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Research article

(Co-)fibration of generalized crossed modules

  • Received: 23 July 2024 Revised: 01 October 2024 Accepted: 01 November 2024 Published: 19 November 2024
  • MSC : 18A35, 18D30, 18G45

  • Crossed modules are algebraic structures that generalize the concept of group extensions. They involve group-like objects (often groups or groupoids) with additional structure and mappings between them that satisfy certain properties. Generalized crossed modules further extend this concept to higher-dimensional settings or more general algebraic contexts. In this paper, we studied the fibration and co-fibration of generalized crossed modules.

    Citation: Hatice Gülsün Akay. (Co-)fibration of generalized crossed modules[J]. AIMS Mathematics, 2024, 9(11): 32782-32796. doi: 10.3934/math.20241568

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  • Crossed modules are algebraic structures that generalize the concept of group extensions. They involve group-like objects (often groups or groupoids) with additional structure and mappings between them that satisfy certain properties. Generalized crossed modules further extend this concept to higher-dimensional settings or more general algebraic contexts. In this paper, we studied the fibration and co-fibration of generalized crossed modules.



    Throughout the paper, all sets are assumed to be finite and nonempty. Let X be a set and Y a subset of X. It is well known that T(X) is a semigroup containing all functions from X into itself under the composition of maps. In fact, T(X) is called the full transformation semigroup on the set X. Define a semigroup Fix(X,Y) as follows:

    Fix(X,Y)={αT(X):yα=yfor allyY}.

    Indeed, Fix(X,Y) is a subsemigroup of T(X) and called a transformation semigroup with fixed set which was first introduced by Honyam and Sanwong [11] in 2013. In this content, we write functions on the right. Particularly, for the composition αβ of α,βT(X), we mean that α is applied first. An algebraic object, relative to graph theory and semigroup theory, our focus in this paper is the Cayley digraph, which is defined as follows. Let A be a subset of Fix(X,Y). The Cayley digraph Cay(Fix(X,Y),A) of Fix(X,Y) with respect to A is defined as a digraph (without multiple arcs) with vertex set Fix(X,Y) and arc set consisting of ordered pairs (α,β) where β=αλ for some λA. The set A is called a connection set of the Cayley digraph Cay(Fix(X,Y),A). For convenience, we write Γ instead of Cay(Fix(X,Y),A). Recently, there are some researches related to Cayley digraphs of Fix(X,Y). For example, in 2020, Nupo and Pookpienlert [15] presented some prominent results on Γ relative to minimal idempotents. In 2021, they investigated in [14] the domination parameters of Γ where the connection sets contain only minimal idempotents or permutations. The major objects focused on in the present paper are related to Cayley digraphs of Fix(X,Y) with respect to minimal idempotents and permutations.

    Graph theory is one of the branches of modern mathematics having experienced the most impressive development in recent years. Several applications can be found in game theory, management sciences, and transportation network theory. The next step of the evolution is related to fractional graph theory. By developing the fractional idea, the purpose of researchers is multiple. First, to enlarge the scope of applications in scheduling, operations research, and various kinds of assignment problems. Second, to simplify those such problems. The fractional version of a theorem is frequently easier to prove than the classical one. Further, a bound for a fractional coefficient of a graph is also a bound for the classical coefficient or suggests a conjecture. In 1987, Hedetniemi et al. [10] introduced the fractional domination concept in graphs. The fractional parameters we consider in this paper are fractional domination and fractional total domination numbers. Many researchers are interested in the concepts of fractional parameters in graphs and digraphs. In 1990, Fricke et al. [7] determined the computational complexity of the fractional total domination number of graphs and described a linear time algorithm to find such parameter for trees. In 1994, Fisher [4] showed the relations of the 2-packing number, fractional domination number, and domination number of the Cartesian product graphs. Moreover, Fisher et al. [6] proved the fractional domination of strong direct products of graphs. In addition, Hare [8] studied the fractional domination number of the m×n complete grid graph, which is isomorphic to the product of two paths Pm×Pn. In 2002, Fisher [5] considered the fractional domination number and the fractional total domination number of graphs and their complements. In 2007, Walsh [21] presented the fractional domination number of prisms. In the same year, Arumugam and Rejikumar [2] proposed several results for the fractional domination chain in graphs. In 2009, Rubalcaba and Walsh [17] investigated some structural properties of the fractional dominating functions of graphs. In 2012, Arumugam et al. [1] studied the fractional version of distance k-domination and related parameters. In 2015, Sangeetha and Maheswari [18] introduced the minimal dominating functions of Euler Totient Cayley graphs. In 2018, Harutyunyan et al. [9] presented an upper bound of the fractional domination number of digraphs in terms of independence number of their underlying graphs. In 2021, Karunambigai and Sathishkumar [13] introduced the dominating functions on fractional graphs and found the fractional domination number of certain graphs by formulating the linear programming problem. In 2023, Shanthi et al. [20] analyzed the effects on the fractional domination number of a graph G after deleting a vertex from G. Additionally, some bounds of the parameter are discussed and proved using the eccentricity value of a vertex of G.

    In this contribution, we are interested to investigate the fractional concepts of domination parameters on Cayley digraphs Γ of transformation semigroups with fixed sets. We divide the paper into six sections. The introduction and preliminaries are provided in Sections 1 and 2, respectively. In Section 3, we present the fractional (total) domination numbers on Γ with respect to minimal idempotents. We prove that if all elements in the connection set of Γ are minimal idempotents, then the fractional domination number of Γ is equal to the domination number. However, the fractional total domination number of Γ does not exist. Moreover, we determine the fractional domination number and fractional total domination number of Γ with respect to permutations in Sections 4 and 5, respectively. Finally, we summarize the work in the conclusion provided in Section 6.

    Let D be a digraph with vertex set V and arc set E. Assume that D admits loops and excludes multiple edges. A dominating set of D is a set S of vertices such that every vertex in VS has an in-neighbor in S. The domination number γ(D) of D is defined to be the cardinality of the smallest dominating set. Further, a total dominating set of D is a set S of vertices such that every vertex in V has an in-neighbor in S. Similarly, the total domination number γt(D) is the cardinality of the smallest total dominating set of D. Let S be a dominating set of D and f:V{0,1} be the characteristic function on S, so that

    f(v)={0ifvVS,1ifvS.

    The dominating property of S is equivalent to

    xN[v]f(x)1,

    for every vV, where N[v]=N(v){v} in which N(v) is the set of in-neighbors of v in D. That is, N(v)={uV{v}:(u,v)E}. Similarly, for each vV, one can define N+[v]=N+(v){v} where N+(v)={uV{v}:(v,u)E} is the set of all out-neighbors of v in D. Moreover, the in-degree d(v) and out-degree d+(v) of v in D are defined to be |N(v)| and |N+(v)|, respectively. Furthermore, if (v,v)E and N(v)==N+(v), then we say that v is an isolated loop in D. Fractional graph theory deals with the generalization of integer-valued graph theoretic concepts such that they take on fractional values. One of the standard methods for converting a graph concept from integer to fractional is to formulate the concept as an integer program and then to consider the linear programming relaxation. A detailed study of fractional graph theory and fractionalization of various graph parameters are given by Scheinerman and Ullman [19] in 1997.

    Let D=(V,E) be a digraph and f:VR be a real-valued function. For any subset S of V, we let f(S)=xSf(x). The weight of f is defined by |f|=f(V). The fractional analogues of invariants for a digraph D=(V,E) may be obtained as the linear relaxation of the associated integer programs or combinatorically as follows.

    A function f:V[0,1] is called a fractional dominating function of D if

    f(N[v])=uN[v]f(u)1,

    for every vV. The fractional domination number of D, denoted by γ(D), is the minimum weight of a fractional dominating function of D, that is,

    γ(D)=min{|f|:f(N[v])1for everyvV}.

    Similarly, one can define a fractional total dominating function of D and the fractional total domination number γt(D) of D by replacing N[v] with N(v). That is,

    γt(D)=min{|f|:f(N(v))1for everyvV}.

    Hereafter, some preliminaries and relevant information about digraphs can be found in [16] and others on semigroups can be seen in [3] and [12]. We now present some concepts and notations related to the semigroup Fix(X,Y). For convenience, we write Y={ai:iI}. For each αFix(X,Y), we then have aiα=ai for all iI. Further, the image of α is denoted by Xα=Y˙{bj:jJ}. Note that the index set J could be empty, that is, Xα=Y. Consequently, α can be written as follows:

    α=(AiBjaibj),

    where Ai=aiα1 for each iI and Bj=bjα1 for each jJ. We can observe that AiY={ai} for each iI and BjXY for each jJ. Moreover, Honyam and Sanwong [11] proved that the set Em of all minimal idempotents in Fix(X,Y) are as follows:

    Em={(Aiai):{Ai:iI}{is a partition of X with aiAi}}.

