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For example, monoids are semigroups with identity. In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity.
The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory. As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system). Every monoid may be presented by a semi-Thue ...
Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product. Any commutative monoid (,,) can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid.
Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element.
A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale. A monoid object in (Ab, ⊗ Z, Z), the category of abelian groups, is a ring. For a commutative ring R, a monoid object in (R-Mod, ⊗ R, R), the category of modules over R, is a R-algebra.
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element.
A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category, A monad as a tool for studying algebraic gadgets; for example, a group can be described by a certain monad. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary ...
Some examples and non-examples of symmetric monoidal categories: The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object. The category of groups. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.