2. Preliminaries

2.3. Monogenic free inverse semigroups

For background on inverse semigroups, we refer the reader to any standard textbook such as [Reference Clifford and Preston3, Reference Howie9, Reference Lawson11, Reference Petrich16]. It is well known that free objects exist for inverse semigroups; for an account, see [Reference Lawson11, Chapter 6]. In this paper, we focus solely on the monogenic free inverse semigroup, which will be denoted by $FI_1$ . It can be represented as follows. The elements of $FI_1$ are certain triples:

\begin{equation*} FI_1 \,:\!=\, \bigl \{({-}a, p, b)\in \mathbb {Z}^3 \,:\, a, b \geq 0, \, a+b \gt 0, \, -a \leq p \leq b \bigr \}, \end{equation*}

and the multiplication and inversion are as follows:

\begin{align*} \begin{aligned} ({-}a_1, p_1, b_1)({-}a_2, p_2, b_2) &= ({-}\max (a_1, a_2-p_1), p_1+p_2, \max (b_1, b_2+p_1)),\\ ({-}a, p, b)^{-1} &= ({-}(a+p), -p, b-p). \end{aligned} \end{align*}

An element $({-}a, p, b)\in FI_1$ is an idempotent if and only if $p = 0$ .

Some readers may be more used to viewing the elements in a free inverse semigroup as Munn trees; see [Reference Munn14] and [Reference Lawson11, Chapter 6]. For $FI_1$ the Munn trees are simply directed paths with inital and terminal vertices, and the translation between the two representations is straightforward. The triple $({-}a,p,b)$ corresponds to the following Munn tree:

In the figure, the initial and terminal vertices are indicated by the ‘loose’ in- and out-arrows, respectively. The sign of $p$ is positive if and only if the terminal vertex lies on the right-hand side of the initial vertex. The numbers of edges between the initial and leftmost vertices, the initial and rightmost vertices, and the initial and terminal vertices correspond to $a$ , $b$ and $\vert p \vert$ , respectively.

We now give a semigroup presentation for $FI_1$ .

Proposition 2.1 ([Reference Schein21, Lemma 1]). The monogenic free inverse semigroup $FI_1$ is defined by:

\begin{equation*} \langle x, x^{-1} \mid \mathfrak {R} \rangle \end{equation*}

via $\phi \colon x\mapsto (0, 1, 1),\ x^{-1}\mapsto ({-}1, -1, 0)$ , with

\begin{equation*} \mathfrak {R} \,:\!=\, \bigl \{(xx^{-1}x, x), (x^{-1}xx^{-1}, x^{-1})\bigr \}\\ \cup \bigl \{ (x^{-i}x^ix^jx^{-j}, x^{j}x^{-j}x^{-i}x^{i})\colon i, j \in \mathbb {N}_0,\ i+j\gt 0)\bigr \}. \end{equation*}

Note that, with the notation as above, we have $(x^{-a}x^{a}x^{b}x^{-b}x^{p})\phi =({-}a, p, b)$ .

We partition $FI_1$ into its $\mathscr{D}$ -classes:

\begin{equation*} D_n\,:\!=\, \bigl \{ ({-}a,p,b)\in FI_1\colon a+b=n\bigr \} \quad (n\in \mathbb {N}). \end{equation*}

These arise from the standard Green theory (see [Reference Howie9, Chapter 2]), but this will not be needed here, nor will any of the other Green’s equivalences $\mathscr{L},\mathscr{R},\mathscr{H},\mathscr{J}$ . For every $n\in \mathbb{N}$ , the union $\bigcup _{i\geq n} D_i$ is an ideal of $FI_1$ .

Figure 1. Semilattice of idempotents of $FI_1$ .

We say that an element $({-}a, p, b)$ is positive, negative or of sign $0$ if $p \gt 0$ , $p \lt 0$ or $p = 0$ , respectively. The elements of sign $0$ are precisely the idempotents of $FI_1$ . The set of all idempotents of $FI_1$ is partially ordered by:

\begin{equation*} ({-}a, 0, b) \leq ({-}c, 0, d) \Leftrightarrow a \geq c \text { and } b \geq d. \end{equation*}

This actually coincides with the natural partial order arising from general theory (see [Reference Howie9, Chapter 5] or [Reference Lawson11, Chapter 1]). The resulting partially ordered set is isomorphic to $\mathbb{N}_0\times \mathbb{N}_0\setminus \{(0,0)\}$ and is depicted in Figure 1. In the diagram, the idempotents are in the same $\mathscr{D}$ -class if and only if they are on the same horizontal level. Two idempotents $({-}a, 0, b)$ and $({-}c, 0, d)$ are incomparable if and only if $(a-c)(b-d)\lt 0$ .

