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Semigroup

From Wikipedia, the free encyclopedia
(Redirected fromSemigroup theory)
Algebraic structure consisting of a set with an associative binary operation
Not to be confused withquasigroups.
Algebraic structures betweenmagmas andgroups: Asemigroup is amagma withassociativity. Amonoid is asemigroup with anidentity element.

In mathematics, asemigroup is analgebraic structure consisting of aset together with anassociative internalbinary operation on it.

The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmeticmultiplication):x⋅y{\displaystyle x\cdot y}, or simplyxy{\displaystyle xy}, denotes the result of applying the semigroup operation to theordered pair(x,y){\displaystyle (x,y)}. Associativity is formally expressed as that(x⋅y)⋅z=x⋅(y⋅z){\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z)} for allx{\displaystyle x},y{\displaystyle y} andz{\displaystyle z} in the semigroup.

Semigroups may be considered a special case ofmagmas, where the operation is associative, or as a generalization ofgroups, without requiring the existence of an identity element or inverses.[a] As in the case of groups or magmas, the semigroup operation need not becommutative, sox⋅y{\displaystyle x\cdot y} is not necessarily equal toy⋅x{\displaystyle y\cdot x}; a well-known example of an operation that is associative but non-commutative ismatrix multiplication. If the semigroup operation is commutative, then the semigroup is called acommutative semigroup or (less often than in theanalogous case of groups) it may be called anabelian semigroup.

Amonoid is an algebraic structure intermediate between semigroups and groups, and is a semigroup having anidentity element, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid. A natural example isstrings withconcatenation as the binary operation, and the empty string as the identity element. Restricting to non-emptystrings gives an example of a semigroup that is not a monoid. Positiveintegers with addition form a commutative semigroup that is not a monoid, whereas the non-negativeintegers 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 withquasigroups, which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroupspreserve from groups the notion ofdivision. Division in semigroups (or in monoids) is not possible in general.

The formal study of semigroups began in the early 20th century. Early results includea Cayley theorem for semigroups realizing any semigroup as atransformation semigroup, in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups isKrohn–Rhodes theory, analogous to theJordan–Hölder decomposition for finite groups. Some other techniques for studying semigroups, likeGreen's relations, do not resemble anything in group theory.

The theory of finite semigroups has been of particular importance intheoretical computer science since the 1950s because of the natural link between finite semigroups andfinite automata via thesyntactic monoid. Inprobability theory, semigroups are associated withMarkov processes.[1] In other areas ofapplied mathematics, semigroups are fundamental models forlinear time-invariant systems. Inpartial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time.

There are numerousspecial classes of semigroups, 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 semigroups,orthodox semigroups,semigroups with involution,inverse semigroups andcancellative semigroups. There are also interesting classes of semigroups that do not contain any groups except thetrivial group; examples of the latter kind arebands and their commutative subclass –semilattices, which are alsoordered algebraic structures.

Algebraic structures

Definition

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A semigroup is asetS{\displaystyle S} together with abinary operation⋅{\displaystyle \cdot } (that is, afunction⋅:S×S→S{\displaystyle \cdot :S\times S\to S}) that satisfies theassociative property:

for alla,b,c∈S{\displaystyle a,b,c\in S}, the equation(a⋅b)⋅c=a⋅(b⋅c){\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)} holds.

More succinctly, a semigroup is an associativemagma.

Examples of semigroups

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Basic concepts

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Identity and zero

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Aleft identity of a semigroupS{\displaystyle S} (or more generally,magma) is an elemente{\displaystyle e} such that for allx{\displaystyle x} inS{\displaystyle S},e⋅x=x{\displaystyle e\cdot x=x}. Similarly, aright identity is an elementf{\displaystyle f} such that for allx{\displaystyle x} inS{\displaystyle S},x⋅f=x{\displaystyle x\cdot f=x}. Left and right identities are both calledone-sided identities. A semigroup may have one or more left identities but no right identity, and vice versa.

Atwo-sided identity (or justidentity) is an element that is both a left and right identity. Semigroups with a two-sided identity are calledmonoids. 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 semigroupS{\displaystyle S} without identity may beembedded in a monoid formed by adjoining an elemente∉S{\displaystyle e\notin S} toS{\displaystyle S} and defininge⋅s=s⋅e=s{\displaystyle e\cdot s=s\cdot e=s} for alls∈S∪{e}{\displaystyle s\in S\cup \{e\}}.[2][3] The notationS1{\displaystyle S^{1}} denotes a monoid obtained fromS{\displaystyle S} by adjoining an identity if necessary (S1=S{\displaystyle S^{1}=S} for a monoid).[3]

Similarly, every magma has at most oneabsorbing element, which in semigroup theory is called azero. Analogous to the above construction, for every semigroupS{\displaystyle S}, one can defineS0{\displaystyle S^{0}}, a semigroup with 0 that embedsS{\displaystyle S}.

Subsemigroups and ideals

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The semigroup operation induces an operation on the collection of its subsets: given subsetsA{\displaystyle A} andB{\displaystyle B} of a semigroupS{\displaystyle S}, their productA⋅B{\displaystyle A\cdot B}, written commonly asAB{\displaystyle AB}, is the set{ab∣a∈A and b∈B}{\displaystyle \{ab\mid a\in A{\text{ and }}b\in B\}}. (This notion is defined identically asit is for groups.) In terms of this operation, a subsetA{\displaystyle A} is called

IfA{\displaystyle A} is both a left ideal and a right ideal then it is called anideal (or atwo-sided ideal).

IfS{\displaystyle S} is a semigroup, then the intersection of any collection of subsemigroups ofS{\displaystyle S} is also a subsemigroup ofS{\displaystyle S}.So the subsemigroups ofS{\displaystyle S} form acomplete lattice.

An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of acommutative semigroup, when it exists, is a group.

Green's relations, a set of fiveequivalence relations that characterise the elements in terms of theprincipal ideals 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 thecenter of the semigroup.[4] The center of a semigroup is actually a subsemigroup.[5]

Homomorphisms and congruences

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Asemigrouphomomorphism is a function that preserves semigroup structure. A functionf:S→T{\displaystyle f:S\to T} between two semigroups is a homomorphism if the equation

f(ab)=f(a)f(b){\displaystyle f(ab)=f(a)f(b)}.

holds for all elementsa{\displaystyle a},b{\displaystyle b} inS{\displaystyle S}, i.e. the result is the same when performing the semigroup operation after or before applying the mapf{\displaystyle f}.

A semigroup homomorphism between monoids preserves identity if it is amonoid homomorphism. But there are semigroup homomorphisms that are not monoid homomorphisms, e.g. the canonical embedding of a semigroupS{\displaystyle S} without identity intoS1{\displaystyle S^{1}}. Conditions characterizing monoid homomorphisms are discussed further. Letf:S0→S1{\displaystyle f:S_{0}\to S_{1}} be a semigroup homomorphism. The image off{\displaystyle f} is also a semigroup. IfS0{\displaystyle S_{0}} is a monoid with an identity elemente0{\displaystyle e_{0}}, thenf(e0){\displaystyle f(e_{0})} is the identity element in the image off{\displaystyle f}. IfS1{\displaystyle S_{1}} is also a monoid with an identity elemente1{\displaystyle e_{1}} ande1{\displaystyle e_{1}} belongs to the image off{\displaystyle f}, thenf(e0)=e1{\displaystyle f(e_{0})=e_{1}}, i.e.f{\displaystyle f} is a monoid homomorphism. Particularly, iff{\displaystyle f} issurjective, then it is a monoid homomorphism.

Two semigroupsS{\displaystyle S} andT{\displaystyle T} are said to beisomorphic if there exists abijective semigroup homomorphismf:S→T{\displaystyle f:S\to T}. Isomorphic semigroups have the same structure.

Asemigroup congruence∼{\displaystyle \sim } is anequivalence relation that is compatible with the semigroup operation. That is, a subset∼⊆S×S{\displaystyle \sim \,\subseteq S\times S} that is an equivalence relation andx∼y{\displaystyle x\sim y} andu∼v{\displaystyle u\sim v} impliesxu∼yv{\displaystyle xu\sim yv} for everyx,y,u,v{\displaystyle x,y,u,v} inS{\displaystyle S}. Like any equivalence relation, a semigroup congruence∼{\displaystyle \sim } inducescongruence classes

[a]={x∈S∣x∼a}{\displaystyle [a]=\{x\in S\mid x\sim a\}}

and the semigroup operation induces a binary operation∘{\displaystyle \circ } on the congruence classes:

[u]∘[v]=[uv]{\displaystyle [u]\circ [v]=[uv]}.

Because∼{\displaystyle \sim } is a congruence, the set of all congruence classes of∼{\displaystyle \sim } forms a semigroup with∘{\displaystyle \circ }, called thequotient semigroup orfactor semigroup, and denotedS/∼{\displaystyle S/\!\sim }. The mappingx↦[x]{\displaystyle x\mapsto [x]} is a semigroup homomorphism, called thequotient map,canonicalsurjection orprojection; ifS{\displaystyle S} is a monoid then quotient semigroup is a monoid with identity[1]{\displaystyle [1]}. Conversely, thekernel of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of thefirst isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study instring rewriting systems.

