In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a semigroup with no elements (the empty semigroup) is a
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...
in which the
underlying set
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
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.
[P. A. Grillet (1995). ''Semigroups''. CRC Press. pp. 3–4 ] 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 to ''S''. This operation
vacuously
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
satisfies the closure and associativity axioms of a semigroup. Not excluding the empty semigroup simplifies 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.
When a semigroup is defined to have additional structure, the issue may not arise. For example, the definition of a
monoid
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 ...
requires an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
, which rules out the empty semigroup as a 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 elements is an
inverse semigroup, since the necessary condition is vacuously satisfied.
See also
*
Field with one element
In 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. The nam ...
*
Semigroup with one element
*
Semigroup with two elements In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements:
* O2, the null semigroup of order two,
* LO2, the left zero ...
*
Semigroup with three elements In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplicati ...
*
Special classes of semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups cons ...
References
{{reflist
Algebraic structures
Semigroup theory