Definitions
from The Century Dictionary.
- In ancient prosody, containing but one kind of foot: noting certain meters. Monoid meters are also called
pure meters or simple meters, and distinguished fromcompound (episynthetic) meters and mixed or logaœdic meters. - noun In mathematics, a surface which possesses a conical point of the highest possible (n —1)th order.
from Wiktionary, Creative Commons Attribution/Share-Alike License.
- noun mathematics A
set which isclosed under anassociative binary operation , and which contains an element which is anidentity for the operation.
Etymologies
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Examples
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Insights from category theory, in particular the expression of a variety as a monad, defined as a monoid object in the category
Algebra Pratt, Vaughan 2007
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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.
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Anything satisfying these laws is called a monoid homomorphism, or just homomorphism for short.
Planet Haskell 2009
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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] 2008
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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 Pratt, Vaughan 2007
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X, whence the semigroup of all functions on a set X forms a monoid.
Algebra Pratt, Vaughan 2007
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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 Pratt, Vaughan 2007
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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 Pratt, Vaughan 2007
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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 Pratt, Vaughan 2007
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