Anuclear congruence onS{\displaystyle S} is one that is the kernel of an endomorphism ofS{\displaystyle S}.[6]

A semigroupS{\displaystyle S} satisfies themaximal condition on congruences if any family of congruences onS{\displaystyle S}, ordered by inclusion, has a maximal element. ByZorn's lemma, this is equivalent to saying that theascending chain condition holds: there is no infinite strictly ascending chain of congruences onS{\displaystyle S}.[7]

Every idealI{\displaystyle I} of a semigroup induces a factor semigroup, theRees factor semigroup, via the congruenceρ{\displaystyle \rho } defined byxρy{\displaystyle x\rho y} if eitherx=y{\displaystyle x=y}, or bothx{\displaystyle x} andy{\displaystyle y} are inI{\displaystyle I}.

Quotients and divisions

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The following notions[8] introduce the idea that a semigroup is contained in another one.

A semigroupT is a quotient of a semigroupS if there is a surjective semigroup morphism fromS toT. For example,(Z/2Z, +) is a quotient of(Z/4Z, +), using the morphism consisting of taking the remainder modulo 2 of an integer.

A semigroupT divides a semigroupS, denotedT ≼S ifT is a quotient of a subsemigroupS. In particular, subsemigroups ofS dividesT, while it is not necessarily the case that there are a quotient ofS.

Both of those relations are transitive.

Structure of semigroups

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For any subsetA ofS there is a smallest subsemigroupT ofS that containsA, and we say thatAgeneratesT. A single elementx ofS generates the subsemigroup{xn |n ∈Z+ }. If this is finite, thenx is said to be offinite order, otherwise it is ofinfinite order.A semigroup is said to beperiodic if all of its elements are of finite order.A semigroup generated by a single element is said to bemonogenic (orcyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positiveintegers with the operation of addition.If it is finite and nonempty, then it must contain at least oneidempotent.It follows that every nonempty periodic semigroup has at least one idempotent.

A subsemigroup that is also a group is called asubgroup. 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 idempotente of the semigroup there is a unique maximal subgroup containinge. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the termmaximal 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 minimalideal 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{a,b}, eight form semigroups[b] whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, seeKrohn–Rhodes theory.

Special classes of semigroups

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Main article:Special classes of semigroups
  • Amonoid is a semigroup with anidentity element.
  • Agroup is a monoid in which every element has aninverse element.
  • A subsemigroup is asubset of a semigroup that is closed under the semigroup operation.
  • Acancellative semigroup is one having thecancellation property:[9]a ·b =a ·c impliesb =c and similarly forb ·a =c ·a. Every group is a cancellative semigroup, and every finite cancellative semigroup is a group.
  • Aband is a semigroup whose operation isidempotent.
  • Asemilattice is a semigroup whose operation is idempotent andcommutative.
  • 0-simple semigroups.
  • Transformation semigroups: any finite semigroupS can be represented by transformations of a (state-) setQ of at most|S| + 1 states. Each elementx ofS then mapsQ into itselfx :Q →Q and sequencexy is defined byq(xy) = (qx)y for eachq inQ. Sequencing clearly is an associative operation, here equivalent tofunction composition. This representation is basic for anyautomaton orfinite-state machine (FSM).
  • Thebicyclic semigroup is in fact a monoid, which can be described as thefree semigroup on two generatorsp andq, under the relationpq = 1.
  • C0-semigroups.
  • Regular semigroups. Every elementx has at least one inversey that satisfiesxyx =x andyxy =y; the elementsx andy 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 tocommutative algebra.

Structure theorem for commutative semigroups

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There is a structure theorem for commutative semigroups in terms ofsemilattices.[10] A semilattice (or more precisely a meet-semilattice)(L, ≤) is apartially ordered set where every pair of elementsa,b ∈L has agreatest lower bound, denoteda ∧b. The operation ∧ makesL into a semigroup that satisfies the additionalidempotence lawa ∧a =a.

Given a homomorphismf :S →L from an arbitrary semigroup to a semilattice, each inverse imageSa =f−1{a} is a (possibly empty) semigroup. Moreover,S becomesgraded byL, in the sense thatSaSb ⊆Sa∧b.

