Closed inverse subsemigroups of graph inverse semigroups

It is easy to see that closed chains of idempotents are indeed closed inverse subsemigroups. For the cycle type, if $L=L_{p,d}$ then any two elements are suffix comparable, and we have

Now consider $(vp^{r}d,vp^{s}d)(wp^{j}d,wp^{k}d)$: write $p=v_{0}v$ and suppose that $s<j$. Then $wp^{j}d=wp^{j-s-1}v_{0}vp^{s}d$ and so

and a similar calculation applies if $s>j$. If $s=j$ and $v$ is a suffix of $w$, say $w=v_{1}v$, then

and a similar calculation applies if $s=j$ and $w$ is a suffix of $v$. Hence $L$ is a subsemigroup of $\mathcal{S}(\Gamma)$, and since the inverse of an element of $L$ is clearly also in $L$, we deduce that $L$ is an inverse subsemigroup of $\mathcal{S}(\Gamma)$. Since we ascend in the natural partial order in $L$ by deleting identical prefixes from the paths $vp^{r}d$ and $vp^{s}d$, or from a given suffix of $d$, it is also clear that $L$ is closed.

If $F$ is a closed inverse subsemigroup of $\mathcal{S}(\Gamma)$ and contains $(d,pd)$, then for any $m,n\geqslant 0$ we have $(pd,d)^{m}(d,pd)^{n}=(p^{m}d,d)(d,p^{n}d)=(p^{m}d,p^{n}d)\in F$. Ascending in the natural partial order, we may obtain any element of $L_{p,d}$, and so $L_{p,d}\subseteq F$.

Let $L$ be a closed inverse subsemigroup of $\mathcal{S}(\Gamma)$. If $w$ and $w^{\prime}$ are paths occurring in elements of $L$ and are not suffix comparable, then the product of the idempotents $(w,w)$ and $(w^{\prime},w^{\prime})$ in $L$ is equal to $0$, and so $0\in L$ and by closure $L=\mathcal{S}(\Gamma)$. Hence if $L$ is proper, any two paths occurring in elements of $L$ are suffix comparable and hence have the same terminal vertex. By definition, if $(u,v)\in\mathcal{S}(\Gamma)$ then $u,v$ have the same initial vertex: hence if $(u,v)\in L$ then $u,v$ have the same initial and the same terminal vertex in $\Gamma$. Suffix comparability then ensures that any proper closed inverse subsemigroup of $\mathcal{S}(\Gamma)$ consisting entirely of idempotents is either a finite or an infinite chain. We note that in the second case, in order to obtain directed paths of arbitrary length, $\Gamma$ must contain a directed circuit.

We shall now describe those closed inverse subsemigroups of $\mathcal{S}(\Gamma)$ which contain non-idempotent elements. Suppose that $L\neq E(L)$ is a closed inverse subsemigroup of $\mathcal{S}(\Gamma)$. Then there exists $(u,v)\in L,$ with $u\neq v$, and we may assume that the path $u$ is shorter than the path $v$. Hence $u$ is a suffix of $v$ and so $v=pu$ for some path $p$. Since $u$ and $v$ have the same initial and terminal vertices, $p$ must be a directed circuit in $\Gamma$. If $p$ and $u$ share a common prefix, with $p=ap_{1}$ and $u=au_{1}$ then

and so by closure, $(u_{1},p_{1}u)\in L$.

There exists $s\in S$ with $s^{-1}Hs\subseteq K\;\text{and}\;sKs^{-1}\subseteq H$. Let $k\in K$: then $k\neq 0$ and $sks^{-1}\in H$ and so $sks^{-1}\in E(S)$. It follows that $s^{-1}(sks^{-1})s=(s^{-1}s)k(s^{-1}s)\in E(S)$ and $(s^{-1}s)k(s^{-1}s)\leqslant k$. Since $S$ is $E^{*}$–unitary, we deduce that $k\in E(S)$.

(a) It follows from Lemma 3.5 that if $L$ has finite chain type then so does every closed inverse subsemigroup conjugate to $L$ .

Suppose that $L$ and $K$ have the same root $x_{0}\in V(\Gamma)$. Then $(u,v)\in\mathcal{S}(\Gamma)$, and for any suffix $w$ of $u$ we have

Similarly, for any suffix $t$ of $v$, $(u,v)(t,t)(v,u)=(u,u)\in L$. Hence $L$ and $K$ are conjugate.

Conversely, suppose that $L$ and $K$ are conjugate, with conjugating element $(p,q)\in\mathcal{S}(\Gamma)$, so that for any suffixes $w$ of $u$ and $t$ of $v$ we have

Then $(q,p)({\mathbf{r}}(u),{\mathbf{r}}(u))(p,q)\in K$, so that $p$ and ${\mathbf{r}}(u)$ are suffix-comparable: hence $p$ also ends at ${\mathbf{r}}(u)$, and $(q,p)({\mathbf{r}}(u),{\mathbf{r}}(u))(p,q)=(q,q)\in K$. Therefore $q$ is a suffix of $v$. Similarly, $p$ is a suffix of $u$.

Let $v=v_{1}q$: then

and so $v_{1}p$ is a suffix of $u$. Let $u=u_{0}v_{1}p$: then

and so $u_{0}v_{1}q$ is a suffix of $v$. But $v=v_{1}q$ and so $u_{0}$ is a vertex (namely the root of $L$), and $u=v_{1}p$. Hence $u$ and $v$ have the same initial vertex, and so $L$ and $K$ have the same root.

(b) By Lemma 3.5 any closed inverse subsemigroup $K$ conjugate to $L$ must be of chain type, and by part (a) $K$ must be infinite. Suppose that $(t,s)L(s,t)\subseteq K\;\text{and}\;(s,t)K(t,s)\subseteq L$. Since $0\not\in K$ we have, for all $(u,u)\in L$, that $s$ is suffix comparable with $u$ and similarly for all $(v,v)\in K$, that $t$ is suffix comparable with $v$. If we consider $u$ with $|u|\geqslant|s|$ then $s$ must be a suffix of $u$ and by closure of $L$ we have $(s,s)\in L$. Similarly $(t,t)\in K$. Suppose that $(ps,ps)\in L$. Then $(t,s)(ps,ps)(s,t)=(pt,pt)\in K$ and similarly if $(pt,pt)\in K$ then $(ps,ps)\in L$.

Conversely, if $s$ and $t$ exist as in the Theorem and $(w,w)\in L$ then $s$ is suffix comparable with $w$.
If $w$ is a suffix of $s$, with $s=hw$, then

and if $s$ is a suffix of $w$ with $w=ps$ then $(ps,ps)\in L$ and so

Similarly $(s,t)K(t,s)\subseteq L$, and $L$ and $K$ are conjugate.

(c) By parts (a) and (b), any closed inverse subsemigroup of $\mathcal{S}(\Gamma)$ that is conjugate to $L_{p,d}$ must be of cycle type. Suppose that the closed inverse subsemigroups $L_{p,d}$ and $L_{q,k}$ are conjugate in $\mathcal{S}(\Gamma)$, and so there exists $(s,t)\in\mathcal{S}(\Gamma)$ such that

Since $L_{q,k}$ is closed and $L_{p,d}$ is the smallest closed inverse subsemigroup of $\mathcal{S}(\Gamma)$ containing $(d,pd)$, then (3.1) is equivalent to $(t,s)\,(d,pd)\,(s,t)\in L_{q,k}$. Also, since $0\not\in L_{q,k}$ we must have $s\,$ suffix-comparable with $u$ and $v$ whenever $(u,v)$ is an element of $L_{p,d}\,$. Hence $(s,s)\in L_{p,d}\,$, and similarly $(t,t)\in L_{q,k}$.

