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Function f : [Z] 3 → [Z] 6 given by [k] 3 ↦ [3k] 6 is a semigroup homomorphism, since [3k ⋅ 3l] 6 = [9kl] 6 = [3kl] 6. However, f([1] 3) = [3] 6 ≠ [1] 6, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be ...
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.
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. For example, if S = {a, b, c}, then T = {a, a −1, b, b −1, c, c −1}, and
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.
a 2-cycle such as (1 2) maps to the product of three 2-cycles such as (1 2)(3 4)(5 6) and vice versa, there being 15 permutations each way; the product of a 2-cycle and a 3-cycle such as (1 2 3)(4 5) maps to a 6-cycle such as (1 2 5 3 4 6) and vice versa, accounting for 120 permutations each way;
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 ...
The free monoid on a given set is the monoid whose elements are all the strings of zero or more elements from that set, with string concatenation as the monoid operation and the empty string as the identity element.