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The free monoid on a set A is usually denoted A ∗. The free semigroup on A is the subsemigroup of A ∗ containing all elements except the empty string. It is usually denoted A +. [1] [2] More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set. [3]
Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. [ 8 ] There is also a notion of monoid object which is an abstract definition of what is a monoid in a category.
The main discussion of these abbreviations in the context of drug prescriptions and other medical prescriptions is at List of abbreviations used in medical prescriptions. Some of these abbreviations are best not used, as marked and explained here.
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).
Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any (small) strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product).
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 =.
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The definition of a variety of finite semigroups is inspired by the notion of a variety of universal algebras. We recall the definition of a variety in universal algebra. Such a variety is, equivalently: a class of structures, closed under homomorphic images, subalgebras and (direct) products. a class of structures satisfying a set of identities.