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| {{about|the algebraic structure|applications to differential equations|C0-semigroup}}
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| In mathematics, a '''semigroup''' is an [[algebraic structure]] consisting of a [[Set (mathematics)|set]] together with an [[associative]] [[binary operation]]. A semigroup generalizes a [[monoid]] in that a semigroup need not have an [[identity element]]. It also (originally) generalized a [[group (mathematics)|group]] (a monoid with all inverses) to a type where every element did not have to have an [[inverse element|inverse]], thus the name '''semigroup'''.
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| The binary operation of a semigroup is most often denoted multiplicatively: <math>x \cdot y</math>, or simply <math>xy</math>, denotes the result of applying the semigroup operation to the ordered pair <math>(x,y)</math>. The operation is required to be associative so that <math>(x \cdot y) \cdot z = x \cdot (y \cdot z)</math> for all ''x'', ''y'' and ''z'', but need not be [[commutativity|commutative]] so that <math>x \cdot y</math> does not have to equal <math>y \cdot x</math> (contrast to the standard multiplication operator on real numbers, where {{nowrap|''xy'' {{=}} ''yx''}}).
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| By definition, a semigroup is an associative [[magma (algebra)|magma]]. A semigroup with an identity element is called a [[monoid]]. A [[group (mathematics)|group]] is then a monoid in which every element has an inverse element. Semigroups must not be confused with [[quasigroup]]s which are sets with a not necessarily associative binary operation such that division is always possible.
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| The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of "memoryless" systems: time-dependent systems that start from scratch at each iteration. In [[applied mathematics]], semigroups are fundamental models for [[linear time-invariant system]]s. In [[partial differential equations]], a semigroup is associated to any equation whose spatial evolution is independent of time. The theory of finite semigroups has been of particular importance in [[theoretical computer science]] since the 1950s because of the natural link between finite semigroups and [[finite automata]]<!-- ({{harvnb|Eilenberg|1973}}, [[#CITEREFEilenberg1976|1976]])-->. In [[probability theory]], semigroups are associated with [[Markov process]]es {{harv|Feller|1971}}.
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| {{Algebraic structures |Group}}
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| == Definition ==
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| A semigroup is a [[set (mathematics)|set]] <math>S</math> together with a [[binary operation]] "<math>\cdot</math>" (that is, a [[function (mathematics)|function]] <math>\cdot:S\times S\rightarrow S</math>) that satisfies the [[associative property]]:
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| '''For all <math>a,b,c\in S</math>, the equation <math>(a\cdot b)\cdot c = a\cdot(b\cdot c)</math> holds.'''
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| More succinctly, a semigroup is an associative [[magma (algebra)|magma]].
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| == Examples of semigroups ==
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| * [[Empty semigroup]]: the empty set forms a semigroup with the empty function as the binary operation.
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| * [[Semigroup with one element]]: there is essentially just one, the singleton {''a''} with operation ''a'' · ''a'' = ''a''.
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| * [[Semigroup with two elements]]: there are five which are essentially different.
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| * The set of positive [[integer]]s with addition.
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| * Square [[Nonnegative matrix|nonnegative matrices]] of a given size with matrix multiplication.
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| * Any [[ring ideal|ideal]] of a [[ring (algebra)|ring]] with the multiplication of the ring.
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| * The set of all finite [[string (computer science)|strings]] over a fixed alphabet Σ with concatenation of strings as the semigroup operation — the so-called "[[free semigroup]] over Σ". With the empty string included, this semigroup becomes the [[free monoid]] over Σ.
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| * A probability distribution F together with all convolution powers of F, with convolution as the operation. This is called a convolution semigroup.
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| * A [[monoid]] is a semigroup with an [[identity element]].
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| * A [[group (mathematics)|group]] is a monoid in which every element has an [[inverse element]].
