Monoid is an English word. Below you'll find 10+ example sentences showing how it's used in practice.
Monoid meaning
A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.
Using Monoid
- The main meaning on this page is: A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.
- In the example corpus, monoid often appears in combinations such as: monoid is, commutative monoid, the monoid.
Context around Monoid
- Average sentence length in these examples: 20.1 words
- Position in the sentence: 7 start, 9 middle, 4 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis for Monoid
- In this selection, "monoid" usually appears in the middle of the sentence. The average example has 20.1 words, and this corpus slice is mostly made up of statements.
- Around the word, commutative, quotient, abelian, object, operation and homomorphisms stand out and add context to how "monoid" is used.
- Recognizable usage signals include a commutative monoid or less and a monoid object in. That gives this page its own corpus information beyond isolated example sentences.
- By corpus frequency, "monoid" sits close to words such as abated, aberrations and activations, which helps place it inside the broader word index.
Example types with monoid
The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:
Monoid homomorphisms are sometimes simply called monoid morphisms. (8 words)
However, not every monoid sits inside a group. (8 words)
A monoid object in Set is just a monoid. (9 words)
A small category with a single object is the same thing as a monoid : the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. (42 words)
Not every semigroup homomorphism is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the element it maps the identity to will be an identity of the image of the mapping. (41 words)
By definition, a ring is a monoid object in the category of abelian groups ; thus, an associative algebra is obtained by replacing the category of abelian groups with the category of modules. (32 words)
Example sentences (20)
Commutative monoid A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid).
A small category with a single object is the same thing as a monoid : the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation.
A monoid object in Set is just a monoid.
Monoid homomorphisms are sometimes simply called monoid morphisms.
Not every semigroup homomorphism is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the element it maps the identity to will be an identity of the image of the mapping.
One does this by extending (finite) binary relations on Σ ∗ to monoid congruences, and then constructing the quotient monoid, as above.
There is also a notion of monoid object which is an abstract definition of what is a monoid in a category.
Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint.
A semigroup without an identity element can be easily turned into a monoid by just adding an identity element.
A transformation semigroup can be made into an operator monoid by adjoining the identity transformation.
Being lattices, Heyting algebras and Boolean algebras are endowed with these monoid structures.
By definition, a ring is a monoid object in the category of abelian groups ; thus, an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
Category theoretical description main Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of -modules ).
Definition Let R be a ring and let G be a monoid.
For example, a quotient of a category with one object is just a quotient monoid.
For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation.
However, a non-commutative cancellative monoid need not be embeddable in a group.
However, if the original monoid has an absorbing element then its Grothendieck group is the trivial group.
However, not every monoid sits inside a group.
If a monoid has the cancellation property and is finite, then it is in fact a group.
Common combinations with monoid
These word pairs occur most frequently in English texts:
- monoid is 5×
- commutative monoid 4×
- the monoid 4×
- monoid homomorphisms 4×
- is monoid 4×
- quotient monoid 4×
- monoid object 3×
- of monoid 3×
- monoid in 3×
- into monoid 3×