3. Infinitely many idempotents

The aim of this section is to prove the following proposition which is the forward direction of the Main Theorem.

The proof is somewhat technical, but the underlying idea is as follows. It can be viewed as a variation of Schein’s original proof of Theorem 1. There, he exhibits two specific partial transformations, namely $\alpha _n$ and $\beta _n$ on $[0,n]$ . The key strategy of Schein’s proof is to assume, aiming for contradiction, that $FI_1$ is finitely presented as a semigroup. Schein then fixes an arbitrary finite presentation wih respect to the generators $\{a,a^{-1}\}$ and shows that, for a sufficiently large $n$ , the semigroup $\langle \alpha _n, \beta _n\rangle$ satisfies all these finitely many relations and yet fails to preserve commutativity of infinitely many pairs of idempotents. The essence of our adaptation is to find a map from a finitely generated subsemigroup of $FI_1$ containing infinitely many idempotents to the semigroup $\langle \alpha _n, \beta _n \rangle$ . We will show that this map respects multiplication in finitely many $\mathscr{D}$ -classes of $FI_1$ , but yet again fails to preserve commutativity of infinitely many pairs of idempotents.

We start our proof by characterising when a finitely generated subsemigroup of $FI_1$ has infinitely many idempotents:

Proof. The equivalence of (ii) and (iii) is clear.

(iii) $\Rightarrow$ (i). Assume there are elements $u_1 = ({-}a_1, p_1, b_1)$ and $u_2 = ({-}a_2, -p_2, b_2)$ in $A$ where $p_1, p_2 \gt 0$ . Let $m$ be a common multiple of $p_1$ and $p_2$ so that $m = n_1p_1 = n_2p_2$ for some $n_1, n_2 \in \mathbb{N}$ . A straightforward calculation shows that

\begin{equation*} u_1^{n_1x}u_2^{n_2x} = ({-}\max (a_1, a_2-p_2), 0, \max (n_1xp_1 + (b_1-p_1), n_1xp_1+b_2)) \end{equation*}

gives distinct idempotents for each $x \in \mathbb{N}$ .

(i) $\Rightarrow$ (iii). Suppose $A$ consists entirely of non-positive or non-negative elements. Without loss of generality, we assume the latter. Hence, the product of elements of $A$ is an idempotent if and only if each factor in the product is an idempotent. There are only finitely many idempotents (sign $0$ elements) in $A$ , which gives finitely many idempotents in $\langle A \rangle =S$ .

Figure 2. The red dots are idempotents of the form $u_1^{n_1x}u_2^{n_2x}u_2^{n_2y}u_1^{n_1y}$ where $x, y \in \mathbb{N}_0$ with $x+y \gt 0$ . The rightmost and leftmost idempotents are $u_1^{n_1x}u_2^{n_2x}$ and $u_2^{n_2y}u_1^{n_1y}$ , where $x, y \in \mathbb{N}$ , respectively.

The following lemma introduces two partial transformations $\alpha _n$ and $\beta _n$ . It was the key technical observation in [Reference Schein21] towards proving the main result there (Theorem 1).

Lemma 3.4 ([Reference Schein21, Lemma 2]). For $n\in \mathbb{N}$ define two partial transformations on $\{0, \dots, n\}$ :

\begin{align*} \alpha _n &={\biggl (\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 1 & 2 & \cdots & n-2 & n-1 & n \\[3pt] 1 & 2 & 3 & \cdots & n-1 & n & - \end{array}\biggr ),} \\ \beta _n &={ \biggl (\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 1 & 2 & \cdots & n-2 & n-1 & n \\[3pt] 0 & 0 & 1 & \cdots & n-3 & n-2 & n-1 \end{array}\biggr )}.\end{align*}

Then, the following hold.