    Further, denote by HidX the group of all permutations in Fix(X,Y) where idX is the identity map of Fix(X,Y). It is clear that Fix(X,Y)={idX} whenever X=Y, so we consider the case YX throughout the paper. Obviously, if idX is an element of a connection set of the Cayley digraph Γ, then Γ contains loops attached to each vertex. Therefore, we consider only the case that all connection sets exclude the identity map idX.

    In this section, we provide results for the fractional domination number and the fractional total domination number of Γ whose connection sets are contained in Em.

    Theorem 3.1. If AEm is a connection set of Γ, then γ(Γ)=γ(Γ).

    Proof. Let AEm be a connection set of Γ. Define the sets A and B as follows.

    A={μEm:(XY)μ(αA(XY)α)}andB=Fix(X,Y)A.

    By [14, Lemma 3.3], we conclude that B is a dominating set of Γ. Define the characteristic function f:Fix(X,Y)[0,1] on B by

    f(λ)={0ifλA,1ifλB.

    It is not hard to verify that f is a fractional dominating function of Γ. Let βB. We now prove that N(β)=. Suppose to the contrary that there exists ηFix(X,Y) such that ηN(β), that is, (η,β)E(Γ). Thus β=ημ for some μAEm. For each xX, we see that xβ=x(ημ)=(xη)μY which yields that βEm. If ηEm, then β=ημ=ηN(β) which is impossible. So we conclude that ηEm. Then there exists zXY in which zηXY. By the fact that β,μEm and zβ=z(ημ)=(zη)μ(XY)μ, we have

    zβ(XY)β(XY)μ(XY)β(αA(XY)α).

    We obtain that βA which is a contradiction. Hence N(β)= for all βB. Furthermore, it is proved in [14, Theorem 3.4] that B is the minimum dominating set of Γ. This implies that |f| must be the minimum size among fractional dominating functions of Γ. Therefore, γ(Γ)=|f|=|B|=γ(Γ).

    Actually, if a connection set of Γ is contained in Em, then every non-minimal idempotent has zero in-degree. Thus we can not define any fractional total dominating function of Γ and we then get the following proposition.

    Proposition 3.2. If AEm is a connection set of Γ, then γt(Γ) does not exist.

    In semigroup theory, the Green's equivalence relations L,R and H on Fix(X,Y) are defined as follows. For convenience, let S=Fix(X,Y) and for α,βS,

    αLβif and only ifSα=Sβ,αRβif and only ifαS=βS,αHβif and only ifαLβandαRβ.

    Moreover, for each αS, we denote the equivalence H-class containing α by Hα. That is,

    Hα={βS:βHα}.

    We now present the fractional domination number of Γ whose connection sets are contained in HidX{idX}. Clearly, if |XY|=1, then HidX={idX} and yields that A=. So we focus on the case |XY|2. By [14, Theorem 4.1], we obtain that Γ is the disjoint union of induced subdigraphs Γ[Em],Γ[HidX], and Γ[G] where G=Fix(X,Y)(EmHidX). Thus we will propose results by considering each induced subdigraph, independently. However, if A=HidX{idX}, then we obtain the fractional domination number of Γ as follows.

    Proposition 4.1. If A=HidX{idX} is a connection set of Γ, then γ(Γ)=γ(Γ).

    Proof. Let A=HidX{idX} be a connection set of Γ. By following the proof of [14, Proposition 4.4], we conclude that Γ is the disjoint union of Γ[Rα] where Γ[Rα] is a complete subdigraph of Γ induced by an R-class of Fix(X,Y) containing αFix(X,Y). By choosing one vertex from each subdigraph and let U be the set of such vertices, we obtain that γ(Γ)=|U|. Further, by considering the characteristic function on the set U, we observe that it is a fractional dominating function of Γ and yields that γ(Γ)=|U|=γ(Γ).

    We next study the fractional domination number of Γ where a connection set A is a proper subset of HidX{idX} by considering Γ[Em],Γ[HidX], and Γ[G], respectively.

    Theorem 4.2. If AHidX{idX} is a connection set of Γ, then γ(Γ[Em])=|Y||X||Y|.

