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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.
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]
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.
Condition (L): Every cycle in the graph has an exit. Condition (K): There is no vertex in the graph that is on exactly one simple cycle. Equivalently, a graph satisfies Condition (K) if and only if each vertex in the graph is either on no cycles or on two or more simple cycles.
Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph. These graphs can be used to generate examples in which the bound of Vizing's conjecture , an unproven inequality between the domination number of the graphs in a different graph product, the ...
A finite monoid is rational. A group is a rational monoid if and only if it is finite.; A finitely generated free monoid is rational. The monoid M4 generated by the set {0,e, a,b, x,y} subject to relations in which e is the identity, 0 is an absorbing element, each of a and b commutes with each of x and y and ax = bx, ay = by = bby, xx = xy = yx = yy = 0 is rational but not automatic.
S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid generated by T. The empty word is the word with no symbols at all.