A monoid is a set that is closed under an associative binary operation and has an identity
element 
such that for all 
,
. 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."Referenced
on Wolfram|Alpha
Monoid
Cite this as:
Weisstein, Eric W. "Monoid." From MathWorld--A
Wolfram Web Resource. https://mathworld.wolfram.com/Monoid.html
Subject classifications