    Proof. Let AHidX{idX} be a connection set of Γ. In Γ[Em], we see that every minimal idempotent is an isolated loop and then N(μ)==N+(μ) for all μEm. By defining the constant map f(μ)=1 for all μEm, we get that f is a fractional dominating function of Γ which is also minimum. Thus γ(Γ[Em])=|f|=|Em|=|Y||X||Y|.

    Theorem 4.3. If AHidX{idX} is a connection set of Γ, then γ(Γ[HidX])=|XY|!|A|+1.

    Proof. Let AHidX{idX} be a connection set of Γ. For convenience, we may assume that A={α1,α2,,αk} for some kN. By the property of the group HidX, we can conclude that, for each αHidX, elements of {αα1,αα2,,ααk} are all distinct. Further, we can observe that N+(α)={αα1,αα2,,ααk} which implies that d+(α)=|N+(α)|=|A|. Similarly, we can summarize that N(α)={αα11,αα12,,αα1k} and so d(α)=|N(α)|=|A|. Therefore, d(α)=|A|=d+(α) for all αHidX. By [9, Proposition 2.7], we have

    γ(Γ[HidX])|HidX||A|+1=|XY|!|A|+1.

    On the other hand, let f:HidX[0,1] be defined by f(λ)=1|A|+1 for all λHidX. We now show that f is a fractional dominating function of Γ[HidX]. Let λHidX. Consider

    f(N[λ])=αN[λ]f(α)=f(λ)+αN(λ)f(α)=1|A|+1+|A||A|+1=1,

    this implies that f is a fractional dominating function of Γ[HidX], immediately. It follows that

    γ(Γ[HidX])|f|=αHidXf(α)=|HidX||A|+1=|XY|!|A|+1.

    Consequently, we have γ(Γ[HidX])=|XY|!|A|+1, as desired.

    Generally, the structure of Γ[G] is more complicated than the other two subdigraphs Γ[Em] and Γ[HidX]. This yields that the fractional domination number of Γ[G] is quite difficult to be determined. Hence we attempt to investigate the lower and upper bounds of γ(Γ[G]). However, the following lemma is needed.

    Lemma 4.4. If AHidX{idX} is a connection set of Γ, then Δ+(Γ[G])=|A| where Δ+(Γ[G]) is the maximum out-degree of Γ[G] without considering loops.

    Proof. Let AHidX{idX} be a connection set of Γ. Further, let yY and zXY. Define λG by

    xλ={xifxz,yifx=z.

    We now prove that elements in the set λA={λα:αA} are all distinct. Assume that λα=λβ for some α,βA. Let xX{z}. Then xα=(xλ)α=x(λα)=x(λβ)=(xλ)β=xβ. Since α,βAHidX{idX}, we obtain that zα=zβ. Hence α=β. Thus elements in the set λA are all distinct. We next show that λλA. If λ=λα for some αA, then xα=(xλ)α=x(λα)=xλ=x for all xX{z}. From αHidX{idX}, we have zα=z. This yields that α=idX which is a contradiction. Consequently, N+(λ)=λA which implies that d+(λ)=|A|. As the fact that N+(μ)μA for all μΓ, we have Δ+(Γ[G])=d+(λ)=|A|.

    Theorem 4.5. If AHidX{idX} is a connection set of Γ, then

    |G||A|+1γ(Γ[G])min{|G|δ+1,γ(Γ[G])},

    where δ:=δ(Γ[G]) is the minimum in-degree of Γ[G] without considering loops.

    Proof. Assume that AHidX{idX} is a connection set of Γ. By [9, Proposition 2.7], we obtain that γ(Γ[G])|G|Δ++1 where Δ+:=Δ+(Γ[G]) is the maximum out-degree of Γ[G]. Moreover, we get by Lemma 4.4 that Δ+(Γ[G])=|A|. Thus the lower bound, γ(Γ[G])|G||A|+1, is proved. In order to verify the upper bound, we define a function f:G[0,1] by f(α)=1δ+1 for all αG. Obviously, δ|N(α)| for all αG. So it is not hard to consider that, for each αG,

    f(N[α])=λN[α]f(λ)=f(α)+λN(α)f(λ)=1δ+1(|N(α)|+1)|N(α)|+1|N(α)|+1=1.

