On Special Inverse Monoids with the Strong $F$-Inverse Property

Abstract

An inverse monoid $S$ is called $F$-inverse if each $σ$-class of $S$, where $σ$ is the minimum group congruence of $S$, has a maximum element with respect to the natural order of $S$. Since the property of an inverse monoid being $F$-inverse immediately implies that it must be $E$-unitary, it follows that every $X$-generated $F$-inverse monoid with canonical maximum group image $G$ must be isomorphic to a quotient of the Margolis-Meakin expansion $M(G,X)$. If this is realised in such a way that all the maximal elements of each $σ$-class of $M(G,X)$ get identified, thus producing the top element of the corresponding $σ$-class of $S$, we say that $S$ is strongly $F$-inverse. Consequently, there is a universal $X$-generated inverse monoid $M_{sF}(G,X)$ with maximum group image $G$ and the strongly $F$-inverse property. We provide a presentation for this inverse monoid and show it can be further simplified upon introducing additional assumptions on the group $G$ (which will include all one-relator groups). We use this to provide a full description of all one-relator special inverse monoids with a cyclically reduced relator word that are strongly $F$-inverse. We also discuss some further examples and non-examples.

Figures (1)

• Figure 1: The full line represents the path $\Pi_1$ and the dotted line represents the path $\Pi_2$. The simple cycles considered in the proof, corresponding to \ref{['eq:1']}, \ref{['eq:2']} and \ref{['eq:3']}, are coloured in blue, red, and green, respectively. The vertices $y_0,y_1,\dots,y_{k_{r-1}},\dots,y_m$ are highlighted in white. In this particular instance, the permutation $\pi^{-1}$ maps $01234567$ to $05231647$.

Theorems & Definitions (33)

• Proposition 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• proof
• Proposition 3.4
• Remark 3.5
• Theorem 4.1
• proof
• Claim