathcal{R}(mathfrak{q}, mathfrak{r}), ; mathfrak q, mathfrak{r} in mathfrak{P},\
0 & text{if} ; f^{-} in mathcal{E^{-}}(mathfrak{q}, mathfrak{r}), ; g^{+} in mathcal{E^{+}}(mathfrak{q}, mathfrak{r}), ; mathfrak{q}, ; mathfrak{q’}, mathfrak{r} in mathfrak{P}, mathfrak{q} ne mathfrak{q’},\
end{array}right.$$

and

begin{array}{cl}
1_{mathfrak{q}}e^{-}= e^{-} 1_{mathfrak{r}} = e^{-}, & e^{-}in mathcal{E}(mathfrak{q}, mathfrak{r}) \
1_{mathfrak{r}}e^{+}= e^{+} 1_{mathfrak{q}} = e^{+}, & e^{+}in mathcal{E}(mathfrak{q}, mathfrak{r}), & mathfrak q, mathfrak{r} in mathfrak{P},
end{array}

and

begin{array}{cl}
1_{mathfrak{q}}
1_{mathfrak{r}} = 0 & mathfrak q, mathfrak{r} in mathfrak{P}, & mathfrak q ne mathfrak{r}.
end{array}

My question is

What is the value of $g^{+}f^{-}$, where $f^{-} in mathcal{E^{-}}(mathfrak{q}, mathfrak{r}), g^{+} in mathcal{E^{+}}(mathfrak{q}, mathfrak{r})$? I think it should be $1_{mathfrak{r}}$ but I don’t know how?