Definition, Betydelse & Anagram | Engelska ordet MONOID

Definition av MONOID

1. (matematik) monoid

Exempel på hur man kan använda MONOID i en mening

• In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.
• To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.
• Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product.
• In concise terms, a monad is a monoid in the category of endofunctors of some fixed category (an endofunctor is a functor mapping a category to itself).
• Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.
• The coequalizer of these two functors is the monoid of natural numbers under addition, considered as a one-object category.
• In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law.
• In general, if there are k functions, then one may visualize the monoid as a full k-ary tree, also known as a Cayley tree.
• It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup).
• We can reinterpret this monoid as a bicategory with a single object x (one 0-cell); this construction is analogous to construction of a small category from a monoid.
• A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.
• A consequence of the Krohn–Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three-element flip-flop monoid, consisting of an identity element and two right zeros.
• The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).
• Lascoux and Schützenberger studied an associative product on the set of all semistandard Young tableaux, giving it the structure called the plactic monoid (French: le monoïde plaxique).
• A unital quantale may be defined equivalently as a monoid in the category Sup of complete join-semilattices.
• I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid.
• An algebraic model of boolean bunched logic is a poset that is a Boolean algebra and that carries an additional residuated commutative monoid structure.
• The most common type of hyperobject is a reducer, which corresponds to the reduction clause in OpenMP or to the algebraic notion of a monoid.
• The monoid of smooth structures on n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere, taken up to orientation-preserving diffeomorphism.
• The bicyclic semigroup is the quotient of the free monoid on two generators p and q by the congruence generated by the relation.

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