1. (i) $\alpha _n\beta _n\alpha _n = \alpha _n$ ;

2. (ii) $\beta _n\alpha _n\beta _n = \beta _n$ ;

3. (iii) $\beta _n^{i}\alpha _n^{i}\alpha _n^{j}\beta _n^{j} = \alpha _n^{j}\beta _n^{j}\beta _n^{i}\alpha _n^{i}$ when $i\gt n$ or $j\gt n$ or $i+j \leq n$ ;

4. (iv) $\beta _n^{i}\alpha _n^{i}\alpha _n^{j}\beta _n^{j} \neq \alpha _n^{j}\beta _n^{j}\beta _n^{i}\alpha _n^{i}$ when $i+j \gt n$ and $i, j \leq n$ .

We will also make use of the following technical lemma concerning presentations and ideals.

Lemma 3.5. Let $S$ be a semigroup defined by $\langle A \mid R \rangle$ via $\theta \colon A \to S$ . Let $I$ be an ideal of $S$ , let $C\,:\!=\,S\setminus I$ , and let

\begin{equation*} A_C \,:\!=\, \{a\in A \,:\, a\theta \in C\},\quad R_C \,:\!=\, \bigl \{(r, s)\in R \,:\, r\theta =s\theta \in C\bigr \}. \end{equation*}

Let $T$ be a semigroup defined by $\langle A_C \mid R_C \rangle$ via $\pi \colon A_C\to T$ . Suppose $u,v\in A^+$ are such that $u\theta = v\theta \in C$ . Then, $u, v \in A_C^{+}$ and $u\pi = v\pi$ .

Proof. Consider an elementary sequence from $u$ to $v$ with respect to $R$ ; let $n$ be its length. That all the words in the sequence are in $A_C^+$ follows from $I$ being an ideal. We claim that only relations from $R_C$ are used in the sequence. Inductively, it is sufficient to prove this in the case of sequences of length $1$ . So, suppose $u=prq$ , $v=psq$ , where $p,q\in A_C^\ast$ and $(r,s)$ or $(s,r)$ is in $R$ . Again, since $I$ is an ideal, we must in fact have $r\theta =s\theta \in C$ , and it follows that $u\pi = v\pi$ , as required.

Recall from Proposition 2.1 that $FI_1$ is defined by $\langle x,x^{-1}\mid \mathfrak{R} \rangle$ via $\phi \colon x \mapsto (0, 1, 1),\ x^{-1}\mapsto ({-}1, -1, 0)$ . For $n\in \mathbb{N}$ , define a map

\begin{equation*} \psi _n \colon \{x, x^{-1}\} \to \langle \alpha _n, \beta _n \rangle, \ x\mapsto \alpha _n,\ x^{-1}\mapsto \beta _n. \end{equation*}

Note that, of course, $\psi _n$ extends to a homomorphism $\{x,x^{-1}\}^+\rightarrow \langle \alpha _n,\beta _n\rangle$ . However, this homomorphism does not factor through $FI_1$ , since $\alpha _n$ and $\beta _n$ do not satisfy all the relations from $\mathfrak{R}$ according to Lemma 3.4. Let $C_n\,:\!=\,\bigcup _{i=1}^{n}D_i$ and $I_n\,:\!=\,\bigcup _{i=n+1}^{\infty }D_i$ . Clearly, these two sets partition $FI_1$ , and, as observed earlier, $I_n$ is an ideal. The following is a special case of Lemma 3.5.

Proof. Using the notation from Lemma 3.5, we have

\begin{align*} A_{C_n} &= \{x, x^{-1}\},\\ \mathfrak{R}_{C_n} &= \bigl \{(r, s)\in \mathfrak{R} \,:\, r\phi =s\phi \in C_n\bigr \}\\ & = \bigl \{(xx^{-1}x, x), \, (x^{-1}xx^{-1}, x^{-1})\bigr \}\cup \bigl \{ (x^{-i}x^ix^jx^{-j}, x^{j}x^{-j}x^{-i}x^{i})\colon 0 \lt i+j \leq n \bigr \}. \end{align*}

Let $T_n$ be the semigroup defined by $\langle A_{C_n} \mid \mathfrak{R}_{C_n}\rangle$ via $\pi _n\colon A_{C_n} \to T_n$ . By Lemma 3.5, we have $u\pi _n = v\pi _n$ . By Lemma 3.4 (i)–(iii), $\langle \alpha _n, \beta _n \rangle$ satisfies the relations in $\mathfrak{R}_{C_n}$ . Hence, $\langle \alpha _n, \beta _n \rangle$ is a homomorphic image of $T_n$ with $\mathfrak{R}_{C_n}^{\sharp } \subseteq \ker \psi _n$ , and so $u\psi _n = v\psi _n$ .

