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2. Monoid - Wikipedia

   en.wikipedia.org/wiki/Monoid

   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 .

3. Monoid ring - Wikipedia

   en.wikipedia.org/wiki/Monoid_ring

   The monoid ring or monoid algebra of G over R, denoted R[G] or RG, is the set of formal sums , where for each and r g = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G.

4. Monoid (category theory) - Wikipedia

   en.wikipedia.org/wiki/Monoid_(category_theory)

   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.

5. Free monoid - Wikipedia

   en.wikipedia.org/wiki/Free_monoid

   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.

6. Monad (category theory) - Wikipedia

   en.wikipedia.org/wiki/Monad_(category_theory)

   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). According to John Baez, a monad can be considered at least in two ways: [1] A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category,

7. Monoidal category - Wikipedia

   en.wikipedia.org/wiki/Monoidal_category

   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.

8. Homomorphism - Wikipedia

   en.wikipedia.org/wiki/Homomorphism

   Monoid homomorphism from the monoid (N, +, 0) to the monoid (N, ×, 1), defined by () =. It is injective, but not surjective. The real numbers are a ring, having both addition and multiplication. The set of all 2×2 matrices is also a ring, under matrix addition and matrix multiplication. If we define a function between these rings as follows:

9. Residuated lattice - Wikipedia

   en.wikipedia.org/wiki/Residuated_lattice

   In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations ...