F1000ResearchF1000Research2046-1402F1000 Research LimitedLondon, UK10.12688/f1000research.173831.1Research ArticleArticlesStructure of the (Total) Transformation Monoids Under Rank N Generators
[version 1; peer review: 2 approved]
M. SulaimanHalaConceptualizationFormal AnalysisMethodologyResourcesValidationWriting – Original Draft Preparation12Al-AadhamiAsawerConceptualizationFormal AnalysisInvestigationResourcesSupervisionWriting – Review & Editinghttps://orcid.org/0000-0002-4414-9144a1
Department of Mathematics, College of Science, University of Baghdad, Baghdad, Baghdad Governorate, Iraq
Department of Administrative and Financial Affairs, University of Technology, Baghdad, Iraqasawer.d@sc.uobaghdad.edu.iq
Throughout this paper our study focuses on transformation semigroups. These kinds of semigroups are the corner stone of semigroup theory. This is because every semigroup is isomorphic to transformation semigroup. The (total) transformation monoid
TXn on a finite set
Xn={1,2,…,n} where
n≥0,
n∈Z, is a semigroup of mapping that takes a set
Xn into itself, under the operation of composition of mapping with identity
IXn. In this paper, we use an algebraic method for considering the monoid
T(Fl)n(G), where an independence algebra
(Fl)n(G) is a disjointed union of sets of the form
Gxifor all 1
≤i≤n. Firstly, particular attention is paid to find the isomorphism between
T(Fl)n(G)and the endomorphism monoid
End(Fℓ)n(G). Secondly, the embeddedness of
T(Fl)n(G)in (full) wreath product of
Tn by
Gn has been found. Finally, the description of Green’s relation of
T(Fl)n(G)has been provided.
A finite groupSemigroupWreath productA free (left) G-actTotal transformation semigroupEndomorphism monoidSemigroup morphism.The author(s) declared that no grants were involved in supporting this work.1. Introduction
In this paper
G will be a finite monoid (group) and
(Fℓ)n(G) be a free (left)-
G-act of rank
n. It is clear that
(Fℓ)n(G)is an independence algebra,
1 which is a class of universal algebras of rank
n generators that includes free (left)-
G-acts for a finite group
G. For a positive integer
n, we write the (total) transformation
TXn to be
Tn which is the monoid of
n degree, under the operation of a compositions of functions, such that for all
σ,τ∈Tn and
x∈Xn we have
x(σ∘τ)=x(στ)=(xσ)τ. As well as, it is a set that maps every transformations into itself (i.e.,
Xn⟶Xn,
∀n≥0,n∈Z). Clearly, if
n=0, then
Xn=∅, and
T0 becomes a monoid that consists of only one map, which is the (empty) map.
All transformations of this paper are composed from left to right, so for any
σ∈Tn and for any
i∈Xn, then
iσ will be written as the image of
i, and for any
σ∈Tn, we define
Im{σ}={iσ:i∈Xn} and
Ker{σ}={(i,j)∈Xn×Xn:iσ=jσ} and
℘(σ)=rank{σ}=|Im{σ}|. A number of authors appeared in
2–
9 obtained and described various transformation semigroups, such as the description of the singular transformation
Snℊn,10 and they study the structure of the partial endomorphism monoids of the independence algebra like a free acts and other set such as vector spaces and modules. When
G is a finite monoids the endomorphism monoids of free (left)-
G-acts played a major part of semigroup theory, and it can be found in.
11
It has long been known that set of the endomorphism monoid of the free (left)-
G-acts of rank
n is isomorphic to the (full) wreath product
G≀nTn,12,
13 or, more generally
S≀nTn for arbitrary semigroup
S. Clearly, if
G={e}, then
G≀nTn≅Tn. Where
S is a finite semigroup, the (total) transformation monoid
T(Fℓ)n(S)of a free (left)
S−act of rank
n has been described in,
12 for further information we recommend the reader see.