We will now define a map $\sigma _n\colon C_n\rightarrow \langle \alpha _n, \beta _n \rangle$ for each $n\in \mathbb{N}$ as follows. For every $s \in FI_1$ , we choose an arbitrary $w_s\in \{x,x^{-1}\}^{+}$ representing $s$ , that is, such that $w_s\phi = s$ . We then let $s\sigma _n \,:\!=\, w_s\psi _n$ . The next lemma plays a significant role in the proof of Proposition 3.1. It presents two situations: one in which $\sigma _n$ behaves like a homomorphism by preserving multiplication in finitely many $\mathscr{D}$ -classes of $FI_1$ , and the other where it does not.

Proof. For brevity, we write $\alpha, \beta, \psi, \sigma$ for $\alpha _m, \beta _m, \psi _m, \sigma _m$ , respectively.

(i) Write $s_1=({-}a_1, p_1, b_1)$ and $s_2=({-}a_2, p_2, b_2)$ where $a_1+b_1, a_2+b_2 \leq n$ . By definition,

\begin{equation*} s_1s_2 = ({-}\max (a_1, a_2-p_1), p_1+p_2, \max (b_1, b_2+p_1)). \end{equation*}

Since

\begin{equation*} \max (a_1, a_2-p_1)+\max (b_1, b_2+p_1)\leq 3n \leq m, \end{equation*}

it follows that $s_1s_2 \in C_m$ .

We now show that $(s_1s_2)\sigma = (s_1\sigma )(s_2\sigma )$ . By the definition of $\sigma$ , we have

\begin{equation*} (s_1s_2)\sigma = (w_{s_1s_2})\psi \quad \text {and}\quad (s_1\sigma )(s_2\sigma ) = (w_{s_1}\psi )(w_{s_2}\psi ) = (w_{s_1}w_{s_2})\psi . \end{equation*}

Note that $w_{s_1s_2}\phi = s_1s_2 = (w_{s_1}w_{s_2})\phi$ . Since $s_1s_2 \in C_m$ , Lemma 3.6 gives $w_{s_1s_2}\psi = (w_{s_1}w_{s_2})\psi$ , and the result follows.

(ii) Let $e = ({-}a, 0, b)$ and $f = ({-}c, 0, d)$ . The assumptions $e\in C_n$ , $f\in D_m$ and incomparability of $e$ and $f$ imply $a+b \leq n$ , $c+d=m$ and $(a-c)(b-d)\lt 0$ . Without loss of generality, we assume that $a \gt c$ and $b \lt d$ . Note that $w_e\phi = e = (x^{-a}x^ax^bx^{-b})\phi$ , $w_f\phi = f = (x^{-c}x^cx^dx^{-d})\phi$ . Using $e, f\in C_m$ and Lemma 3.6 again, we deduce that

\begin{equation*} e\sigma =w_e\psi =(x^{-a}x^ax^bx^{-b})\psi =\beta ^a\alpha ^a\alpha ^b\beta ^b\quad \text {and similarly} \quad f\sigma =\beta ^c\alpha ^c\alpha ^d\beta ^d. \end{equation*}

Now we consider the mappings $(e\sigma )(f\sigma )$ and $(f\sigma )(e\sigma )$ . We compute the following:

\begin{align*} (e\sigma )(f\sigma ) &= \beta ^a\alpha ^a\alpha ^b\beta ^b\beta ^c\alpha ^c\alpha ^d\beta ^d\\ &=\beta ^a\alpha ^a\beta ^c\alpha ^c\alpha ^b\beta ^b\alpha ^d\beta ^d && \text{by Lemma 3.4 (iii) since } b+c\leq m\\ &=\beta ^a\alpha ^a\alpha ^d\beta ^d&& \text{since } \alpha ^c\beta ^c\alpha ^c = \alpha ^c \text{ and } \alpha ^b\beta ^b\alpha ^b = \alpha ^b,\\ (f\sigma )(e\sigma ) &= \beta ^c\alpha ^c\alpha ^d\beta ^d\beta ^a\alpha ^a\alpha ^b\beta ^b\\ &=\alpha ^d\beta ^d\alpha ^b\beta ^b\beta ^c\alpha ^c\beta ^a\alpha ^a &&\text{by Lemma 3.4 (iii) since } c+d,a+b,b+c\leq m\\ &=\alpha ^d\beta ^d\beta ^a\alpha ^a&&\text{since $\beta ^b\alpha ^b\beta ^b = \beta ^b$ and $\beta ^c\alpha ^c\beta ^c = \beta ^c$.} \end{align*}

