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Commutative Monoid


A monoid that is commutative i.e., a monoid M such that for every two elements a and b in M, ab=ba. This means that commutative monoids are commutative, associative, and have an identity element.

For example, the nonnegative integers under addition form a commutative monoid. The integers under the operation mod(x+y,n) with n in Z^+ all form a commutative monoid. This monoid collapses to a group only if x and y are restricted to the integers 0, 1, ..., n-1, since only then do the elements have unique additive inverses. Similarly, the integers under the operation max(x+y,n) also forms a commutative monoid.

The numbers of commutative monoids of orders n=1, 2, ... are 1, 2, 5, 19, 78, 421, 2637, ... (OEIS A058131).


See also

Monoid, Semigroup, Submonoid

Portions of this entry contributed by Todd Rowland

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References

Sloane, N. J. A. Sequence A058131 in "The On-Line Encyclopedia of Integer Sequences."Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 952, 2002.

Referenced on Wolfram|Alpha

Commutative Monoid

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Commutative Monoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CommutativeMonoid.html

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