Adv. Inequal. Appl. 2019, 2019:5 https://doi.org/10.28919/aia/3960 ISSN: 2050-7461
ON GENERATING SEMIGROUPS FROM A FUNCTION
LEJO J. MANAVALAN∗, P.G. ROMEO
Department of Mathematics, Cochin University of Science and Technology, Kalamassery 682022, India
Copyright c2019 the authors. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. A semigroup is a well studied structure so is a function. This paper describes a way in generating a
semigroup from a function on a finite set, using its directed graph. Further this paper also studies the properties of
the semigroup generated.
Keywords:directed graphs; functions; semigroups.
2010 AMS Subject Classification:20M99.
1.
INTRODUCTIONLetα be a function on a setX. Recent studies have shown when a function forms an inner
translation (also known called Cayley functions) of a semigroup. In 1972 Zupnik characterised
all Cayley functions algebraically(in powers of β) [7]. In 2016, Araoujo et all characterised
the Cayley functions using functional digraphs[1]. He also described a process(a problem half
solved) on generating a semigroup from Cayley functions by constructing a graph with
Cay-ley functions as the vertices and identifying the maximal cliques of the common semigroup
graphGS( two Cayley functions are edge connected if appear on the same semigroup). There
are many ways to construct a semigroup from graphs, for example Endomorphism semigroup,
∗Corresponding author
E-mail address: [email protected]
Received November 21, 2018
graph semigroup, commutative graph semigroup, inverse graph semigroup, path semigroup, etc.
There is no instance of a semigroup from a function. This paper describes a way in generating
a semigroup from a single function on a finite set, using directed graphs, studies the properties
of the semigroup generated.
In the sequelα will denote a function mapping a non-empty finite setX onto itself. For any
positive integern, αn denotes thenth iterate ofα. Byα0 we mean the identity function on S, soα0(x) =x. LetX be a finite set, thenT(X)denotes the set of all transformations (functions) fromX toX.
2.
DIRECTEDGRAPHSA directed graph is an ordered pairD= (S,ρ)where the elements ofSare called vertices and
Ais a set of ordered pairs of vertices (binary relation), called arrows, directed edges.
Any pair(a,b)∈ρ is called an arc ofD, which we will write asa→b. A vertexais called
is called an initial vertex inDif there is nobinρ such thatb→a; it is called a terminal vertex
inDif there is nob∈Ssuch thata→b
Definition: A digraphD is called a functional digraph if there is α :S−→S such that for all
x,y∈S,x→yis an arc inDif and only ifα(x) =y.
Such a functional digraph is denoted asDα as there is one and only one function that
repre-sents a functional di-graph.
LetDbe a digraph and if there exists pairwise distinct vertices...,x1,x0,x1, ...ofDsuch that x0−→x1−→...−→xk−1−→x0then the graph is said to have a cycle of lengthkdenoted as
(x0x1...xk−1).Dhas a chain of lengthm, denoted[x0x1...xm]if there are pairwise distinct vertices such thatx0−→x1−→...−→xm inD. SimilarlyDhas a right ray [left ray or double ray] denoted[x0x1x2...i; [h...x2x1x0],h...x1x0x1...i] if there exist pairwise distinct vertices such that x0→x1→x2→...[(...→x2→x1→x0)or(....→x−1→x0→x1→...)]
LetSbe a non-empty set, thenT(S)denoted the set of all functions onS, α ∈T(S)andDα
the directed graph that representsα, then a right ray[x0x1x2....iinDα is called a maximal right
Definition:LetDα be a functional digraph, whereα ∈T(S)
• A left rayL=h...x2x1x0]inDα is called an infinite branch of a cycleC[double rayW] inDα ifx0lies onC[W] andx1does not lie onC[W]. We will refer to any suchLas an infinite branch inDα.
• A chain P= [x0x1...xm] of length m≥1 in Dα is called a finite branch of a cycle C [double rayW, maximal right rayR, infinite branchL] inDα ifx0is an initial vertex of Dα,xmlies onC[W,R,L] andxm≥1 does not lie onC[W,R,L]. Ifxmlies on an infinite
branchL=h...y2y1y0], we also require thatxm6=y0.
The components ofDα correspond to the connected components of the underlying undirected
graph ofDα where the component containingxis defined as follows
Definition:Letα∈T(S),x∈S. The subgraph ofDα induced by the set
{y∈S:αk(y) =αm(x)for some integersk,m≥0}
is called the component ofDα containingx.
The following two propositions is due to [4] shows the character of functional digraphs.
Proposition: LetDα be a functional digraph. Then for every componentAofDα , exactly one
of the following three conditions holds:
(1) A has a unique cycle but not a double ray or right ray;
(2) A has a double ray but not a cycle ; or
(3) A has a maximal right ray but not a cycle or double ray.