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| * [[Transformation semigroup]]s and [[transformation monoid|monoids]]
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| * The set of [[continuous]] functions from a [[topological space]] to itself
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| ==Basic concepts==
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| === Identity and zero ===
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| Every semigroup, in fact every [[Magma (algebra)|magma]], has at most one [[identity element]]. A semigroup with identity is called a [[monoid]]. A semigroup without identity may be [[embedding|embedded]] into a monoid simply by adjoining an element <math>e \notin S</math> to <math>S\ </math> and defining <math>e \cdot s = s \cdot e = s</math> for all <math>s \in S \cup \{e\}</math>.<ref>Jacobson (2009), p. 30, ex. 5</ref><ref name="lawson98">Lawson (1998), {{Google books quote|id=2805q4tFiCkC|page=20|text=adjoining an identity|p. 20}}</ref> The notation S<sup>1</sup> denotes a monoid obtained from S by adjoining an identity ''if necessary'' (''S''<sup>1</sup> = ''S'' for a monoid).<ref name="lawson98"/>
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| Similarly, every magma has at most one [[absorbing element]], which in semigroup theory is called a '''zero'''. Analogous to the above construction, for every semigroup ''S'', one defines ''S''<sup>0</sup>, a semigroup with 0 that embeds ''S''.
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| === Subsemigroups and ideals ===
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| The semigroup operation induces an operation on the collection of its subsets: given subsets ''A'' and ''B'' of a semigroup ''S'', their product ''A*B'', written commonly as ''AB'', is the set { ''ab'' | ''a'' in ''A'' and ''b'' in ''B'' }. In terms of this operations, a subset ''A'' is called
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| * a '''subsemigroup''' if ''AA'' is a subset of ''A'',
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| * a '''right ideal''' if ''AS'' is a subset of ''A'', and
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| * a '''left ideal''' if ''SA'' is a subset of ''A''.
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| If ''A'' is both a left ideal and a right ideal then it is called an '''ideal''' (or a '''two-sided ideal''').
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| If ''S'' is a semigroup, then the intersection of any collection of subsemigroups of ''S'' is also a subsemigroup of ''S''.
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| So the subsemigroups of ''S'' form a [[complete lattice]].
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| An example of semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a [[commutative]] semigroup, when it exists, is a group.
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| [[Green's relations]], a set of five [[equivalence relation]]s that characterise the elements in terms of the [[principal ideal]]s they generate, are important tools for analysing the ideals of a semigroup and related notions of structure.
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| === Homomorphisms and congruences ===
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| A '''semigroup [[homomorphism]]''' is a function that preserves semigroup structure. A function {{nowrap begin}}''f'': ''S'' → ''T''{{nowrap end}} between two semigroups is a homomorphism if the equation
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| :{{nowrap begin}}''f''(''ab'') = ''f''(''a'')''f''(''b'').{{nowrap end}}
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| 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''.
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| A semigroup homomorphism between monoids preserves identity if it is a [[monoid homomorphism]]. But there are semigroup homomorphisms which are not monoid homomorphisms, e.g. the canonical embedding of a semigroup <math>S</math> without identity into <math>S^1</math>. Conditions characterizing monoid homomorphisms are discussed further. Let <math>f:S_0\to S_1</math> be a semigroup homomorphism. The image of <math>f</math> is also a semigroup. If <math>S_0</math> is a monoid with an identity element <math>e_0</math>, then <math>f(e_0)</math> is the identity element in the image of <math>f</math>. If <math>S_1</math> is also a monoid with an identity element <math>e_1</math> and <math>e_1</math> belongs to the image of <math>f</math>, then <math>f(e_0)=e_1</math>, i.e. <math>f</math> is a monoid homomorphism. Particularly, if <math>f</math> is [[surjective]], then it is a monoid homomorphism.
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| Two semigroups ''S'' and ''T'' are said to be [[isomorphism|isomorphic]] if there is a [[bijection]] ''f'' : ''S'' ↔ ''T'' with the property that, for any elements ''a'', ''b'' in ''S'', ''f''(''ab'') = ''f''(''a'')''f''(''b''). Isomorphic semigroups have the same structure.
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| A '''semigroup congruence''' <math>\sim</math> is an [[equivalence relation]] that is compatible with the semigroup operation. That is, a subset <math>\sim\;\subseteq S\times S</math> that is an equivalence relation and <math>x\sim y\,</math> and <math>u\sim v\,</math> implies <math>xu\sim yv\,</math> for every <math>x,y,u,v</math> in ''S''. Like any equivalence relation, a semigroup congruence <math>\sim</math> induces [[equivalence class|congruence class]]es
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| :<math>[a]_\sim = \{x\in S\vert\; x\sim a\}</math>
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| and the semigroup operation induces a binary operation <math>\circ</math> on the congruence classes:
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| :<math>[u]_\sim\circ [v]_\sim = [uv]_\sim</math>
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| Because <math>\sim</math> is a congruence, the set of all congruence classes of <math>\sim</math> forms a semigroup with <math>\circ</math>, called the '''quotient semigroup''' or '''factor semigroup''', and denoted <math>S/\sim</math>. The mapping <math>x \mapsto [x]_\sim</math> is a semigroup homomorphism, called the '''quotient map''', '''canonical [[surjection]]''' or '''projection'''; if S is a monoid then quotient semigroup is a monoid with identity <math>[1]_\sim</math>. Conversely, the [[Kernel (set theory)|kernel]] of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of the [[Isomorphism theorems#First Isomorphism Theorem 4|first isomorphism theorem in universal algebra]]. Congruence classes and factor monoids are the objects of study in [[string rewriting system]]s.
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| A '''nuclear congruence''' on ''S'' is one which is the kernel of an endomorphism of ''S''.<ref name=LotII463>Lothaire (2011) p.463</ref>
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| 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]], this is equivalent to saying that the [[ascending chain condition]] holds: there is no infinite strictly ascending chain of congruences on ''S''.<ref ma,e=LotII465>Lothaire (2011) p.465</ref>
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| Every ideal ''I'' of a semigroup induces a subsemigroup, the [[Rees factor semigroup]] via the congruence ''x'' ρ ''y'' ⇔ either ''x'' = ''y'' or both ''x'' and ''y'' are in ''I''.
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| == Structure of semigroups ==
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| 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 { ''x''<sup>''n''</sup> | ''n'' is a positive integer }.
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| If this is finite, then ''x'' is said to be of '''finite order''', otherwise it is of '''infinite order'''.
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| A semigroup is said to be '''periodic''' if all of its elements are of finite order.
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| A semigroup generated by a single element is said to be [[monogenic semigroup|monogenic]] (or [[Cyclic semigroup|cyclic]]). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive [[integer]]s with the operation of addition.
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| If it is finite and nonempty, then it must contain at least one [[idempotent]].
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| It follows that every nonempty periodic semigroup has at least one idempotent.
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| 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.
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| 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. For more on the structure of finite semigroups, see [[Krohn–Rhodes theory]].
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| == Special classes of semigroups ==
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| {{Main|Special classes of semigroups}}
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| * A [[monoid]] is a semigroup with identity.
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| * A subsemigroup is a [[subset]] of a semigroup that is closed under the semigroup operation.
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| * A [[band (algebra)|band]] is a semigroup the operation of which is [[idempotent]].
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| * A [[cancellative semigroup]] is one having the [[cancellation property]]:<ref>{{harv|Clifford|Preston|1967|p=3}}</ref> ''a'' · ''b'' = ''a'' · ''c'' implies ''b'' = ''c'' and similarly for ''b'' · ''a'' = ''c'' · ''a''.
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| * A [[semilattice]] is a semigroup whose operation is idempotent and [[commutativity|commutative]].
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| * [[0-simple]] semigroups.
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| * [[Transformation semigroup]]s: any finite semigroup ''S'' can be represented by transformations of a (state-) set ''Q'' of at most |''S''|+1 states. Each element ''x'' of ''S'' then maps ''Q'' into itself ''x'': ''Q'' → ''Q'' and sequence ''xy'' is defined by ''q''(''xy'') = (''qx'')''y'' 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).
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| * The [[bicyclic semigroup]] is in fact a monoid, which can be described as the [[free semigroup]] on two generators ''p'' and ''q'', under the relation ''p'' ''q'' = 1.
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| * [[C0-semigroup|C<sub>0</sub>-semigroups]].
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| * [[Regular semigroup]]s. Every element ''x'' has at least one inverse ''y'' satisfying ''xyx''=''x'' and ''yxy''=''y''; the elements ''x'' and ''y'' are sometimes called "mutually inverse".
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| * [[Inverse semigroup]]s are regular semigroups where every element has exactly one inverse. Alternatively, a regular semigroup is inverse if and only if any two idempotents commute.
