Monoids

Definition

A semigroup with an (identity) element e satisfying, for all x,

	xe=x   and  ex=x
	

Examples

• Any group
• n x n matrices over a commutative ring, under matrix multiplication
• Natural numbers under ordinary addition
• Odd integers under ordinary multiplication
• Mappings on some set under operation of composition
• This finite example, which is not a group:

  	  	    * | 1 2 3 4 5 6
                      ---------------
  		    1 | 1 2 3 4 5 6
                      2 | 2 3 4 5 6 3 
                      3 | 3 4 5 6 3 4
                      4 | 4 5 6 3 4 4
                      5 | 5 6 3 4 4 6
                      6 | 6 3 4 5 6 3
  	

Structure

Representation

Decision problems

Identity problem:

Word problem:

Spectra and growth

Finite spectrum:

Free spectrum:

Growth series: 1/(1-rz) for the free monoid on r generators

History/Importance

References

Subsystems

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A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu