As an example of a substructure, we show that the consecutive performance of the binary operation "$\ast$" on the same element creates a substructure in a monoid.

Proposition: Subset of Powers is a Submonoid

If $x$ is an element of a monoid $(X,\ast)$, then the subset of all powers $x^n$, $(n=0,1,2,\ldots)$ is a substructure of $X$, called its submonoid.

Proofs: 1


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References

Bibliography

  1. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition