Search results
Results from the WOW.Com Content Network
For example, monoids are semigroups with identity. In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity.
The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory. As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system). Every monoid may be presented by a semi-Thue ...
A free monoid is equidivisible: if the equation mn = pq holds, then there exists an s such that either m = ps, sn = q (example see image) or ms = p, n = sq. [9] This result is also known as Levi's lemma. [10] A monoid is free if and only if it is graded (in the strong sense that only the identity has gradation 0) and equidivisible. [9]
A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale. A monoid object in (Ab, ⊗ Z, Z), the category of abelian groups, is a ring. For a commutative ring R, a monoid object in (R-Mod, ⊗ R, R), the category of modules over R, is a R-algebra.
Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product. Any commutative monoid (,,) can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid.
A more general class of examples is given by the Boolean algebra of all binary relations on a set X, namely the power set of X 2, made a residuated lattice by taking the monoid multiplication • to be composition of relations and the monoid unit to be the identity relation I on X consisting of all pairs (x,x) for x in X.
The syntactic monoid is the group of order 2 on {,}. [9] For the language (+), the minimal automaton has 4 states and the syntactic monoid has 15 elements. [10] The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).
An example of a commutative monoid that is not naturally ordered is (, +,), the commutative monoid of the integers with usual addition, as for any , there exists such that + =, so holds for any ,, so is not a partial order. There are also examples of monoids that are naturally ordered but are not semirings with monus.