    Thus, f is a fractional dominating function of Γ[G] which yields that γ(Γ[G])|f|=αGf(α)=|G|δ+1. Clearly, γ(Γ[G])γ(Γ[G]). Therefore,

    γ(Γ[G])min{|G|δ+1,γ(Γ[G])}.

    We now describe some terminologies related to the expression of permutations. Let αHidX. In fact, every permutation can be written as the product of disjoint cycles. Therefore, we can write α as follows:

    α=δ1δ2δr,

    such that, for each k=1,2,,r, there exist c1,c2,,csX in which c1δk=c2,c2δk=c3,,cs1δk=cs, and csδk=c1. For convenience, δk is called a subcycle of length s in the permutation α. Furthermore, if there exists l{1,2,,r} such that δl is a subcycle of odd length in α, then we say that α contains an odd subcycle. Otherwise, it does not contain an odd subcycle.

    To investigate the equality of bounds stated in Theorem 4.5, we consider a singular connection set and obtain the parameter γ(Γ[G]) as follows.

    Proposition 4.6. Let A={α}HidX{idX} be a connection set of Γ. If no element in XY is a fixed point of α, then

    γ(Γ[G])=12|G|={γ(Γ[G])ifαdoes not contain an odd subcycle of length greater than 1 ,γ(Γ[G])t2ifαcontains an odd subcycle of length greater than 1 ,

    where t is the number of odd directed cycles in Γ[G].

    Proof. Assume that no element in XY is a fixed point of α. We first show that Γ[G] is the disjoint union of directed cycles. To show this, it suffices to prove that d(λ)=1=d+(λ) for all λG. Let λG. Clearly, d+(λ)=1. Further, we easily observe that λ1β=λ2β if and only if λ1=λ2 whenever λ1,λ2G and βHidX. Thus d(λ)=1. Hence Γ[G] is the disjoint union of directed cycles.

    Case 1. α does not contain an odd subcycle of length greater than 1. We need to show that those directed cycles in Γ[G] are of even length. Suppose that there exists a directed cycle Ck of odd length k in Γ[G]. For convenience, let Ck:=λ1,λ2,,λk,λ1. We obtain that λk=λk1α=λk2α2==λ2αk2=λ1αk1 and λ1=λkα. Then λk=λ1αk1=(λkα)αk1=λkαk. Since λkG, there exists bXY such that bλkXY. By assumption, we have (bλk)αbλk. Let p be the smallest positive integer such that (bλk)αp=bλk. As the fact that α does not contain an odd subcycle of length greater than 1, we obtain that p is even. If p<k, then by Division's algorithm, k=qp+r for some qN and 1r<p. Thus bλk=bλkαk=(bλk)αk=(bλk)αqp+r=(bλk)αqpαr=(bλk)αr which is impossible since r<p. If p>k, then (bλk)αk=b(λkαk)=bλk. This also leads to a contradiction. Hence every induced directed cycle in Γ[G] has even length. We now define the function f:G[0,1] by f(β)=12 for all βG. It is not complicated to conclude that f is a minimum fractional dominating function of Γ[G]. Therefore, γ(Γ[G])=|f|=12|G|=γ(Γ[G]).

    Case 2. α contains an odd subcycle of length greater than 1. Let (a1a2ak) be an odd subcycle of α for some odd positive integer k3. Define λFix(X,Y) by

    xλ={xifxY,a1ifxXY.

    We obtain that λ,λα,λα2,,λαk1,λαk=λ form a directed cycle of length k. In this case, we observe that λ,λα,λα2,,λαk1G. That is, Γ[G] contains an induced directed cycle of odd length. Let t be the number of odd directed cycles in Γ[G]. It is easily investigated that if r is an odd number, then γ(Cr)=r+12. Let g:V(Cr)[0,1] be defined by g(β)=12 for all βV(Cr). Then γ(Cr)r2. By [9, Proposition 2.7], we have γ(Cr)r2 and hence γ(Cr)=r2=r+1212=γ(Cr)12. Consequently,

    γ(Γ[G])=γ(C)=iis evenγ(Ci)+jis oddγ(Cj)=iis evenγ(Ci)+jis odd[γ(Cj)12]=γ(Γ[G])t2.

    Using the minimum fractional dominating function f of Γ[G] defined in Case 1, we also have γ(Γ[G])=|f|=12|G|. Hence the result is proved.