Since $a+d \gt c+d = m$ , $(e\sigma )(f\sigma ) \neq (f\sigma )(e\sigma )$ by Lemma 3.4 (iv).

We are now ready to prove Proposition 3.1. A key idea is that if a finitely generated subsemigroup of $FI_1$ has infinitely many idempotents, then we can find large enough natural numbers $n$ and $m$ as in Lemma 3.7.

Proof of Proposition 3.1. Let $S$ be a finitely generated subsemigroup of $FI_1$ which contains infinitely many idempotents. Aiming for a contradiction, suppose that $S$ is defined by a finite presentation $\langle A \mid R \rangle$ via $\theta \colon A \to S$ . Let $\eta \colon A^+ \to \{x, x^{-1}\}^+$ be the homomorphism defined by $a\eta = w_{a\theta }$ for $a\in A$ . Note that $a\eta \phi = w_{a\theta }\phi = a\theta$ , which shows $\eta \phi = \theta$ .

By Lemma 3.2, $S$ contains both positive and negative elements. Recall that $S$ particularly contains some $u_1 = ({-}a_1, p, b_1)$ and $u_2 = ({-}a_2, -p, b_2)$ with $p \gt 0$ (Remark 3.3 (1)). Let $n\in \mathbb{N}$ be such that $C_n$ contains all of the following elements:

• • an idempotent of the form $u_1^xu_2^xu_2^yu_1^y$ where $x,y\in \mathbb{N}_0$ with $x+y \gt 0$ ;

• • $a\theta$ for each $a\in A$ ;

• • $r\theta \,\, (= s\theta )$ for each $(r,s)\in R$ .

Let $m\in \mathbb{N}$ be such that $m \geq 3n$ and $D_m$ contains an idempotent of the form $u_1^xu_2^xu_2^yu_1^y$ . Recall the map $\sigma _m \colon C_m \to \langle \alpha _m, \beta _m \rangle$ and the natural homomorphism $\psi _m\colon \{x, x^{-1}\}^+ \to \langle \alpha _m, \beta _m \rangle$ . Figure 3 shows the diagram of maps and homomorphisms we have defined.

Figure 3. Homomorphisms and maps in the proof of Proposition 3.1. Here, $\iota$ denotes the inclusion mapping. The upper part of the diagram is commutative, that is, $\theta \iota = \eta \phi$ . Note that the domain of $\sigma _m$ is $C_m \subseteq FI_1$ .

From now on, we take $m$ and $n$ to be fixed and we write $\alpha, \beta, \psi$ and $\sigma$ for $\alpha _m, \beta _m, \psi _m$ and $\sigma _m$ , respectively. We claim that

(1) \begin{equation} \ker \theta \subseteq \ker \eta \psi . \end{equation}

Suppose $(r, s)\in R$ . Note that $r\eta \phi = (w_{r\theta })\phi = r\theta = s\theta = (w_{s\theta })\phi = s\eta \phi$ . Moreover, $r\theta \,(=s\theta ) \in C_n\subseteq C_m$ . Hence, $r\eta \psi = s\eta \psi$ by Lemma 3.6 and so indeed we have (1).

By our construction, there are idempotents of the form $u_1^xu_2^xu_2^yu_1^y$ in $C_n\cap S$ and $D_m\cap S$ . Pick two idempotents $e$ and $f$ of this form such that $e\in C_n\cap S$ , $f\in D_m\cap S$ and $e$ and $f$ are incomparable (see Remark 3.3 (3)). Now, let $v_e,v_f\in A^+$ be such that $v_e\theta = e$ and $v_f\theta = f$ . Since $ef = fe$ in $S$ , we have

\begin{equation*} (v_ev_f)\theta = (v_e\theta )(v_f\theta ) = ef=fe= (v_f\theta )(v_e\theta )=(v_fv_e)\theta, \end{equation*}

i.e. $(v_ev_f, v_fv_e)\in \ker \theta$ . Therefore, $(v_ev_f)\eta \psi = (v_fv_e)\eta \psi$ by (1).