Iff is onto, the semilatticeL is isomorphic to thequotient ofS by the equivalence relation ~ such thatx ~y if and only iff(x) =f(y). 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 semigroupS, there is a finest congruence ~ such that the quotient ofS by this equivalence relation is a semilattice. Denoting this semilattice byL, we get a homomorphismf fromS ontoL. As mentioned,S becomes graded by this semilattice.

Furthermore, the componentsSa are allArchimedean semigroups. An Archimedean semigroup is one where given any pair of elementsx,y, there exists an elementz andn > 0 such thatxn =yz.

The Archimedean property follows immediately from the ordering in the semilatticeL, since with this ordering we havef(x) ≤f(y) if and only ifxn =yz for somez andn > 0.

Group of fractions

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Thegroup of fractions orgroup completion of a semigroupS is thegroupG =G(S) generated by the elements ofS as generators and all equationsxy =z that hold true inS asrelations.[11] There is an obvious semigroup homomorphismj :S →G(S) that sends each element ofS to the corresponding generator. This has auniversal property for morphisms fromS to a group:[12] given any groupH and any semigroup homomorphismk :S →H, there exists a uniquegroup homomorphismf :G →H withk =fj. We may think ofG as the "most general" group that contains a homomorphic image ofS.

An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, takeS to be the semigroup of subsets of some setX withset-theoretic intersection as the binary operation (this is an example of a semilattice). SinceA.A =A holds for all elements ofS, this must be true for all generators ofG(S) as well, which is therefore thetrivial group. It is clearly necessary for embeddability thatS have thecancellation property. WhenS is commutative this condition is also sufficient,[13] and the group of fractions can be constructed as theGrothendieck group of the semigroup, or via a minor variant of the standard construction of thefield of fractions of an integral domain.[14] The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups.[15][16]Anatoly Maltsev gave necessary and sufficient conditions for embeddability in 1937.[17]

Semigroup methods in partial differential equations

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Further information:C0-semigroup

Semigroup theory can be used to study some problems in the field ofpartial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as anordinary differential equation on a function space. For example, consider the following initial/boundary value problem for theheat equation on the spatialinterval(0, 1) ⊂R and timest ≥ 0:

{∂tu(t,x)=∂x2u(t,x),x∈(0,1),t>0;u(t,x)=0,x∈{0,1},t>0;u(t,x)=u0(x),x∈(0,1),t=0.{\displaystyle {\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}}}

LetX =L2((0, 1)R) be theLp space of square-integrable real-valued functions with domain the interval(0, 1) and letA be the second-derivative operator withdomain

D(A)={u∈H2((0,1);R)|u(0)=u(1)=0},{\displaystyle D(A)={\big \{}u\in H^{2}((0,1);\mathbf {R} ){\big |}u(0)=u(1)=0{\big \}},}

whereH2{\displaystyle H^{2}} is aSobolev space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the spaceX:

{u˙(t)=Au(t);u(0)=u0.{\displaystyle {\begin{cases}{\dot {u}}(t)=Au(t);\\u(0)=u_{0}.\end{cases}}}

On an heuristic level, the solution to this problem "ought" to beu(t)=exp⁡(tA)u0.{\displaystyle u(t)=\exp(tA)u_{0}.}However, for a rigorous treatment, a meaning must be given to theexponential oftA. As a function oft, exp(tA) is a semigroup of operators fromX to itself, taking the initial stateu0 at timet = 0 to the stateu(t) = exp(tA)u0 at timet. The operatorA is said to be theinfinitesimal generator of the semigroup.

History

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The study of semigroups trailed behind that of other algebraic structures with more complex axioms such asgroups orrings. A number of sources[18][19] attribute the first use of the term (in French) to J.-A. de Séguier inÉlements 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'sTheory 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 finitesimple semigroups and showed that the minimal ideal (orGreen's relations J-class) of a finite semigroup is simple.[19] From that point on, the foundations of semigroup theory were further laid byDavid Rees,James Alexander Green,Evgenii Sergeevich Lyapin [fr],Alfred H. Clifford andGordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical calledSemigroup Forum (currently published bySpringer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory.

Therepresentation theory of semigroups was developed in 1963 byBoris Schein usingbinary relations on a setA andcomposition of relations for the semigroup product.[20] At an algebraic conference in 1972 Schein surveyed the literature on BA, the semigroup of relations onA.[21] In 1997 Schein andRalph McKenzie proved that every semigroup is isomorphic to a transitive semigroup of binary relations.[22]

In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, likeinverse semigroups, as well as monographs focusing on applications inalgebraic automata theory, particularly for finite automata, and also infunctional analysis.

