Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.
from The Century Dictionary and Cyclopedia
 In ancient prosody, containing but one kind of foot: noting certain meters. Monoid meters are also called pure meters or simple meters, and distinguished from compound (episynthetic) meters and mixed or logaœdic meters.
 n. In mathematics, a surface which possesses a conical point of the highest possible (n —1)th order.
Etymologies
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Examples

Insights from category theory, in particular the expression of a variety as a monad, defined as a monoid object in the category

In Haskell, a monoid is a type with a rule for how two elements of that type can be combined to make another element of the same type.

Anything satisfying these laws is called a monoid homomorphism, or just homomorphism for short.

The technical term being used here isn't actually "monoid", but "monoid object" which is a generalisation of the definition of a monoid from the category of sets to an arbitrary

We are creating the world's most trusted encyclopedia and knowledge base. log in, you'll be able to edit this page instantly! contribs) (new entry, just a stub) algebra, a monoid is a set equipped with a binary operation satisfying certain properties similar to but less stringent than those of a group.

With no generators the free monoid, free group, and free ring are all the oneelement algebra consisting of just the additive identity 0.

A monoid H is a submonoid of a monoid G when it is a subsemigroup of G that includes the identity of G. 2.2 Groups

Hence any of the above examples of semigroups for which the operation is addition forms a monoid if and only if it contains zero.

X, whence the semigroup of all functions on a set X forms a monoid.

The monoids of natural numbers and of even integers are both submonoids of the monoid of integers under addition, but only the latter submonoid is a subgroup, being closed under negation, unlike the natural numbers.
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