System of Garside

Definition

A system of Garside is the data of a couple $(M,\; \backslash \; Delta)$ where

1. $M$ is a Monoïde that can be simplified, without invertible element other than 1 and admitting PPCM and PGCD (for divisibility);
2. $\backslash \; Delta$ is an element of Garside:
3. * $\backslash \; rm\; \{Div\}\; \_L\; \backslash \; \backslash \; Delta\; =\; \backslash \; rm\; \{Div\}\; \_R\; \backslash \; \backslash \; Delta$ (and one writes thereafter $\backslash \; rm\; \{Div\}\; \backslash \; \backslash \; Delta$ for this unit);
4. * $\backslash \; rm\; \{Div\}\; \backslash \; \backslash \; Delta$ generates $M$;
5. * $\backslash \; rm\; \{Div\}\; \backslash \; \backslash \; Delta$ is finished.

where $\backslash \; rm\; \{Div\}\; \_L\; \backslash \; \backslash \; Delta$ (resp. $\backslash \; rm\; \{Div\}\; \_R\; \backslash \; \backslash \; Delta$) indicates the whole of the dividers on the left (resp. on the right) of $\backslash \; Delta$.

Example

The monoid of the braids with three bits $B^+\_3$ presented by $\backslash \; langle\; \backslash \; sigma\_1,\; \backslash \; sigma\_2;\; \backslash \; sigma\_1\; \backslash \; sigma\_2\; \backslash \; sigma\_1=\; \backslash \; sigma\_2\; \backslash \; sigma\_1\; \backslash \; sigma\_2\; \backslash \; rangle^+$ and the element $\backslash \; sigma\_1\; \backslash \; sigma\_2\; \backslash \; sigma\_1$ forms a system of Garside.

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