Generalizations

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Group-like structures
TotalAssociativeIdentityDivisible
Partial magmaUnneededUnneededUnneededUnneeded
SemigroupoidUnneededRequiredUnneededUnneeded
Small categoryUnneededRequiredRequiredUnneeded
GroupoidUnneededRequiredRequiredRequired
MagmaRequiredUnneededUnneededUnneeded
QuasigroupRequiredUnneededUnneededRequired
Unital magmaRequiredUnneededRequiredUnneeded
LoopRequiredUnneededRequiredRequired
SemigroupRequiredRequiredUnneededUnneeded
AssociativequasigroupRequiredRequiredUnneededRequired
MonoidRequiredRequiredRequiredUnneeded
GroupRequiredRequiredRequiredRequired

If the associativity axiom of a semigroup is dropped, the result is amagma, which is nothing more than a setM equipped with abinary operation that is closedM ×M →M.

Generalizing in a different direction, ann-ary semigroup (alson-semigroup,polyadic semigroup ormultiary semigroup) is a generalization of a semigroup to a setG with an-ary operation instead of a binary operation.[23] The associative law is generalized as follows: ternary associativity is(abc)de =a(bcd)e =ab(cde), i.e. the stringabcde with any three adjacent elements bracketed.n-ary associativity is a string of lengthn + (n − 1) with anyn adjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to ann-ary group.

A third generalization is thesemigroupoid, 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.[c]

See also

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Notes

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  1. ^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.
  2. ^Namely: the trivial semigroup in which (for allx andy)xy =a and its counterpart in whichxy =b, the semigroups based on multiplication modulo 2 (choosinga orb as the identity element 1), the groups equivalent to addition modulo 2 (choosinga orb to be the identity element 0), and the semigroups in which the elements are either both left identities or both right identities.
  3. ^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 termsemimodule in place ofsemigroup.

Citations

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  1. ^Feller 1971
  2. ^Jacobson 2009, p. 30, ex. 5
  3. ^abLawson 1998, p. 20
  4. ^Kilp, Mati; Knauer, U.; Mikhalev, Aleksandr V. (2000).Monoids, Acts, and Categories: With Applications to Wreath Products and Graphs : a Handbook for Students and Researchers. Walter de Gruyter. p. 25.ISBN 978-3-11-015248-7.Zbl 0945.20036.
  5. ^Li͡apin, E. S. (1968).Semigroups. American Mathematical Soc. p. 96.ISBN 978-0-8218-8641-0.
  6. ^Lothaire 2011, p. 463
  7. ^Lothaire 2011, p. 465
  8. ^Pin, Jean-Éric (November 30, 2016).Mathematical Foundations of Automata Theory(PDF). p. 19.
  9. ^Clifford & Preston 2010, p. 3
  10. ^Grillet 2001
  11. ^Farb, B. (2006).Problems on mapping class groups and related topics. Amer. Math. Soc. p. 357.ISBN 978-0-8218-3838-9.
  12. ^Auslander, M.; Buchsbaum, D. A. (1974).Groups, rings, modules. Harper & Row. p. 50.ISBN 978-0-06-040387-4.
  13. ^Clifford & Preston 1961, p. 34
  14. ^Cain 2012, Theorem 6.1.
  15. ^Suschkewitsch 1928
  16. ^Preston, G. B. (1990).Personal reminiscences of the early history of semigroups. Archived fromthe original on 2009-01-09. Retrieved2009-05-12.
  17. ^Maltsev, A. (1937). "On the immersion of an algebraic ring into a field".Math. Annalen.113:686–691.doi:10.1007/BF01571659.S2CID 122295935.
  18. ^"Earliest Known Uses of Some of the Words of Mathematics".
  19. ^ab"An account of Suschkewitsch's paper by Christopher Hollings"(PDF). Archived fromthe original(PDF) on 2009-10-25.
  20. ^B. M. Schein (1963) "Representations of semigroups by means of binary relations" (Russian),Matematicheskii Sbornik 60: 292–303MR 0153760
  21. ^B. M. Schein (1972)Miniconference on semigroup Theory,MR 0401970
  22. ^B. M. Schein & R. McKenzie (1997) "Every semigroup is isomorphic to a transitive semigroup of binary relations",Transactions of the American Mathematical Society 349(1): 271–85MR 1370647
  23. ^Dudek, W.A. (2001)."On some old problems inn-ary groups".Quasigroups and Related Systems.8:15–36. Archived fromthe original on 2009-07-14.

References

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General references

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Specific references

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