In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group .[1] Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic study of these geometric objects.
^Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Monographs in Mathematics. Springer. ISBN 0-387-76356-2.
In abstract algebra, a branch of mathematics, an affinemonoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of...
multiplicative monoids called the structure sheaf. An affinemonoid scheme is a monoidal space that is isomorphic to the spectrum of a monoid, and a monoid scheme...
algebraic structure is a monoid, usually called the full linear monoid, but occasionally also full linear semigroup, general linear monoid etc. It is actually...
the analogous case of groups) it may be called an abelian semigroup. A monoid is an algebraic structure intermediate between semigroups and groups, and...
mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in...
In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements...
which the set of nonzero elements is a commutative monoid under multiplication (because a monoid must be closed under multiplication). An integral domain...
are given by the monoid that describes the symmetries of the infinite binary tree or Cantor space. This so-called period-doubling monoid is a subset of...
groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism...
{\displaystyle x\cdot x=x} for all x ∈ S {\displaystyle x\in S} . In the monoid ( N , × ) {\displaystyle (\mathbb {N} ,\times )} of the natural numbers...
arises as the function composition of endomorphisms over any commutative monoid. The theory of (associative) algebras over commutative rings can be generalized...
associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian...
be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of tensor product of OX-modules. The identity element...
we say that M is a normal monoid. For example, the monoid Nn consisting of n-tuples of natural numbers is a normal monoid, with the Grothendieck group...
f i {\displaystyle f_{i}} generates a monoid under composition. If there are only two such functions, the monoid can be visualized as a binary tree, where...
point. For example, the point associated to the zero ideal for any integral affine scheme. F(n), F(D) 1. If X is a projective scheme with Serre's twisting...
completing a semigroup to a monoid, taking the corresponding opposite category, and then possibly removing the unit from that monoid. The category of Boolean...
generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. Representation theory is a branch of mathematics that draws heavily...
As conjectured by Knop, every "smooth" affine spherical variety is uniquely determined by its weight monoid. This uniqueness result was proven by Losev...
endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R. For any fractional ideal...
over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are forgetful functors A : Ring → Ab M : Ring → Mon...
equivalent to affine group schemes. (Every affine group scheme over a field k is pro-algebraic in the sense that it is an inverse limit of affine group schemes...
a right adjoint to F. From monoids and groups to rings. The integral monoid ring construction gives a functor from monoids to rings. This functor is left...
numbers can be replaced by the more general notion of a monoid. Let M {\displaystyle M} be a monoid with identity element 1 ∈ M , {\displaystyle 1\in M,}...
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