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| Let $\doubles{X}$ be the free monoid |
Let $\doubles{X}$ be the free monoid |
| with involution on $X$, and $T\subseteq \doubles X\times \doubles X$ be a binary relation between words. We denote by $T^\e$ [resp. $T^\co$] the equivalence relation [resp. congruence] generated by $T$. |
with involution on $X$, and $T\subseteq \doubles X\times \doubles X$ be a binary relation between words. We denote by $T^\e$ [resp. $T^\co$] the equivalence relation [resp. congruence] generated by $T$. |
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| A \emph{presentation (for an inverse monoid)} is a couple $(X;T)$. We use this couple of objects to define an inverse monoid $\mipres{X}{T}$. Let $\rho_X$ be the Wagner congruence on $X$, we define the inverse monoid $\mipres{X}{T}$ \emph{presented} by $(X;T)$ as $$\mipres{X}{T}=\doubles{X}/(T\cup\rho_X)^\co.$$ |
A \emph{presentation (for an inverse monoid)} is a couple $(X;T)$. We use this couple of objects to define an inverse monoid $\mipres{X}{T}$. Let $\rho_X$ be the Wagner congruence on $X$, we define the inverse monoid $\mipres{X}{T}$ \emph{presented} by $(X;T)$ as $$\mipres{X}{T}=\doubles{X}/(T\cup\rho_X)^\co.$$ |
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| In the previous dicussion, if we replace everywhere $\doubles X$ with $\doublep X$ we obtain a \emph{presentation (for an inverse semigroup)} $(X;T)$ and an inverse semigroup $\sipres{X}{T}$ \emph{presented} by $(X;T)$. |
In the previous dicussion, if we replace everywhere $\doubles X$ with $\doublep X$ we obtain a \emph{presentation (for an inverse semigroup)} $(X;T)$ and an inverse semigroup $\sipres{X}{T}$ \emph{presented} by $(X;T)$. |
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| A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on $X$, that is usually denoted by $\fim(X)$ [resp. $\fis(X)$] and is defined by $$\fim(X)=\mipres{X}{\varnothing}=\doubles{X}/\rho_X,\ \ \mbox{[resp. $\fis(X)=\sipres{X}{\varnothing}=\doublep{X}/\rho_X$]}.$$ |
A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on $X$, that is usually denoted by $\fim(X)$ [resp. $\fis(X)$] and is defined by $$\fim(X)=\mipres{X}{\varnothing}=\doubles{X}/\rho_X,\ \ \mbox{[resp. $\fis(X)=\sipres{X}{\varnothing}=\doublep{X}/\rho_X$]}.$$ |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{b:petrich} N. Petrich, \emph{Inverse Semigroups}, Wiley, New York, 1984. |
\bibitem{b:petrich} N. Petrich, \emph{Inverse Semigroups}, Wiley, New York, 1984. |
| \bibitem{b:steph} J.B. Stephen, \emph{Presentation of inverse monoids}, J. Pure Appl. Algebra 63 (1990) 81-112. |
\bibitem{b:steph} J.B. Stephen, \emph{Presentation of inverse monoids}, J. Pure Appl. Algebra 63 (1990) 81-112. |
| \end{thebibliography} |
\end{thebibliography} |
