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Monoid


A monoid is a set that is closed under an associative binary operation and has an identity element I in S such that for all a in S, Ia=aI=a. Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element.

A monoid must contain at least one element.

A monoid that is commutative is, not surprisingly, known as a commutative monoid.


See also

Binary Operator, Commutative Monoid, Free Idempotent Monoid, Group, Semigroup, Submonoid

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References

Lallement, G. Semigroups and Combinatorial Applications. New York: Wiley, 1979.Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.Sloane, N. J. A. Sequence A005345/M1820 in "The On-Line Encyclopedia of Integer Sequences."

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Monoid

Cite this as:

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

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