TheInfoList

In mathematics, a semigroup is an
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
consisting of a set together with an
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The binary operation of a semigroup is most often denoted : ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of
magmas 300px, Hawaii Hawaii ( ; haw, Hawaii or ) is a U.S. state in the Western United States, in the Pacific Ocean about 2,000 miles (3,200 km) from the U.S. mainland. It is the only state outside North America, the only island state, and ...
, where the operation is associative, or as a generalization of
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is associative but non-commutative is
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup'' or (less often than in the analogous case of groups) it may be called an ''abelian semigroup''. A
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
is an algebraic structure intermediate between groups and semigroups, and is a semigroup having an
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid. A natural example is
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
with
concatenation In formal language theory and computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a specific task. Programming involves ...
as the binary operation, and the empty string as the identity element. Restricting to non-empty
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
gives an example of a semigroup that is not a monoid. Positive
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s with addition form a commutative semigroup that is not a monoid, whereas the non-negative
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s do form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with
quasigroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, which are a generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups a notion of
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
. Division in semigroups (or in monoids) is not possible in general. The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as
transformation semigroupIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
, in which arbitrary functions replace the role of bijections from group theory. A deep result in the classification of finite semigroups is
Krohn–Rhodes theoryIn mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata theory, automata that seeks to decompose them in terms of elementary components. These compone ...
, analogous to the
Jordan–Hölder decompositionIn abstract algebra, a composition series provides a way to break up an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
for finite groups. Some other techniques for studying semigroups, like
Green's relationsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, do not resemble anything in group theory. The theory of finite semigroups has been of particular importance in
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ...

since the 1950s because of the natural link between finite semigroups and
finite automata A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of ...
via the
syntactic monoid In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. In
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
, semigroups are associated with
Markov process A Markov chain or Markov process is a stochastic model In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory tre ...
es. In other areas of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
, semigroups are fundamental models for
linear time-invariant system In system analysis, among other fields of study, a linear time-invariant system (LTI system) is a system that produces an output signal from any input signal subject to the constraints of Linear system#Definition, linearity and Time-invariant sy ...
s. In
partial differential equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, a semigroup is associated to any equation whose spatial evolution is independent of time. There are numerous
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 (set theory), class of semigroups satisfying additional property (philosophy), properties or conditions. Thus the ...
, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention:
regular semigroup In mathematics, a regular semigroup is a semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denote ...
s,
orthodox semigroupIn mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup, ''E''-semigroup. The term ''orthodox semigroup'' was coined by T. ...
s, semigroups with involution,
inverse semigroupIn 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 ''x = xyx'' and ''y = yxy'', ...
s and
cancellative semigroupIn mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality (mathematics), equality of the form ''a''·'' ...
s. There are also interesting classes of semigroups that do not contain any groups except the
trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
; examples of the latter kind are
bands Band or BAND may refer to: Places *Bánd, a village in Hungary *Band, Iran, a village in Urmia County, West Azerbaijan Province, Iran *Band, Mureș, a commune in Romania *Band-e Majid Khan, a village in Bukan County, West Azerbaijan Province, Ira ...
and their commutative subclass—
semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...
s, which are also ordered algebraic structures.

# Definition

A semigroup is a set $S$ together with a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
"$\cdot$" (that is, a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
$\cdot:S\times S\rightarrow S$) that satisfies the
associative property In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
: :For all $a,b,c\in S$, the equation $\left(a\cdot b\right)\cdot c = a\cdot\left(b\cdot c\right)$ holds. More succinctly, a semigroup is an associative
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
.

# Examples of semigroups

* Empty semigroup: the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

forms a semigroup with the
empty function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
as the binary operation. * Semigroup with one element: there is essentially only one (specifically, only one
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

), the singleton with operation . * Semigroup with two elements: there are five which are essentially different. * The "flip-flop" monoid: a semigroup with three elements representing the three operations on a switch - set, reset, and do nothing. * The set of positive
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s with addition. (With 0 included, this becomes a
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
.) * The set of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s with minimum or maximum. (With positive/negative infinity included, this becomes a monoid.) * Square
nonnegative matrices In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of a given size with matrix multiplication. * Any
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
of a ring with the multiplication of the ring. * The set of all finite
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
over a fixed alphabet Σ with
concatenation In formal language theory and computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a specific task. Programming involves ...
of strings as the semigroup operation — the so-called "
free semigroupIn abstract algebra, the free monoid on a set is the monoid In abstract algebra, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, st ...
over Σ". With the empty string included, this semigroup becomes the
free monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
over Σ. * A
probability distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
F together with all convolution powers of F, with convolution as the operation. This is called a convolution semigroup. *
Transformation semigroupIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
s and monoids. * The set of
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s from a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
to itself with composition of functions forms a monoid with the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

acting as the identity. More generally, the
endomorphisms In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
of any object of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
form a monoid under composition. * The product of faces of an arrangement of hyperplanes.

# Basic concepts

## Identity and zero

A left identity of a semigroup $S$ (or more generally,
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
) is an element $e$ such that for all $x$ in $S$, $ex = x$. Similarly, a right identity is an element $f$ such that for all $x$ in $S$, $xf = x$. Left and right identities are both called one-sided identities. A semigroup may have one or more left identities but no right identity, and vice versa. A two-sided identity (or just identity) is an element that is both a left and right identity. Semigroups with a two-sided identity are called
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
s. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity). A semigroup $S$ without identity may be embedded in a monoid formed by adjoining an element $e \notin S$ to $S$ and defining $e \cdot s = s \cdot e = s$ for all $s \in S \cup \$. The notation $S^1$ denotes a monoid obtained from $S$ by adjoining an identity ''if necessary'' ($S^1 = S$ for a monoid). Similarly, every magma has at most one
absorbing elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, which in semigroup theory is called a zero. Analogous to the above construction, for every semigroup $S$, one can define $S^0$, a semigroup with 0 that embeds $S$.

## Subsemigroups and ideals

The semigroup operation induces an operation on the collection of its subsets: given subsets ''A'' and ''B'' of a semigroup ''S'', their product , written commonly as ''AB'', is the set (This notion is defined identically as it is for groups.) In terms of this operation, a subset ''A'' is called * a subsemigroup if ''AA'' is a subset of ''A'', * a right ideal if ''AS'' is a subset of ''A'', and * a left ideal if ''SA'' is a subset of ''A''. If ''A'' is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal). If ''S'' is a semigroup, then the intersection of any collection of subsemigroups of ''S'' is also a subsemigroup of ''S''. So the subsemigroups of ''S'' form a
complete lattice In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
semigroup, when it exists, is a group.
Green's relationsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, a set of five
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s that characterise the elements in terms of the
principal ideal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s they generate, are important tools for analysing the ideals of a semigroup and related notions of structure. The subset with the property that every element commutes with any other element of the semigroup is called the
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of the semigroup. The center of a semigroup is actually a subsemigroup.

## Homomorphisms and congruences

A semigroup
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
is a function that preserves semigroup structure. A function between two semigroups is a homomorphism if the equation :. holds for all elements ''a'', ''b'' in ''S'', i.e. the result is the same when performing the semigroup operation after or before applying the map ''f''. A semigroup homomorphism between monoids preserves identity if it is a
monoid homomorphism In abstract algebra, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
. But there are semigroup homomorphisms which are not monoid homomorphisms, e.g. the canonical embedding of a semigroup $S$ without identity into $S^1$. Conditions characterizing monoid homomorphisms are discussed further. Let $f:S_0\to S_1$ be a semigroup homomorphism. The image of $f$ is also a semigroup. If $S_0$ is a monoid with an identity element $e_0$, then $f\left(e_0\right)$ is the identity element in the image of $f$. If $S_1$ is also a monoid with an identity element $e_1$ and $e_1$ belongs to the image of $f$, then $f\left(e_0\right)=e_1$, i.e. $f$ is a monoid homomorphism. Particularly, if $f$ is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, then it is a monoid homomorphism. Two semigroups ''S'' and ''T'' are said to be
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