Proposition: LetDα be a functional digraph. Then for every componentAofDα :
(1) if A has a (unique) cycle C, then A is the join of C and its branches;
(2) if A has a double ray W, then A is the join of W and its branches;
(3) if A has a maximal right ray R but not a double ray , then A is the join of R and its
(finite) branches(typerro).
As we are dealing only with finite functions every component will be a join of a cycleCand
3.
CONSTRUCTING A SEMIGROUP FROM A FUNCTIONNow consider a functionα ∈T(X), letA1,...Atbe the connected components ofDα now fix
a relation≤onA1,...At such that≤is a total ordering onA1,...At. Without loss of generality,
letA1≤...≤At (otherwise renumbering as required). Now for eachAifix aai∈Aisuch that
aiis the initial vertex of the longest branch of the cycle (if the cycle has no branches then choose
any one vertex in the cycle ), if there two or more branches of the same length choose any one.
For x,y∈X define N(x,y) = {(r,q) :αr(x) = αq(y)}. If N(x,y) is empty then x, y are
in different connected components. If N(x,y) is non-empty then x, y are in same connected
component and in this case we defineδxy=ψxy−τxy whereτxy=min{q:(r,q)∈N(x,y)}and
ψxy=min{r:αr(x) =ατxy(y)}. If you take two verticesx,yin the directed graphDα, thenτxy
is the minimum distance from y to the path travelled by x asα is composed again and again
which is zero if y is on the path, that isτxy=0 if there is an integernsuch thatαn(x) =y, again
ify0=ατxy(y)thenψxyis the minimum distance fromxtoy0.
The following lemma is required to prove the associativity of the semigroup constructed.
Lemma 1:LetXbe a finite set andα∈T(X). Letδxybe as defined above, thenδxαl(y)=δxy+l
Proof. The proof is by induction, whenl=1 we have to prove that
δxα(y)=δxy+1
butδxy=ψxy−τxyandδxα(y)=ψxα(y)−τxα(y)by definition, alsoτxy=min{q:(r,q)∈N(x,y)}
andψxy=min{r:αr(x) =ατxy(y)}
ifτxy=0, τxα(y)=0 andψxα(y)=ψxy+1
ifτxy≥0, τxα(y)=τxy+1 andψxα(y)=ψxy
so either way
Now assume thatδxαl(y)=δxy+lto proveδxαl+1(y)=δxy+l+1,
δxαl+1(y) =δxαl(y)+1 from (1)
=δxy+l+1 from assumption
Hence proved
Now for two elementsx,y∈X,α ∈T(X)define
(2) x∗y=
α|δAaix+1
i (y) ifxandyare in the same connected componentAi.
x ifx∈Ai,y∈AjandAi≥Aj
y ifx∈Ai,y∈AjandAi≤Aj.
To show that(X,∗)forms a semigroup we need to show that its an associative binary
opera-tion(x∗y)∗z=x∗(y∗z)∀x,y,z∈X. For this we have five cases,
Case 1: x,y,zare in 5 different components sayAiAjAk respectively, then
x∗(y∗z) = (x∗y)∗z=xifAi≥Aj,Ai≥Ak
x∗(y∗z) = (x∗y)∗z=yifAj≥Ai,Aj≥Ak
x∗(y∗z) = (x∗y)∗z=zifAk≥Ai,Ak≥Aj
Case 2: x,yare in one component sayAiandzinAj then
x∗(y∗z) = (x∗y)∗z=x∗y ifAi≥Aj
Case 3: x,zare in one component sayAiandyinAj
x∗(y∗z) = (x∗y)∗z=x∗z ifAi≥Aj
x∗(y∗z) = (x∗y)∗z=y ifAi≤Aj
Case 4: y,zare in one component sayAiandxinAj
x∗(y∗z) = (x∗y)∗z=x ifAi≥Aj
x∗(y∗z) = (x∗y)∗z=y∗z ifAi≤Aj
Case 5: x,y,zare in one component sayAi,
Letk=δaiy+1 andl=δaix+1 then
x∗(y∗z) =x∗αδaiy+1(z) =αδaix+1(αδaiy+1(z)) =αk+l(z)
(x∗y)∗z=αδaix+1(y)∗z =α
δ
ai(αδaix+1(y))+1(z) =αδai(αl(y))+1(z) =αδaiy+l+1(z)by lemma 1
=αk+l(z)
Hence∗is a associative operation onX, and so(X,∗)is a semigroup, and this summaries to the
following theorem,
Theorem 1: LetX be a finite set and α ∈T(X)then there exist a semigroup operation on X
derived from the functionα.
So now one can construct a semigroup from any function α on a finite set which I would
rather call a functional semigroup and denote bySα the semigroup constructed from a function
in the above manner. Sα changes with a change in the total order though there is a similarity in
them begin that the principal sub-semigroups are isomorphic.