14–
21 Through this work, the discerption of
T(Fℓ)n(M), has established in particular when
M is a finite group, and many properties of this concepts have been studied such as the isomorphic of
T(Fℓ)n(M) to
End(Fℓ)n(M) and the embeddedness of it in the (total) wreath product
M≀nTn. Furthermore, in the last section of this work the discerption of Green’s relation of
T(Fℓ)n(M) has been found.
2. Preliminaries
12 A mapping
σfrom a monoid
A to a monoid
B is named (morphism) if
(
aa¯)
σ=(a)σ(a¯)σ, for all
a,
a¯∈A;
(
1A)σ=1B.
12 Let
B be a non-empty set and
A be a monoid, then
B is refed to a (left)-
A-act if there is a mapping
α:A×B⟶B,defined by
α(a,b)=ab, such the following conditions are satisfied:
a(rb)=(ar)b.
1Ab=b for all
b∈B and
a,r∈A.
12 Suppose that
Y and
Tare two (left)-
A-acts, then a mapping
α:Y⟶T is named an
A- acts morphism if
(ay)α=a(yα),
∀a∈A,y∈Y.
12,
13 The (total) transformation semigroup on a non-empty set
X is denoted by
TX={σ|σ:X→X}. If
n≥0, such that
n∈Z, where
X=Xn={1,2,…,n}, we write
TX to be
Tn. With respect to the semigroup operation of composition that
x(δ∘γ)=x(
δγ)
=(xδ)γ, for all
x∈X and
δ,γ∈Tx. Clearly,
Tn is monoid with identity transformation
IX=(12…n12…n).
12,
13
If
σ∈Tn,
n≥0,
n∈Z, then
σ can be written as
σ=(1…n1σ…nσ).
For any
σ∈Tn, the second row of
σ is not a permutation for the first row. Since not all element
σ∈Tnis injective.
The number of elements of
Tn is equal to
nn.
Recall that, if
S is a semigroup and if
t∈S, the (principal) left ideal generated by
t is the smallest left ideal of
S containing
t, and it is denoted by
S1t. Dually, for
tS1.
22
Now, let’s have the following definition:
22 Let
S be any semigroup, the Green’s preorder binary relations are defined as follows:
s≤Lt if and only if
s∈S1t.
s≤Rt if and only if
s∈tS1.
s≤Jt if and only if
s∈S1tS1.
From the above definition we can define the Green’s relations by:
22 Let
S be any semigroup, the Green’s relations are defined by:
R=≤R∩≥R.
L=≤L∩≥L.
J=≤J∩≥J.
H=R∩L.
D=R∘L.
Clearly, if
S is finite, then
D=J. So, the above definition can be translated as follows:
22 Let
S be a finite semigroup, the Green’s relations can be defined as:
This section is devoted to give an explicit description of the set of the free (left)-
G-acts, when
G is a finite group.
12,
22 Let
G be a finite group and
X be a non-empty set, then
(Fℓ)X(G) is known as a free (left)-
G-acts on
X if:
For any mapping
σ:X⟶(Fℓ)X(G);
For all
G-acts
Y and every map
δ:X⟶Y, there exists a unique morphism
ϕ:(Fℓ)X(G)⟶Y such that the following diagram is commute:
If
X=Xn={1,2,…,n} where
n≥0 and
n∈Z, the free (left)-
G-act can be written as
(Fℓ)n(G).
It is clear that
(Fℓ)n(G) consists elements of the form
gx, where
g∈G and
x∈X i.e.,
(Fℓ)n(G)=∪̇i=1i=nGxi, such that
Gxi={gxi:g∈G,1≤i≤n,xi∈X}.1,
12 It is well-known that the set of all the morphisms
σ:B⟶B is called the endomorphism monoid, and it is denoted by
End(B).Therefore, the set of all morphisms from the free (left)-
G-acts into itself is an endomorphism monoid and it is denoted by
End(Fℓ)n(G) such that
End(Fℓ)n(G)={σ|σ:(Fℓ)n(G)⟶(Fℓ)n(G); and
σ is a
G−morphism}. If
σ∈End(Fℓ)n(G), we define
σ to be
xjσ=gjσxjσ´, where 1
≤j≤n,
σ´∈Tn, and
gjσ∈Gn. As
σ is a
G−morphism and for
g∈G and 1
≤j≤n we have
(gxj)σ=g(xjσ)=g(gjσxjσ´).
