Let be a presentation for the inverse monoid [resp. inverse semigroup ]. In what follows, the argument for inverse semigroups and inverse monoids is exactly the same, so we concentrate on the last one.
Given , let be the equivalence class of with respect to the Right Green relation . The Right Schützenberger graph of with respect to the presentation is defined as the -inverse word graph with vertex and edge set respectively
This is a good definition, in fact it can be easily shown that given with we have .
Analogously we can define the Left Schützenberger graph using the Left Green relation instead of the Right Green relation , but this notion is not used in literature.
Schützenberger graphs play in combinatorial inverse semigroups theory the role that Cayley graphs play in combinatorial group theory. In fact, if happen to be a group (with identity ), then the Schützenberger graph of its unique -class is exactly the Cayley graph of the group .
- 1 N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
- 2 J.B. Stephen, Presentation of inverse monoids, J. Pure Appl. Algebra 63 (1990) 81-112.
|Date of creation||2013-03-22 16:10:50|
|Last modified on||2013-03-22 16:10:50|
|Last modified by||Mazzu (14365)|
|Defines||left Schützenberger graph|
|Defines||right Schützenberger graph|