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The monoids from AND and OR are also idempotent while those from XOR and XNOR are not. The set of natural numbers N = {0, 1, 2, ...} is a commutative monoid under addition (identity element 0) or multiplication (identity element 1). A submonoid of N under addition is called a numerical monoid.
For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit. The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as ...
In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
tensor graph product (or direct graph product, categorical graph product, cardinal graph product, Kronecker graph product): it is a commutative and associative operation (for unlabelled graphs), zig-zag graph product; [3] graph product based on other products: rooted graph product: it is an associative operation (for unlabelled but rooted ...
First, one takes the symmetric closure R ∪ R −1 of R. This is then extended to a symmetric relation E ⊂ Σ ∗ × Σ ∗ by defining x ~ E y if and only if x = sut and y = svt for some strings u, v, s, t ∈ Σ ∗ with (u,v) ∈ R ∪ R −1. Finally, one takes the reflexive and transitive closure of E, which then is a monoid congruence.
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In mathematics, the Grothendieck group, or group of differences, [1] of a commutative monoid M is a certain abelian group.This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M.