4. Properties of the (full) wreath product multiplication
The (full) wreath product multiplication in semigroup theory is a multiplication comes from a semidirect product between two semigroups. The next definition gives an abstract construction of the (full) wreath product multiplication.
12,
13 Let
Gbe a finite group and
Tn be the (total) transformation on
Xn, where
n≥0,n∈Z.
Then, the (full) wreath product of
Gnby
Tn is
(g1δ,…,gnδ,δ)(g1σ,…,gnσ,σ)=(g1δg1δσ,…,gnδgnδσ,δσ),where
δ,σ∈Tn and
gjδ,gjσ∈Gn.Under this multiplication, the set (
Gn×Tn) is a monoid which is denoted by
G≀nTn, with identity
(e,e,…,en,In), when
Inis the identity of
Tn.
Suppose
B be an algebra and
B be a universe of
B,
1,
12,
13 The sets
TB={σ|σ:B⟶B,∀σ∈TB}, and
TB={τ|τ:B⟶B, such that
τ is a morphism
,∀τ∈TB} are monoids. Furthermore,
TB is submonoid of
TB, this result has been proved by Al-Aadhami in,
12 and this can be shown by the following lemma:
In this subsection we give an explicit description to the (total) transformation monoid
T(Fℓ)n(G).
The set of all morphisms
T(Fℓ)n(G)={σ|σ:(Fℓ)n(G)⟶(Fℓ)n(G), such that
σ is morphism} is a monoid. If
γ∈T(Fℓ)n(G), then
γ can be expressed as:
γ=(xi1…ximℊi1γxi1γ´…ℊimγximγ´).
Where
γ´∈Tn,ℊi1γ,…,ℊimγ∈G such that 1
≤i1<…<im≤n,m≥0 and
xikγ=ℊikγxikγ´. Notice, for every selection of
σ´∈Tn with
Dom(σ´)={ℓ1,….,ℓr} for
1≤ℓ1<…<ℓr≤n,r≥0 and
ℊℓ1σ,…,ℊℓrσ∈G this gives
σ=(xℓ1…xℓrℊℓ1σxℓ1σ´…ℊℓrσxℓrσ´)∈T(Fℓ)n(G).
Clearly, where
G={e}, i.e., (
G is a trivial set), then
T(Fℓ)n(G) will be isomorphic to
Tn.
The next lemma is analogous to Lemma 4.1.
12 However, the following lemma a semigroup
S is assumed to be a finite group
G.
If
n≥0,
n∈Z, then
T(Fℓ)n(G)≅End(Fℓ)n(G).
Suppose
σ be a mapping such that
σ:T(Fℓ)n(G)⟶End(Fℓ)n(G) defined by
λσ=λ¯ when
(gxi)λ¯=(gxi)λ, for all
g∈G. We must show that
λ¯ is a
G-acts morphism, i.e., we want to show that
g(bλ¯)=(gb)λ¯, for all
g∈G and
b∈(Fℓ)n(G). Let
b=kxi. Clearly, an element
xi∈Dom(λ)if and only if
kxi∈Dom(λ) for all
k∈G. Where
kxi∈Dom(λ), that means for all
g∈G,
gkxi∈Dom(λ)then
(gkxi)λ¯=(gkxi)λ=g((kxi)λ)=g((kxi)λ¯),since
λ is a
G-acts morphism.
To show
σ is injective. Assume that
λσ=γσ, then for any
gxi∈(Fℓ)n(G) obtaining (
gxi)λσ=(gxi)γσ,∀i. Notice that, (
gxi)λ=(gxi)λ¯=(gxi)λσ=(gxi)γσ=(gxi)γ¯=(gxi)γ, so
λ=γ, as required.