Note that we have $v_e\eta \phi = v_e\theta = e = w_e\phi$ and $e\in C_m$ . By Lemma 3.6, $v_e\eta \psi = w_e\psi$ . Similarly, $v_f\eta \psi = w_f\psi$ . Then,

\begin{equation*} (v_ev_f)\eta \psi = (v_e\eta \psi )(v_f\eta \psi ) = (w_e\psi )(w_f\psi ) = (e\sigma )(f\sigma ), \end{equation*}

and, analogously, $(v_fv_e)\eta \psi = (f\sigma )(e\sigma )$ . However, $(e\sigma )(f\sigma ) \neq (f\sigma )(e\sigma )$ by Lemma 3.7 (ii). We have obtained a contradiction from the assumption that $S$ is finitely presented, and hence the proposition follows.

4. Finitely many idempotents

In this section, we prove the following proposition and thus complete the proof of the Main Theorem.

In our proof, we will make use of the following three results:

An equivalence relation $\rho$ on a semigroup $S$ is called a congruence if it is compatible with the multiplication, that is if $(x/\rho )(y/\rho )\subseteq (xy)/\rho$ for all $x,y\in S$ ; see [Reference Howie9, Section 1.5]. The index of an equivalence relation is the number of its equivalence classes.

The following proposition has already been discussed in the Introduction.

To prove Proposition 4.1, we first consider a finitely generated subsemigroup $S$ of $FI_1$ wholly consisting of positive elements, that is,

\begin{equation*} S\leq P\,:\!=\, \bigl \{({-}a, p, b)\in FI_1 \colon p\gt 0 \bigr \} \: (\leq FI_1). \end{equation*}

For such an $S$ , we will establish a decomposition into a disjoint union of subsemigroups of $\mathbb{N}$ , in such a way that Propositions4.3 and 4.4 can be applied. Finally, an appeal to Proposition 4.2 and Lemma 3.2 will enable us to extend to an arbitrary $S$ .

We begin to implement this programme. The first two lemmas discuss a decomposition of $P$ into infinitely many copies of $\mathbb{N}$ .

Lemma 4.5. For $x \in \mathbb{N}_0$ and $y\in \mathbb{N}$ define

\begin{equation*} N_{x,y} \,:\!=\, \bigl \{({-}x, p, p+(y-1)) \colon p\in \mathbb {N}\bigr \}. \end{equation*}

1. (i) Each $N_{x,y}$ is a subsemigroup of $P$ isomorphic to $\mathbb{N}$ .

2. (ii) The subsemigroups $N_{x,y}$ are pairwise disjoint.

3. (iii) $P=\bigcup \bigl \{ N_{x,y}\colon x\in \mathbb{N}_0,\ y\in \mathbb{N}\bigr \}$ .

Proof. (i) follows by observing that $N_{x,y}$ is generated by $({-}x,1,y)$ , while (ii) and (iii) are obvious.

Note that the last component of an element of $N_{x,y}$ is the sum of the middle component and $y-1$ . To emphasise this, we will sometimes write $({-}x, p, p+(y-p))$ for $({-}x, p, y)$ in what follows.

Lemma 4.6. For $n\in \mathbb{N}_0$ define

\begin{equation*} T_n\,:\!=\,\bigcup \bigl \{N_{x,y} \colon 0 \leq x \leq n,\, 1\leq y \leq n+1\bigr \}. \end{equation*}

1. (i) Each $T_n$ is a subsemigroup of $P$ .

2. (ii) $T_0\leq T_1\leq T_2\leq \dots$ .

3. (iii) $P=\bigcup _{n\in \mathbb{N}_0} T_n$ .

