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A Group-like object which fails to be a Group because elements need not have an inverse within the object. A monoid $S$ must also be Associative and have an Identity Element $I\in S$ such that for all $a\in S$, $1a=a1=a$. A monoid is therefore a Semigroup with an Identity Element. A monoid must contain at least one element.

The numbers of free idempotent monoids on $n$ letters are 1, 2, 7, 160, 332381, ... (Sloane's A005345).

See also Binary Operator, Group, Semigroup


Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.

Sloane, N. J. A. Sequence A005345/M1820 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

© 1996-9 Eric W. Weisstein