To prove
σ is onto. Let
β¯∈End(Fℓ)n(G) and let
β∈T(Fℓ)n(G) defined from
(Fℓ)n(G) to
(Fℓ)n(G) by
(gxi)β=(gxi)β¯, for each
gxi∈(Fℓ)n(G). As
β∈T(Fℓ)n(G)that implies
β is a
G-acts morphism, therefore for some
gxi∈Dom(β)=(Fℓ)n(G) and
k∈G, we have
k((gxi)β)=k((gxi)β¯)=(kgxi)β¯=(kgxi)β.
Since,
(gxi)β=(gxi)β¯=(gxi)βσ, for all
gxi∈(Fℓ)n(G),
g∈G. Therefore,
βσ=β¯, that gives
σis onto.
To prove
σ is a homomorphism. Let
δ,γ∈T(Fℓ)n(G).We need
(δγ)σ=δσγσ. From the definition of
σ and from our assumption we have
(gxi)(δγ)σ=(gxi)δγ¯=(gxi)δγ∀i. Conversely,
(gxi)(δσγσ)=(gxi)δ¯γ¯.
Obviously,
δ,¯γ¯∈End(Fℓ)n(G), so
δ,¯and
γ¯ are a
G-acts morphisms. Then,
(gxi)δ¯γ¯=((gxi)δ¯)γ¯=((gxi)δ)γ¯=((gxi)δ)γ=(gxi)δγ=(gxi)δγ¯. So,
(δγ)σ=δγ¯=δ¯γ¯=δγ, and this complete the proof.
For all
n≥0,n∈Z,T(Fℓ)n(G)↪G≀nTn.
Assume that
σ∈T(Fℓ)n(G), then
σ can be written as
=(xi1…ximgi1σxi1σ´…gimσximσ´), such that
σ´∈Tn,gi1σ,…,gimσ∈G for 1
≤i1<…<im≤n,m≥0. Let
α:T(Fℓ)n(G)⟶G≀nTndefined by
μα=(g1μ,…,gnμ,μ´). In order to prove
α is embedding, we have to prove
α is one to one and a homomorphism map. Let
δ,γ∈T(Fℓ)n(G), we need show (
δγ)
α=δαγα. Now,
δαγα=(g1δ,…,gnδ,δ´)(g1γ,…,gnγ,γ´)=(g1δg1δ´γ,…,gnδgnδ´γ,δ´γ´). The other side gives (
δγ)
α=(g1δγ,…,gnδγ,(δγ)′). As
δ,γ∈T(Fℓ)n(G) then
δ,γ are
G-acts morphisms such that they take
T(Fℓ)n(G) to itself. Hence,
(xi)δγ=(xiδ)γ=(giδxiδ´)γ=giδ(xiδ´γ)=giδgiδ´γxiδ´γ´∀i, and (
xi)δγ=giδγxi(δγ)′. From that,
∀i we get
giδγxi(δγ)′=giδgiδ´γxiδ´γ´, so we obtain
giδγ=giδgiδ´γ and
(δγ)′=δ´γ´. This implies,
αis homomorphism.
To prove
α is one-to-one. Let
δ,γ∈T(Fℓ)n(G)such that
δα=γα that leads to
(g1δ,…,gnδ,δ´)=(g1γ,…,gnγ,γ´).
Then for any
gxi∈(Fℓ)n(G), we have
(gxi)δ=g(xiδ)=g(giδxiδ´)=g(giγxiγ´)=g(xiγ)=(gxi)γ, and
Where
Bis an independence algebra V. Gould,
1 gave the characterization of Green’s relations on
End(B), after that A. Al-Aadhami,
12 described the Green’s relations on
End(Fℓ)n(S), where
S is a finite semigroup. In this section the characterization of Green’s relations on
T(Fℓ)n(G) have been illustrated.
For all
σ,τ∈T(Fℓ)n(G), we get the following identities:
σ≤Lτif and only if
Im{σ}⊆Im{τ}.
σ≤Rτ if and only if
Ker{τ}⊆Ker{σ}.
℘(στ)≤℘(σ),℘(τ).