Proof. (i) Note that

\begin{equation*} ({-}a, p, b) \in T_n\quad \Leftrightarrow \quad p\gt 0\text { and } a,b-p \in [0,n]. \end{equation*}

Suppose $({-}a, p, b), ({-}c, q, d) \in T_n$ . Then

\begin{equation*} ({-}a, p, b)({-}c, q, d) =({-}\max (a, c-p), p+q, \max (b, d+p)). \end{equation*}

Since $a,c\in [0,n]$ it follows that $\max (a, c-p) \in [0,n]$ as well. Next, from $b-p,d-q\in [0,n]$ it follows that $b-(p+q)\leq n$ and $(d+p)-(p+q)=d-q\in [0,n]$ . Hence, $\max (b,d+p)-(p+q)\in [0,n]$ . Since $p+q\gt 0$ , it follows that $({-}a, p, b)({-}c, q, d)\in T_n$ , and therefore $T_n\leq P$ .

(ii) is obvious, and (iii) follows from Lemma 4.5 (iii).

We now turn to investigate a finitely generated subsemigroup of $P$ . The first result follows easily from the previous lemma:

Proof. As $S$ is finitely generated, there must exist $n\in \mathbb{N}_0$ such that $S\subseteq T_n$ because of Lemma 4.6 (ii), (iii). But $T_n$ in turn is a union of finitely many $N_{x,y}$ by definition, and the result follows.

As a key technical step, we will now show that a finitely generated subsemigroup $S$ of $P$ with an additional technical condition is finitely presented.

Lemma 4.8. Let $S \leq P$ be generated by $A = \bigl \{({-}a_1, p_1, b_1), \dots, ({-}a_n, p_n, b_n)\bigr \}$ . Suppose that for all $i, j \in [n]$ we have

(2) \begin{equation} a_i \geq a_j - p_i \;\;\text{ and }\;\; b_i \leq b_j + p_i. \end{equation}

For each $(x,y)\in I\,:\!=\,\{a_1, \dots, a_n\}\times \{b_1-p_1, \dots, b_n-p_n\}$ , define

\begin{equation*} S_{x,y} \,:\!=\, N_{x,y+1}\cap S. \end{equation*}

Then the following hold.

1. (i) The sets $S_{x,y}$ , $(x,y)\in I$ , partition $S$ .

2. (ii) The equivalence relation with the equivalence classes $S_{x,y}$ , $(x,y)\in I$ , is a congruence on $S$ .

3. (iii) $S$ is finitely presented.

Proof. We first prove the following:

Claim. We have

\begin{equation*} ({-}c_1, q_1, d_1)\dots ({-}c_m, q_m, d_m) = \bigl ({-}c_1, q_1+\dots +q_m, q_1+\dots +q_m+(d_m-q_m)\bigr )\in S_{c_1,d_m-q_m}, \end{equation*}

where each $({-}c_i, q_i, d_i)\in A$ .

Proof. We prove the claim by induction on $m$ . The case $m=1$ is trivial. Consider $m\gt 1$ , and inductively assume the claim is valid for $m-1$ . Then

\begin{align*} &({-}c_1, q_1, d_1)\dots ({-}c_{m}, q_{m}, d_{m})\\ =&\bigl (({-}c_1, q_1+\dots +q_{m-1}, q_1+\dots +q_{m-1}+(d_{m-1}-q_{m-1})\bigr )({-}c_{m}, q_{m}, d_{m})=({-}c, q, d), \end{align*}

where

\begin{align*} c&=\max \bigl (c_1, c_{m}-(q_1+\dots +q_{m-1})\bigr ),\\ q&=q_1+\dots + q_m,\\ d&=\max \bigl (q_1+\dots +q_{m-1}+(d_{m-1}-q_{m-1}), q_1+\dots +q_{m-1}+d_{m}\bigr ). \end{align*}

Using (2), we have

\begin{equation*} c_1 \geq c_{m}-q_1 \geq c_{m}-(q_1+\dots +q_m), \end{equation*}

and so $c=c_1$ . Similarly,

\begin{equation*} q_1+\dots +q_{m-1}+(d_{m-1}-q_{m-1})\leq q_1+\dots +q_{m-1}+d_m=q_1+\dots +q_m+(d_m-q_m), \end{equation*}

implying $d=q_1+\dots +q_m+(d_m-q_m)$ . This completes the inductive step and proof of the claim.

(i) First note that $S= \bigcup _{(x,y)\in I}S_{x,y}$ : indeed, ( $\subseteq$ ) follows from Claim, whereas ( $\supseteq$ ) holds by the definition of the $S_{x,y}$ . That each $S_{x,y}$ is non-empty also follows from Claim: indeed, products of elements of $A$ of length $2$ already yield members in each $S_{x,y}$ . Finally, that the $S_{x,y}$ are mutually disjoint follows from Lemma 4.5 (ii).