Theorem 2: LetX be a finite set and α ∈T(X), S1α S2α be two semigroups constructed from α as described in (2), with two different total ordering but same fixed points, then a principal
Proof. Letx∈X, be a vertex ofAia connected component ofα thenx∗x=αδaix+1(x)in both S1α andS2α, invariant of change in the total order, hence the theorem.
Sα changes with a change in the choice of the fixed points ai as well, though there is a
similarity in them being that the sub-semigroups generated byai,andbi,are isomorphic.
Theorem 3: LetX be a finite set andα ∈T(X), Suppose thatα has t connected components
and a connected component (sayAk )such that the length of the longest branch iss and there
is at-least two branches of length s, let the initial vertex say of these branches be ak, bk, ...
LetS1α S2α be two semigroups constructed fromα one with fixed pointsa1, ...aK, ...,at and the
other with fixed points a1, ..bk, ...atwith two different choice of fixed points then a principal
sub-semigroup generated byakS1α is isomorphic to a principal sub-semigroup generated bybk ofSα2. Also the principal sub-semigroupaiis a commutative sub-semigroup.
Proof. Letaibe a fixed point as described, then
ai∗ai=αδaiai+1(ai) =α(ai),
ai∗α(ai) =α(α(ai)) =α2(ai)
α(ai)∗ai=αδaiα(ai)+1(ai) =α2(ai)by lemma 1
and hence we get the sub-semigroup generated byai={αn(ai);n∈N}Also,αn(ai)∗αm(ai) =
αδaiαn(ai)+1(
αm(ai)) =αδaiai+n+1(αm(ai))by lemma 1 =αm+n+1(ai) =αm(ai)∗αn(ai), hence
the commutativity is proved.
To prove thatSak={αn(ak);n∈N}is isomorphic toSbk ={α n(b
k);n∈N}, Letφ :Sak−→
Sbkbe such thatφ(αn(ak)) =αn(bk), then forx=αn(ak),y=αm(ak)∈Sak
φ(x∗y) = φ(αn(ak)∗αm(ak)) = φ(αm+n+1(ak)) = αm+n+1(bk)
φ(x)∗φ(y) = φ(αn(ak))∗φ(αm(ak)) = αn(bk)∗αm(bk) = αm+n+1(bk)
Henceφ is an isomorphism
Hence if the directed graph Dα has at-most one branch in each connected component then
the semigroup generated from α is commutative. Moreover if α is a permutation then the
components thenSα hast maximal-subgroups. Precisely the vertices of a cycle inDα form a
group.
Now comparing the number of semigroup on a finite set and the number of function on a
finite set one can see that [3]
n functions semigroups commutative semigroup
2 4 4 3
3 27 18 12
4 256 126 58
5 3,125 1,160 325
6 46,656 15,973 2,143
7 823,543 836,021 1,7291
8 16,777,216 1,843,120,128 221,805
So now one concludes that most of the semigroups constructed from a function in a manner
described above must be isomorphic.
Theorem 4: LetX be a finite set and α andβ two functions on X, then for anySα there is a
ordering on the directed graphDβ and a choice of verticesbisuch thatSα andSβ are isomorphic
if and only if the directed graphsDα andDβ are digraph isomorphic.
Proof. Suppose thatDα and Dβ are digraph isomorphic with isomorphismφD andSα is
con-structed fromα such thatA1≤...≤Atand for eachAi,ai∈Aiis the chosen fixed vertex, then
letbi=φD(ai)be the fixed vertices andBibe the connected component such thatbi∈Bi. Then
the semigroup constructed with the orderB1≤...≤Btandbithe fixed vertices is isomorphic to
Sα as whena,b∈Ai,φD(a∗b) =φD(αδaia+1(b)) =βδφD(ai)φD(a)+1(φD(b))(by digraph isomorphism) =
φD(a)∗φD(b), whena,bbelong to different components its trivial .
Conversely suppose thatSα andSβ are isomorphic and thatφSis the semigroup-isomorphism,
then φS is also the digraph isomorphism for when α(a) =b then by semigroup isomorphism
φS(ai)∗φS(a) =φS(ai∗a)
φS(ai∗a) =φS(α(a)) =φS(b)
which shows that if there is an edge froma to bin Dα (α(a) =b) thenβ(φS(a)) =φS(b)so
there is an edge fromφS(a)toφS(b)and vice-versa. HenceDα andDβ are digraph isomorphic.
Remark:If in equation (2)α|δAaI x+1
i (y)is replaced byα
δaI x
|Ai (y)then the isomorphism in theorem
4 is not obtained.
This construction could be extended to the case whenX is countably infinite
Conflict of Interests
The authors declare that there is no conflict of interests.
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