⟹)Let
σ≤Lτ in
T(Fℓ)n(G) that implies
σ=δτ for some
δ∈T(Fℓ)n(G)and so by,
12 [Proposition 5.5] we have
Im{σ}=Im{δτ}⊆Im{τ}, and hence
Im{σ}⊆Im{τ}.
⇐)Assume that
Im{σ}⊆Im{τ}, so for all
j∈{1,2,…,n}, we get
xjσ∈Im{σ}⊆Im{τ}.
If we select
bj∈(Fℓ)n(G) such that
xjσ=bjτ, then by defining
υ∈T(Fℓ)n(G) such that
xjυ=bj for
j∈{1,2,…,n}. It is clear that
xjυτ=bjτ=xjσand therefore we can obtain
υτ=σ that means
σ≤Lτ.
⟹(Where
σ≤Rτ in
T(Fℓ)n(G) that implies
σ=τλ for any
λ∈T(Fℓ)n(G). Assume
(x,y)∈Ker{τ}, hence
xτ=yτ. Therefore,
xσ=x(τλ)=(xτ)λ(asτ,λareG−acts morphism)=(yτ)λ=y(τλ)(asτ,λareG−acts morphism)=yσ.
That means,
(x,y)∈Ker{σ},and so
Ker{τ}⊆Ker{σ}.
⇐) If
Ker{τ}⊆Ker{σ}, let
ϖ∈T(Fℓ)n(G) such that
ϖ:
(Fℓ)n(G)⟶(Fℓ)n(G). Since
Im{τ}=Gxj1∪̇⋯∪̇Gxjn by,
12 [Lemma 5.1], define
xjıϖ =
yıσ,such that
yıτ =
xjı, and
xjϖ=xj, for all
j∉{j1,j2,…,
jm}.
Now, let
yıτ =
yi´τ=xjı then (
yı,
yi´)
∈Ker{τ}⊆Ker{σ} means
yıσ=yi´σ that implies
yıϖ=yi´ϖ, therefore,
ϖ is well –defined.
Since
(Fℓ)n(G) is free on
Xn={x1,….,xn}, and as
ϖ∈T(Fℓ)n(G) then
ϖ must be a
G-acts morphism. Let
z∈(Fℓ)n(G) be such that
z=gxk,also let
xkτ =
hxjı, that implies
zτ=(gxk)τ=g(xkτ)=g(hxjı)=gh(xjı)=gh(yıτ)=(ghyı)τ. Now,
zτϖ =
(ghxji)ϖ=(gh)(
xjıϖ) =
(gh)(yıσ)=(ghyı)σ.
Because
zτ=(ghyı)τ, and
Ker{τ}⊆Ker{σ}we must have z
σ=(ghyı)σ=zτϖ. Hence,
σ=τϖ.
If we suppose that for any
σ,τ∈T(Fℓ)n(G) then
℘(στ)≤℘(σ),and
℘(στ)≤℘(τ), by,
12 [Remark 5.2]. Recall,
℘(σ)=℘(Im{σ}) and let
Im{σ}=∪̇z∈ZGz, where
Z⊆Xn, so that,
℘(σ)=|
Z|. Because
Im{στ}=Im{σ}τ=(∪̇z∈ZGz)τ=∪̇z∈ZG(zτ)this implies that
℘(στ)≤|Z|=℘(σ). Notice,
℘(τσ)=℘(Im{τσ})=℘((Im{τ})σ). As
Im{τσ}⊆Im{σ} we obtain
℘(Im{τσ})≤℘(Im{σ}) and then
℘(τσ)≤℘(τ). Therefore,
℘(στ)≤℘(σ),℘(τ).
For all
σ,τ∈T(Fℓ)n(G), we have the following:
σLτif and only if
Im{σ}=Im{τ}.
σRτ if and only if
Ker{τ}=Ker{σ}.
σHτif and only if
Im{σ}=Im{τ} and
Ker{τ}=Ker{σ}.
σDτ if and only if
℘(σ)=℘(τ).
σ≤Jτif and only if
℘(σ)≤℘(τ).
σJτ if and only if
℘(σ)=℘(τ).
D=J.
The point (1) and (2) can be easily be verified using
Theorem 6.1.