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| * Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Z<sup>d</sup>. These semigroups have applications to [[commutative algebra]].
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| ==Group of fractions==
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| The '''group of fractions''' of a semigroup ''S'' is the group ''G'' = ''G(S)'' generated by the elements of ''S'' as generators and all equations ''xy''=''z'' which hold true in ''S'' as relations.<ref>B. Farb, ''Problems on mapping class groups and related topics'' (Amer. Math. Soc., 2006) page 357. ISBN 0-8218-3838-5</ref> This has a universal property for morphisms from ''S'' to a group.<ref>M. Auslander and D.A. Buchsbaum, ''Groups, rings, modules'' (Harper&Row, 1974) page 50. ISBN 0-06-040387-X</ref> There is an obvious map from ''S'' to ''G(S)'' by sending each element of ''S'' to the corresponding generator.
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| 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 ''A''.''A'' = ''A'' holds for all elements of ''S'', this must be true for all generators of ''G(S)'' as well: which is therefore the [[trivial group]]. It is clearly necessary for embeddability that ''S'' have the [[cancellation property]]. When ''S'' is commutative this condition is also sufficient<ref>{{harv|Clifford|Preston|1961|p=34}}</ref> 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, {{harv|Suschkewitsch|1928}}.<ref>{{cite web|url=http://www.gap-system.org/~history/Extras/Preston_semigroups.html|title=Personal reminiscences of the early history of semigroups|author=G. B. Preston|year=1990|accessdate=2009-05-12|authorlink=Gordon Preston}}</ref> [[Anatoly Maltsev]] gave necessary and conditions for embeddability in 1937.<ref>{{Citation | doi=10.1007/BF01571659 | last=Maltsev | first=A. | authorlink=Anatoly Maltsev | title=On the immersion of an algebraic ring into a field | journal=Math. Annalen | volume=113 | year=1937 | pages=686–691 | postscript=.}}</ref>
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| ==Semigroup methods in partial differential equations==
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| {{Further2|[[C0-semigroup]]}}
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| Semigroup theory can be used to study some problems in the field of [[partial differential equations]]. 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]] (0, 1) ⊂ '''R''' and times ''t'' ≥ 0:
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| :<math>\begin{cases} \partial_{t} u(t, x) = \partial_{x}^{2} u(t, x), & x \in (0, 1), t > 0; \\ u(t, x) = 0, & x \in \{ 0, 1 \}, t > 0; \\ u(t, x) = u_{0} (x), & x \in (0, 1), t = 0. \end{cases}</math>
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| Let ''X'' be the [[Lp space|''L''<sup>''p''</sup> space]] ''L''<sup>2</sup>((0, 1); '''R''') and let ''A'' be the second-derivative operator with [[domain (mathematics)|domain]]
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| :<math>D(A) = \big\{ u \in H^{2} ((0, 1); \mathbf{R}) \big| u(0) = u(1) = 0 \big\}.</math>
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| Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space ''X'':
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| :<math>\begin{cases} \dot{u}(t) = A u (t); \\ u(0) = u_{0}. \end{cases}</math>
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| On an heuristic level, the solution to this problem "ought" to be ''u''(''t'') = exp(''tA'')''u''<sub>0</sub>. 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''<sub>0</sub> at time ''t'' = 0 to the state ''u''(''t'') = exp(''tA'')''u''<sub>0</sub> at time ''t''. The operator ''A'' is said to be the [[C0 semigroup#Infinitesimal generator|infinitesimal generator]] of the semigroup.
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| ==History==
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| The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as [[group (mathematics)|groups]] or [[ring (algebra)|rings]]. A number of sources<ref>[http://jeff560.tripod.com/s.html Earliest Known Uses of Some of the Words of Mathematics]</ref><ref name=Hollings>[http://www.webcitation.org/query?url=http://uk.geocities.com/cdhollings/suschkewitsch3.pdf&date=2009-10-25+04:13:15 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''.
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| [[Anton Suschkewitsch]] 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 semigroup]]s and showed that the minimal ideal (or [[Green's relations]] J-class) of a finite semigroup is simple.<ref name=Hollings/> 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.