First suppose that $s=up^{a}d\,$ and $t=vq^{b}k\,$ for some $a,b\geqslant 0$, where $u$ is a suffix of $p$ and $v$ is a suffix of $q$. Write $p=hu$: then

It follows that $uhvq^{b}k=vq^{m}k\,$ for some $m\geqslant 0$. Comparing lengths of these directed paths, we see that $m>b$, and then after cancellation we obtain $uhv=vq^{m-b}$. Hence $uh$ is conjugate to some power of $q$, and since $uh$ is a conjugate of $p$, we conclude that $p$ is conjugate to some power of $q$.

Now suppose that $s$ is a suffix of $d$ and write $d=cs$. With $t$ as before, we now obtain

It follows that $pcvq^{b}k=cvq^{m}k$ for some $m\geqslant 0$. Again $m>b$ and after cancellation we obtain $pcv=cvq^{m-b}$. Here we see directly that $p$ is conjugate to a power of $q$.

Now suppose that $s$ is a suffix of $d$ and write $d=cs$, and that $t$ is a suffix of $k$ and write $k=jt$. We now obtain

Since by assumption $p\,$ is not the empty path, we have $ct=wq^{a}k\,$ and $pct=wq^{b}k\,$ for some suffix $w$ of $q$ and some $a,b\geqslant 0$. Again comparing lengths, we see that $b>a$, and then $pct=pwq^{a}k=wq^{b}k$. After cancellation we obtain $pw=wq^{b-a}$ and again $p$ is conjugate to a power of $q$.

Hence for each possibility of $s$, we deduce from (3.1) that $p\,$ is conjugate to some power of $q\,$. Using equation (3.2) we deduce similarly that $q$ is conjugate to a power of $p\,$. Again comparing lengths, we conclude that $p$ and $q$ are conjugate.

(a) Let $L$ have finite chain type. A coset representative of $L$ has the form $(q,u)$ where $q$ is some suffix of $w$, and $(q,u)$ have the same initial vertex. If $L$ has infinite index, then there are infinitely many distinct choices for $(q,u)$ and since $\Gamma$ is finite, there must be a directed circuit in $\Gamma$ as described.

Conversely, suppose that $g,c\,$ exist. Let $q\,$ be the suffix of $w\,$ that has initial vertex $v_{0}\,$. Then $q\in L$ and so for any $k\geqslant 0\,$ the coset $C_{k}=L(q,gc^{k})^{\uparrow}$ exists. Now for $k>l,\,$ we have if $g$ is non-empty, that

and so by part (c) of Proposition 2.1, the cosets $C_{k}\,$ and $C_{l}\,$ are distinct. If $g$ is empty then we have

and again the cosets $C_{k}\,$ and $C_{l}\,$ are distinct.

(b) By part (a) there are no directed cycles accessible from any vertex of $w$, and so $N^{\Gamma}_{v,w}$ is finite for each vertex $v$ of $w$. A coset representative of $L$ has the form $(s,t)$ where $s$ is a suffix of $w$. Suppose that two such elements, $(s_{1},t_{1})$ and $(s_{2},t_{2})$, represent the same coset. Then $(s_{1},t_{1})(t_{2},s_{2})\in L$: in particular the product is non-zero and so $t_{1},t_{2}$ are suffix comparable. We may assume that $t_{2}=ht_{1}$: then $(s_{1},t_{1})(ht_{1},s_{2})=(hs_{1},s_{2})$ and this is in $L$ if and only if $s_{2}=hs_{1}$. Therefore $L(s_{1},t_{1})^{\uparrow}=L(s_{2},t_{2})^{\uparrow}$ if and only if $(s_{2},t_{2})=(hs_{1},ht_{1})$, and so the distinct coset representatives are the pairs $(s,t)$ where $s$ is a suffix of $w$, $s$ and $t$ have the same initial vertex, but do not share the same initial edge. It follows that the number of distinct cosets is $\sum_{v\in V(w)}N^{\Gamma}_{v,w}$, and $L$ itself is represented by $({\mathbf{r}}(w),{\mathbf{r}}(w))$.