3) It is direct result from (1) and (2).
4)⟹)Let
σDτthat means
σRδLτfor any
δ∈T(Fℓ)n(G). From (1) and (2) we get
Im{δ}=Im{τ} and
Ker{σ}=Ker{δ}. Now,
Im{σ}≅(Fℓ)n(G)/Ker{σ} (by the Fundamental Theorem of Semigroup), therefore,
m{σ}≅(Fℓ)n(G)/Ker{δ}≅Im{δ}, and so
℘(σ)=℘(Im{σ})=℘(Im{δ})=℘(δ), and then we get
℘(σ)=℘(δ), furthermore,
℘(δ)=℘(Im{δ})=℘(Im{τ})=℘(τ). Hence,
℘(σ)=℘(τ).
⇐) Let
℘(σ)=℘(τ). As
Im{σ}=∪̇y∈YGy,Im{τ}=∪̇z∈ZGzfor some
Y,Z⊆Xn. With |
Y|=|
Z| =
℘(σ)=℘(τ).
If we let
φ:
Y⟶Ζbe a bijection and by defining
ϕ:Im{σ}⟶Im{τ} by
(gy)
ϕ=g(yφ), for all
g∈G and
y∈Y. Clearly,
ϕis one to one as where
(gy)
ϕ=(hy′)ϕ that implies
g(yφ) =
h(y′φ), for all
g,h∈Gand
y,y´∈Y. Then we must have
g=h and
(yφ) =
(y´φ), this is because they are in
Im{τ} and as
Im{τ} is free (left)-
G-act and
φ is one-one, then
y=y´. Furthermore,
ϕ is onto since from definition of
ϕ we get,
(gy)ϕ=g(yφ) for all
g∈G,y∈Y and because
φ is bijection hence for all
gz∈Im{τ}, choose
y∈Y with
yφ=z,thengy∈Im{σ}andgz=g(y)φ=(gy)
ϕ.
Suppose
ξ=σϕ,whereξ∈T(Fℓ)n(G). We have
Im{ξ} =
Im{σϕ}=
Im{σ}ϕ=Im{τ}, therefore,
τLξ.
Let
c,d∈(Fℓ)n(G). Clearly,
cσ=dσ if and only if
(cσ)ϕ=(dσ)ϕ. Since
ϕ is one to one, then
Ker{σ}=Ker{σϕ}=Ker{ξ}, therefore,
σRξ. Hence,
σDτ.
5)
⟹) Let
σ≤Iτthat means
σ=ζτη, for any
ζ,η∈T(Fℓ)n(G). From
Theorem 6.1, (III), we have
℘(σ)=℘(ζτη)≤℘(ζτ)≤℘(τ).
⇐)If℘(σ)≤℘(τ),Im{σ}=∪̇y∈YGyand
Im{τ}=∪̇z∈ZGzfor some
Y,Ζ⊆Xn, with
℘(σ)= |
Y| and
℘(τ)=|Z|. Because
℘(σ)≤℘(τ) then by,
12 [Remark 5.7] there is a one to one map
ϕ:Y⟶Ζ. Let
P=Im{ϕ}, that means
P⊆Z and
|Y|=|P|.
Fix
p0∈P and define
π:Z⟶P by z
π=z, for all z
∈P, and
zπ=p0 for all z
∈Z\P, then
Im{π}=P. Define
β:∪̇z∈ZGz⟶∪̇p∈PGpby
zβ=zπ. Obviously,
β extends to a
G-acts morphism therefore,
βπ∈T(Fℓ)n(G). Since
Im{βπ}
=
(Im{β})π
=
(∪̇z∈ZGz)π=∪̇z∈ZGzπ
=
∪̇z∈ZGzβ
=
∪̇p∈PGp, we obtain
℘(βπ)=|P|=|Y|=℘(σ) so, by (4) we get
βπDσand then
βπJσ because
D⊆J. Therefore,
σ≤Iτ.
