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 trivial semigroup (a semigroup with one element) 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 ...
for which the
cardinality of 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
one
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
. The number of
distinct nonisomorphic semigroups with one element is one. If ''S'' = is a semigroup with one element, then the
Cayley table of ''S'' is
:
The only element in ''S'' is the
zero element 0 of ''S'' and is also the
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 ...
1 of ''S''. However not all semigroup theorists consider the unique element in a semigroup with one element as the zero element of the semigroup. They define zero elements only in semigroups having at least two elements.
In spite of its extreme triviality, the semigroup with one element is important in many situations. It is the starting point for understanding the
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
of semigroups. It serves as a
counterexample in illuminating many situations. For example, the semigroup with one element is the only semigroup in which 0 = 1, that is, the zero element and the identity element are equal.
Further, if ''S'' is a semigroup with one element, the semigroup obtained by adjoining an identity element to ''S'' is isomorphic to the semigroup obtained by adjoining a zero element to ''S''.
The semigroup with one element is also a
group.
In the language of
category theory, any semigroup with one element is a
terminal object in the category of semigroups.
See also
*
Trivial group
*
Zero ring
*
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 ...
*
Empty semigroup
*
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