Search results
Results from the WOW.Com Content Network
History monoids were first presented by M.W. Shields. [1] History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.
Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. [8]
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.
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.
Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example: Set, the category of sets with the Cartesian product, any particular one-element set serving as the unit.
In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi.The underpinning is provided by an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation, analogous to the way that the free ...
Let denote the free monoid on a set of generators , that is, the set of all strings written in the alphabet .The asterisk is a standard notation for the Kleene star.An independency relation on the alphabet then induces a symmetric binary relation on the set of strings : two strings , are related, , if and only if there exist ,, and a pair (,) such that = and =.
Let T, η, μ be a monad over a category C.The Kleisli category of C is the category C T whose objects and morphisms are given by = (), (,) = (,).That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in C T (but with codomain Y).