6)
⟹)If
σJτ, then
σ=δτλ,τ=ασυ,for someδ,λ,α,υ∈T(Fℓ)n(G). By
Theorem 6.1, (III),
℘(σ) =
℘(δτλ)≤℘(δτ)≤℘(τ), as well as
℘(τ)=℘(ασυ)≤℘(ασ)≤℘(σ), that gives
℘(σ)=℘(τ).
⇐) Let
℘(σ)=℘(τ), then
σDτ by using (4), so that
σJτ as
D⊆J.
7) This is an immediate consequence of (4) and (6).
7. Conclusions
Throughout this paper, the independence algebra
(Fℓ)n(G) has been studied, and we show that it is free (left)-
G-acts under specific conditions. Moreover, the structure of
T(Fℓ)n(G) have been considered, and the we show that
T(Fℓ)n(G) isomorphic to the endomorphism monoid
End(Fℓ)n(G), with embeddedness in the (full) wreath product of
Tn by
Gn. For future work we may consider the structure of
PT(Fℓ)n(S), where
S is a finite semigroup.
Data availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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10.1093/oso/9780198511946.001.000110.5256/f1000research.191682.r470859Reviewer response for version 1FuentesKatrina Belleza1Refereehttps://orcid.org/0000-0001-8375-6302
University of San Carlos, Cebu City, Central Visayas, Philippines
Competing interests: No competing interests were disclosed.
The paper discusses the structure of transformation monoids and how they relate to endomorphisms, wreath products, and Green’s relations.
Comments:
1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections.
2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term.
3) Perhaps include an end of proof symbol to formally mark the end of a proof.
4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format
5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Algebra, Algebra and Music Theory, Topological Algebra, Algebraic Topology
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
Duraid HamdiAsawer
Competing interests: No competing interests were disclosed.
1642026
The comments and suggestion made by the referee are gratefully acknowledged
1) It is weird to have sections 3,4, and 5 containing only 1 entry each: a definition. Recommend including these in the preliminaries instead and introduce each definition by a paragraph instead of having each of them in separate sections.
Answer: we prefere to unchange the order of the definitons becouse we aim the reader to be connected with the context and the proof.
2) Definition 5.2.1 - what is the term being defined here? The writeup doesn't look like defining a term.
Answer: Done
3) Perhaps include an end of proof symbol to formally mark the end of a proof.
Answer: Done
4) Revise format of the proof of Theorem 6.1 part II: align qualities or rewrite to a better format
Answer: We check the format and it is correct.
5) I have no comments on the technicalities of the results and proof; they can be followed except that symbol used looks complicated
Answer: Thank you for your opinion.
10.5256/f1000research.191682.r469306Reviewer response for version 1Ahmed BajriDr. Sanaa1Referee
Department of Mathematical Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia
Competing interests: No competing interests were disclosed.
This paper addresses a study on an interesting topic in semigroup theory which is the transformation semigroups. It provides all the required facts and properties concerning transformation semigroups. The presentation and the structure of the paper is clear and accurate. Thus, I have found it excellent paper, however, I have some minor comments throughout the paper to improve it.
The following are the minor comments that I have suggested:
First Section (Preliminaries):
Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab".
Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it.
Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following".
Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ...
Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
My area is algebra and its applications.
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
Duraid HamdiAsawer
Competing interests: No competing interests were disclosed.
1642026
The comments and suggestions made by the referee are gratefully acknowledged
First Section (Preliminaries):
Page 3: Definition 2.2: since the action throughout the paper is on the right, the statement " α(a,b)=ab " should be written " (a,b)α =ab".
Answer: Done
Page 4: before Definition 2.6, the following parse is unnecessary " Now, let's have the following definition:" it is better to be deleted it.
Answer: Done
Section Five: page 5: before Theorem 5.2.2, written " the next Lemma" So, the next is it a lemma or theorem?? check this please. Also, at the same line there is a grammar mistake: It should be " in the following".
Answer: Done
Section six: page 7; at the beginning of the proof of Theorem 6.2 the word "be" is repeated twice, and it should be: the point can be easily verified ...
Answer: Done
Throughout the paper: it will be better for the mathematical statement using the curly braces for the set to be displayed in one line.