Factor Monoid and Monoid Presentations
Since is a congruence, we can define the factor monoid of the free monoid by the Thue congruence in the usual manner. If a monoid is isomorphic with, then the semi-Thue system is called a monoid presentation of .
We immediately get some very useful connections with other areas of algebra. For example, the alphabet {a, b} with the rules { ab → ε, ba → ε }, where ε is the empty string, is a presentation of the free group on one generator. If instead the rules are just { ab → ε }, then we obtain a presentation of the bicyclic monoid.
The importance of semi-Thue systems as presentation of monoids is made stronger by the following:
Theorem: Every monoid has a presentation of the form, thus it may be always be presented by a semi-Thue system, possibly over an infinite alphabet.
In this context, the set is called the set of generators of, and R is called the set of defining relations . We can immediately classify monoids based on their presentation. is called
- finitely generated if is finite.
- finitely presented if both and R are finite.
Read more about this topic: Semi-Thue System
Famous quotes containing the word factor:
“In his very rejection of art Walt Whitman is an artist. He tried to produce a certain effect by certain means and he succeeded.... He stands apart, and the chief value of his work is in its prophecy, not in its performance. He has begun a prelude to larger themes. He is the herald to a new era. As a man he is the precursor of a fresh type. He is a factor in the heroic and spiritual evolution of the human being. If Poetry has passed him by, Philosophy will take note of him.”
—Oscar Wilde (18541900)