# monoid

## Definitions

• 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

Sorry, no etymologies found.

## Examples

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

Algebra

• 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.

A Neighborhood of Infinity

• 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

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• 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.

Citizendium, the Citizens' Compendium - Recent changes [en]

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

Algebra

• 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

Algebra

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

Algebra

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

Algebra

• 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.

Algebra

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