(ii) Claim gives that $S_{x,y}S_{z,t}\subseteq S_{x,t}$ for all $(x,y),(z,t)\in I$ , which readily implies this part.

(iii) Each $S_{x,y}$ is a subsemigroup of $N_{x,y+1}$ , and $N_{x,y+1}\cong \mathbb{N}$ by Lemma 4.5 (i). Hence, each $S_{x,y}$ is finitely presented by Proposition 4.4. It now follows that $S$ is finitely presented by combining (i), (ii) and Proposition 4.3.

Next, we extend to an arbitrary finitely generated subsemigroup of $P$ .

Proof. By Lemma 4.7, there are only finitely many $N_{x,y}$ which have non-empty intersection with $S$ ; suppose they are $N_{x_1,y_1}, \dots, N_{x_m, y_m}$ . For each $i = 1, \dots, m$ , let

\begin{equation*} q_i \,:\!=\, \max (x_i, y_i-1),\quad Q \,:\!=\, \max _{1\leq i \leq m}q_i,\quad U_i \,:\!=\, \bigl \{({-}a, p, b)\in N_{x_i,y_i} \,:\, p \gt Q\bigr \} \cap S. \end{equation*}

Since the multiplication in $P$ always increases the middle component, it follows that $U_i \leq S \cap N_{x_i,y_i}$ and that $U \,:\!=\, \bigcup _{i=1}^{m}U_i\leq S$ . Since $U_i \leq N_{x_i,y_i} \cong \mathbb{N}$ , it follows that each $U_i$ is finitely generated by Proposition 4.4. Hence, $U$ is also finitely generated and we let $A = \{({-}a_1, p_1, b_1), \dots, ({-}a_n, p_n, b_n)\}\subseteq P$ be its generating set.

We claim that arbitrary $({-}a_i, p_i, b_i), ({-}a_j, p_j, b_j) \in A$ satisfy the condition (2) in Lemma 4.8. We have $p_i, p_j \gt Q$ , $a_i, a_j \in \{x_1, \dots, x_m\}$ and $b_i-p_i, b_j-p_j \in \{y_1-1, \dots, y_m-1\}$ . By our choice of $Q$ , we have $a_j \leq Q \lt p_i$ which implies $a_j - p_i \lt 0 \leq a_i$ . Also, $b_i - p_i \leq Q \lt p_j \leq b_j$ which implies $b_i \leq b_j + p_i$ , and (2) indeed holds.

Lemma 4.8 (iii) now implies that $U$ is finitely presented. Since $S\setminus U$ is finite, $S$ is also finitely presented by Proposition 4.2.

Finally, we can prove the main result of this section.

Proof of Proposition 4.1. Let $S$ be a finitely generated subsemigroup of $FI_1$ with only finitely many idempotents. By Lemma 3.2, $S$ consists entirely of non-negative or non-positive elements. Without loss of generality suppose the former is the case. Notice that $S\cap P$ is an ideal of $S$ , and that $S\setminus P$ is the set of idempotents of $S$ . Hence, $S\setminus (S\cap P)$ is finite. By Proposition 4.2, $S\cap P$ is finitely generated. Then it follows that it is finitely presented by Lemma 4.9. And finally another application of Proposition 4.2 gives that $S$ itself is finitely presented, completing the proof.

The Corollary can be quickly derived from the Main Theorem:

Proof of the Corollary. If $S$ contains a non-idempotent element $a$ , it also contains infinitely many idempotents $a^{-n}a^n$ , $n\in \mathbb{N}$ , so $S$ cannot be finitely presented by our Main Theorem.

We conclude the paper by recalling the results from [Reference Cutting and Solomon5] and [Reference Gray and Steinberg8], and ask whether they extend to subsemigroups of the monogenic free inverse semigroup, in the sense that our Main Theorem extends the original theorem by Schein (Theorem 1). Specifically, we ask whether it is true that each of the following two properties concerning a finitely generated subsemigroup $S$ of $FI_1$ is equivalent to $S$ having only finitely many idempotents: (a) having a regular language of normal forms and (b) the property $\mathrm{FP}_2$ ?