In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a semigroup with no elements (the empty semigroup) is a
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
in which the
underlying set is the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
. 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
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
binary operation. However not all authors insist on the underlying set of a semigroup being non-empty.
[P. A. Grillet (1995). ''Semigroups''. ]CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technol ...
. 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
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called ...
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.
It is sometimes said that a s ...
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 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 .
Monoids are semigroups with identity ...
requires an
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
, which rules out the empty semigroup as a monoid.
In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, the empty semigroup is always admitted. It is the unique
initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element) ...
of the category of semigroups.
A semigroup with no elements is an
inverse semigroup In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigr ...
, 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 ...
*
Semigroup with three elements
*
Special classes of semigroups
References
{{reflist
Algebraic structures
Semigroup theory