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| In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like [[inverse semigroup]]s, as well as monographs focusing on applications in [[algebraic automata theory]], particularly for finite automata, and also in [[functional analysis]].
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| ==Generalizations==
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| {{Group-like structures}}
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| 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]] ''M'' × ''M'' → ''M''.
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| 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.<ref>{{Citation| last=Dudek |first=W.A. |title=On some old problems in ''n''-ary groups |url=http://www.quasigroups.eu/contents/contents8.php?m=trzeci |journal=Quasigroups and Related Systems |year=2001 |volume=8 |pages= 15–36}}</ref> The associative law is generalized as follows: ternary associativity is {{nowrap|(''abc'')''de'' {{=}} ''a''(''bcd'')''e'' {{=}} ''ab''(''cde'')}}, i.e. the string ''abcde'' with any three adjacent elements bracketed. ''N''-ary associativity is a string of length {{nowrap|''n'' + (''n'' − ''1'')}} 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]].
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| 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.
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| ==See also==
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| * [[Absorbing element]]
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| * [[Biordered set]]
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| * [[Empty semigroup]]
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| * [[Identity element]]
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| * [[Light's associativity test]]
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| * [[Semigroup ring]]
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| * [[Weak inverse]]
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| ==Notes==
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| <references/>
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| | |
| ==References==
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| ;General references
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| *{{citation|last= Howie|first= John M.|authorlink=John Mackintosh Howie|title=Fundamentals of Semigroup Theory|year=1995|publisher=[[Clarendon Press]]|isbn=0-19-851194-9}}.
| |
| * {{citation|title=The algebraic theory of semigroups, volume 1|first1=A. H.|last1=Clifford|authorlink1= Alfred H. Clifford |first2=G. B.|last2=Preston|authorlink2=Gordon Preston|publisher=American Mathematical Society|year=1961}}.
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| * {{citation|title=The algebraic theory of semigroups, volume 2|first1=A. H.|last1=Clifford|authorlink1= Alfred H. Clifford |first2=G. B.|last2=Preston|authorlink2=Gordon Preston|publisher=American Mathematical Society|year=1967}}.
| |
| * {{citation|title=Semigroups: an introduction to the structure theory|first=Pierre Antoine|last=Grillet|publisher=Marcel Dekker, Inc.|year=1995}}
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| | |
| ;Specific references
| |
| <!-- there's not enough here about connections with automata to use these as references.
| |
| *{{citation|last=Eilenberg|first=Samuel|authorlink=Samuel Eilenberg|title=Automata, Languages, and Machines (Vol.A)|publisher=[[Academic Press]]|year= 1973|isbn=0-12-234001-9}}
| |
| *{{citation|last=Eilenberg|first=Samuel|authorlink=Samuel Eilenberg|title=Automata, Languages, and Machines (Vol.B)|publisher=Academic Press|year= 1976|isbn=0-12-234002-7}}
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| -->
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| * {{Citation | last1=Feller | first1=William | author1-link=William Feller | title=An introduction to probability theory and its applications. Vol. II. | publisher=[[John Wiley & Sons]] | location=New York | series=Second edition | mr=0270403 | year=1971}}.
| |
| *{{Citation | last1=Hille | first1=Einar | authorlink1=Einar Hille | last2=Phillips | first2=Ralph S. | authorlink2=Ralph Phillips (mathematician) | title=Functional analysis and semi-groups | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0423094 | year=1974}}.
| |
| * {{Citation | last1=Suschkewitsch | first1=Anton | title=Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit | doi=10.1007/BF01459084 | mr=1512437 | year=1928 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=99 | issue=1 | pages=30–50}}.
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| * {{Citation | last1=Kantorovitz | first1=Shmuel | title=Topics in Operator Semigroups.|
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| publisher=Birkhauser| location=Boston, MA |year=2010}}.
| |
| * {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189-1}}
| |
| * {{citation |last1=Lawson |first1=M.V. |authorlink1= |last2= |first2= |authorlink2= |title=Inverse semigroups: the theory of partial symmetries |url= |edition= |series= |volume= |year=1998 |publisher=World Scientific |location= |isbn=978-981-02-3316-7 }}
| |
| * {{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
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| [[Category:Semigroup theory]]
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| [[Category:Algebraic structures]]
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