Empty semigroup
Encyclopedia
In mathematics
, a semigroup with no elements (the empty semigroup) is a semigroup
in which the underlying set is the empty set
. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist on the underlying set of a semigroup being non-empty. One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function
from S × S to S. This operation vacuously satisfies the associativity property. The postulation of the existence of the empty semigroup gives more generality for certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup T is a subsemigroup of T becomes valid even when the intersection is empty.
For more restrictively defined structures, the issue often does not come up. For example, the definition of a monoid
requires the existence of an identity element
, and so rules out the possibility of an empty monoid.
In category theory
, the empty semigroup is always admitted. It is the unique initial object
of the category of semigroups.
A semigroup with no element is an inverse semigroup
, since the necessary condition is vacuously satisfied.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a semigroup with no elements (the empty semigroup) is a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
in which the underlying set is the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist on the underlying set of a semigroup being non-empty. One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function
Empty function
In mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty functionf_A: \varnothing \rightarrow A....
from S × S to S. This operation vacuously satisfies the associativity property. The postulation of the existence of the empty semigroup gives more generality for certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup T is a subsemigroup of T becomes valid even when the intersection is empty.
For more restrictively defined structures, the issue often does not come up. For example, the definition of a monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
requires the existence of an identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
, and so rules out the possibility of an empty monoid.
In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, the empty semigroup is always admitted. It is the unique initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
of the category of semigroups.
A semigroup with no element is an inverse semigroup
Inverse semigroup
In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
, since the necessary condition is vacuously satisfied.
See also
- Field with one elementField with one elementIn mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French-English pun, Fun...
- Semigroup with one element
- Semigroup with two elementsSemigroup with two elementsIn mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five distinct nonisomorphic semigroups having two elements:* O2, the null semigroup of order two,...