if there exists a
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
semigroup homomorphism . Isomorphic semigroups have the same structure. A semigroup congruence $\sim$ is an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
that is compatible with the semigroup operation. That is, a subset $\sim\;\subseteq S\times S$ that is an equivalence relation and $x\sim y\,$ and $u\sim v\,$ implies $xu\sim yv\,$ for every $x,y,u,v$ in ''S''. Like any equivalence relation, a semigroup congruence $\sim$ induces
congruence class #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathem ...
es : and the semigroup operation induces a binary operation $\circ$ on the congruence classes: :\sim Because $\sim$ is a congruence, the set of all congruence classes of $\sim$ forms a semigroup with $\circ$, called the quotient semigroup or factor semigroup, and denoted $S/\!\!\sim$. The mapping is a semigroup homomorphism, called the quotient map, canonical
surjection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

or projection; if ''S'' is a monoid then quotient semigroup is a monoid with identity . Conversely, the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of the first isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study in
string rewriting systemIn theoretical computer science An artistic representation of a Turing machine. Turing machines are used to model general computing devices. Theoretical computer science (TCS) is a subset of general computer science that focuses on mathematical a ...
s. A nuclear congruence on ''S'' is one which is the kernel of an endomorphism of ''S''. A semigroup ''S'' satisfies the maximal condition on congruences if any family of congruences on ''S'', ordered by inclusion, has a maximal element. By
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
, this is equivalent to saying that the
ascending chain conditionIn mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings.Jacobson (2009), p. 1 ...
holds: there is no infinite strictly ascending chain of congruences on ''S''. Every ideal ''I'' of a semigroup induces a factor semigroup, the Rees factor semigroup, via the congruence ρ defined by if either , or both ''x'' and ''y'' are in ''I''.

## Quotients and divisions

The following notions introduce the idea that a semigroup is contained in another one. A semigroup T is a quotient of a semigroup S if there is a surjective semigroup morphism from S to T. For example, $\left(\mathbb Z/2\mathbb Z,+\right)$ is a quotient of $\left(\mathbb Z/4\mathbb Z,+\right)$, using the morphism consisting of taking the remainder modulo 2 of an integer. A semigroup T divides a semigroup S, noted $T\preceq S$ if T is a quotient of a subsemigroup S. In particular, subsemigroups of S divides T, while it is not necessarily the case that there are a quotient of S. Both of those relation are transitive.

# Structure of semigroups

For any subset ''A'' of ''S'' there is a smallest subsemigroup ''T'' of ''S'' which contains ''A'', and we say that ''A'' generates ''T''. A single element ''x'' of ''S'' generates the subsemigroup . If this is finite, then ''x'' is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...
). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has at least one idempotent. A subsemigroup which is also a group is called a subgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent ''e'' of the semigroup there is a unique maximal subgroup containing ''e''. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term ''maximal subgroup'' differs from its standard use in group theory. More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal (ring theory), ideal and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements eight form semigroupsNamely: the trivial semigroup in which (for all ''x'' and ''y'') and its counterpart in which , the semigroups based on multiplication modulo 2 (choosing a or b as the identity element 1), the groups equivalent to addition modulo 2 (choosing a or b to be the identity element 0), and the semigroups in which the elements are either both left identities or both right identities. whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, see
Krohn–Rhodes theoryIn mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata theory, automata that seeks to decompose them in terms of elementary components. These compone ...
.

# Special classes of semigroups

* A
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
is a semigroup with an
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. * A group (mathematics), group is a semigroup with an
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and an inverse element. * A subsemigroup is a subset of a semigroup that is closed under the semigroup operation. * A
cancellative semigroupIn mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality (mathematics), equality of the form ''a''·'' ...
is one having the cancellation property: implies and similarly for . * A band (algebra), band is a semigroup whose operation is idempotent. * A
semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...
is a semigroup whose operation is idempotent and
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

. * 0-simple semigroups. *
Transformation semigroupIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
s: any finite semigroup ''S'' can be represented by transformations of a (state-) set ''Q'' of at most states. Each element ''x'' of ''S'' then maps ''Q'' into itself and sequence ''xy'' is defined by for each ''q'' in ''Q''. Sequencing clearly is an associative operation, here equivalent to function composition. This representation is basic for any automaton or finite-state machine (FSM). * The bicyclic semigroup is in fact a monoid, which can be described as the
free semigroupIn abstract algebra, the free monoid on a set is the monoid In abstract algebra, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, st ...
on two generators ''p'' and ''q'', under the relation . * C0-semigroup, C0-semigroups. * Regular semigroups. Every element ''x'' has at least one inverse ''y'' satisfying and ; the elements ''x'' and ''y'' are sometimes called "mutually inverse". * Inverse semigroups are regular semigroups where every element has exactly one inverse. Alternatively, a regular semigroup is inverse if and only if any two idempotents commute. * Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Zd. These semigroups have applications to commutative algebra.

# Structure theorem for commutative semigroups

There is a structure theorem for commutative semigroups in terms of
semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...
s. A semilattice (or more precisely a meet-semilattice) $\left(L, \le\right)$ is a partially ordered set where every pair of elements $a,b \in L$ has a greatest lower bound, denoted $a \wedge b$. The operation $\wedge$ makes $L$ into a semigroup satisfying the additional idempotence law $a \wedge a = a$. Given a homomorphism $f: S \to L$ from an arbitrary semigroup to a semilattice, each inverse image $S_a = f^ \$ is a (possibly empty) semigroup. Moreover, $S$ becomes graded by $L$, in the sense that :$S_a S_b \subseteq S_.$ If $f$ is onto, the semilattice $L$ is isomorphic to the quotient of $S$ by the equivalence relation $\sim$ such that $x \sim y$ if and only if $f\left(x\right) = f\left(y\right)$. This equivalence relation is a semigroup congruence, as defined above. Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroup $S$, there is a finest congruence $\sim$ such that the quotient of $S$ by this equivalence relation is a semilattice. Denoting this semilattice by $L$, we get a homomorphism $f$ from $S$ onto $L$. As mentioned, $S$ becomes graded by this semilattice. Furthermore, the components $S_a$ are all Special classes of semigroups, Archimedean semigroups. An Archimedean semigroup is one where given any pair of elements $x, y$, there exists an element $z$ and $n > 0$ such that $x^n = y z$. The Archimedean property follows immediately from the ordering in the semilattice $L$, since with this ordering we have $f\left(x\right) \le f\left(y\right)$ if and only if $x^n = y z$ for some $z$ and $n > 0$.

# Group of fractions

The group of fractions or group completion of a semigroup ''S'' is the group (mathematics), group generated by the elements of ''S'' as generators and all equations which hold true in ''S'' as presentation of a group, relations. There is an obvious semigroup homomorphism which sends each element of ''S'' to the corresponding generator. This has a universal property for morphisms from ''S'' to a group: given any group ''H'' and any semigroup homomorphism , there exists a unique group homomorphism with ''k''=''fj''. We may think of ''G'' as the "most general" group that contains a homomorphic image of ''S''. An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, take ''S'' to be the semigroup of subsets of some set ''X'' with set-theoretic intersection as the binary operation (this is an example of a semilattice). Since holds for all elements of ''S'', this must be true for all generators of ''G''(''S'') as well: which is therefore the
trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. It is clearly necessary for embeddability that ''S'' have the cancellation property. When ''S'' is commutative this condition is also sufficient and the Grothendieck group of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups. Anatoly Maltsev gave necessary and sufficient conditions for embeddability in 1937.

# Semigroup methods in partial differential equations

Semigroup theory can be used to study some problems in the field of
partial differential equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial/boundary value problem for the heat equation on the spatial interval (mathematics), interval and times : :$\begin \partial_ u\left(t, x\right) = \partial_^ u\left(t, x\right), & x \in \left(0, 1\right), t > 0; \\ u\left(t, x\right) = 0, & x \in \, t > 0; \\ u\left(t, x\right) = u_ \left(x\right), & x \in \left(0, 1\right), t = 0. \end$ Let be the Lp space, ''L''''p'' space of square-integrable real-valued functions with domain the interval and let ''A'' be the second-derivative operator with domain of a function, domain :$D\left(A\right) = \big\,$ where ''H''2 is a Sobolev space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space ''X'': :$\begin \dot\left(t\right) = A u \left(t\right); \\ u\left(0\right) = u_. \end$ On an heuristic level, the solution to this problem "ought" to be . However, for a rigorous treatment, a meaning must be given to the exponentiation, exponential of ''tA''. As a function of ''t'', exp(''tA'') is a semigroup of operators from ''X'' to itself, taking the initial state ''u''0 at time to the state at time ''t''. The operator ''A'' is said to be the C0 semigroup#Infinitesimal generator, infinitesimal generator of the semigroup.

# History

The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
or ring (algebra), rings. A number of sources An account of Suschkewitsch's paper by Christopher Hollings
/ref> attribute the first use of the term (in French) to J.-A. de Séguier in ''Élements de la Théorie des Groupes Abstraits'' (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's ''Theory of Groups of Finite Order''. Anton Sushkevich obtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finite simple semigroups and showed that the minimal ideal (or
Green's relationsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
J-class) of a finite semigroup is simple. From that point on, the foundations of semigroup theory were further laid by David Rees (mathematician), David Rees, James Alexander Green, Evgenii Sergeevich Lyapin, Alfred H. Clifford and Gordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical called ''Semigroup Forum'' (currently edited by Springer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory. The representation theory of semigroups was developed in 1963 by Boris Schein using binary relations on a set ''A'' and composition of relations for the semigroup product. At an algebraic conference in 1972 Schein surveyed the literature on B''A'', the semigroup of relations on ''A''. In 1997 Schein and Ralph McKenzie proved that every semigroup is isomorphic to a transitive semigroup of binary relations. In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like
inverse semigroupIn 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 ''x = xyx'' and ''y = yxy'', ...
s, as well as monographs focusing on applications in algebraic automata theory, particularly for finite automata, and also in functional analysis.

# Generalizations

If the associativity axiom of a semigroup is dropped, the result is a magma (mathematics), magma, which is nothing more than a set ''M'' equipped with a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
that is closed . Generalizing in a different direction, an ''n''-ary semigroup (also ''n''-semigroup, polyadic semigroup or multiary semigroup) is a generalization of a semigroup to a set ''G'' with a arity, ''n''-ary operation instead of a binary operation. The associative law is generalized as follows: ternary associativity is , i.e. the string ''abcde'' with any three adjacent elements bracketed. ''N''-ary associativity is a string of length with any ''n'' adjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to an n-ary group, ''n''-ary group. A third generalization is the semigroupoid, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities. Infinitary generalizations of commutative semigroups have sometimes been considered by various authors.See references in Udo Hebisch and Hanns Joachim Weinert, ''Semirings and Semifields'', in particular, Section 10, ''Semirings with infinite sums'', in M. Hazewinkel, Handbook of Algebra, Vol. 1, Elsevier, 1996. Notice that in this context the authors use the term ''semimodule'' in place of ''semigroup''.

* Absorbing element * Biordered set * Empty semigroup * Generalized inverse * Identity element * Light's associativity test * Quantum dynamical semigroup * Semigroup ring * Weak inverse

# References

* * * * * * * *

## Specific references

* * * * * * * {{Cite book, last=Lothaire , first=M. , author-link=M. Lothaire , title=Algebraic combinatorics on words , orig-year=2002 , series=Encyclopedia of Mathematics and Its Applications , volume=90, publisher=Cambridge University Press , year=2011 , isbn=978-0-521-18071-9 , zbl=1221.68183 Semigroup theory Algebraic structures