    Next, we will study the fractional total domination number of Γ for a connection set A which is a subset of HidX{idX}. Recall that a function f:Fix(X,Y)[0,1] is called a fractional total dominating function of Γ if

    f(N(α))=λN(α)f(λ)1,

    for every αFix(X,Y). The fractional total domination number of Γ is the minimum weight of a fractional total dominating function of Γ which is denoted by γt(Γ). That is,

    γt(Γ)=min{|f|:f(N(α))1for everyαFix(X,Y)}.

    We observed in the proof of Theorem 4.2 that, for a connection set AHidX{idX}, all minimal idempotents have zero in-degree in Γ. Consequently, we cannot define any fractional total dominating function of Γ. Therefore, the fractional total domination number of Γ does not exist. Hence, we aim to investigate this parameter for the induced subdigraphs Γ[HidX],Γ[Em] and Γ[G].

    As we mentioned above, all minimal idempotents have zero in-degree in Γ which implies that γt(Γ[Em]) does not exist. For results on the fractional total domination numbers of Γ[HidX] and Γ[G], we present as follows. By [15, Theorem 4.5], we have known that Γ is locally connected. Hence N(α) for all αHidX. Consequently, the fractional total domination number of Γ[HidX] always exists.

    Theorem 5.1. If AHidX{idX} is a connection set of Γ, then γt(Γ[HidX])=|XY|!|A|.

    Proof. Assume that A={α1,α2,,αk}. Let λHidX. Clearly, N(λ)={λα11,λα12,,λα1k} where λα1iλα1j for all ij{1,2,,k}. That is, d(λ)=|N(λ)|=|A|. We now define f:HidX[0,1] by f(β)=1|A| for all βHidX. Hence

    f(N(λ))=μN(λ)f(μ)=|N(λ)|1|A|=1.

    Next, let g be arbitrary fractional total dominating function of Γ[HidX]. Then g(N(δ))1=f(N(δ)) for all δHidX which implies that

    |g|=g(HidX)=ηHidXg(η)ηHidXf(η)=|f|.

    Therefore, we can conclude that f is the minimum fractional total dominating function of Γ[HidX]. Consequently, γt(Γ[HidX])=|f|=|HidX||A|=|XY|!|A|, as required.

    We now provide the characterization for an existence of the fractional total domination number of Γ[G]. For each αFix(X,Y), define Fix(α)={xX:xα=x}.

    Theorem 5.2. Let AHidX{idX} be a connection set of Γ. Then the following statements are equivalent.

    (i) The fractional total domination number of Γ[G] exists.

    (ii) N(η) for all ηG.

    (iii) αAFix(α)=Y.

    Proof. Obviously, the statements (ⅰ) and (ⅱ) are equivalent. We now prove that (ⅱ) and (ⅲ) are equivalent.

    (ⅱ)(ⅲ): Assume that N(η) for all ηG. Suppose that there exists xXY such that xαAFix(α). Consider λG defined by

    λ=(aiXYaix).

    By assumption, there exists βN(λ), that means βλ. Then (β,λ)E(Γ), that is, λ=βμ for some μA. Let zXY. Thus x=zλ=z(βμ)=(zβ)μ. Since xFix(μ) and μAHidX, we get that zβ=x. Hence β=λ which is a contradiction.

    (ⅲ)(ⅱ): Assume that αAFix(α)=Y. Let ηG. Then we can write

    η=(AiBjaibj).

    Since αAFix(α)=Y, there exists λA such that (XY)ηFix(λ). Hence there exists bk(XY)η in which bkFix(λ). That is, bkλbk which implies that bkλ1bk. Define μFix(X,Y) by

    μ=(AiBjaibjλ1).

    We can obtain that μG{η} and μλ=η. Thus (μ,η)E(Γ[G]) and so μN(η). Therefore, N(η).

    To present results on the fractional total domination number of Γ[G], we need the following lemma.

    Lemma 5.3. Let λG and αA. Then XλFix(α) if and only if λ=λα.

    Proof. Assume that XλFix(α). Let xX. Then xλXλFix(α). Hence xλ=(xλ)α=x(λα) which implies that λ=λα.

    Conversely, assume that λ=λα. Let aXλ. Then a=bλ for some bX. Thus aα=(bλ)α=b(λα)=bλ=a. Therefore, aFix(α) which yields that XλFix(α).

    Moreover, we aim to investigate some prominent properties of Γ[G] via an equivalence relation defined as follows.

    Let A be a connection set of Γ in which αAFix(α)=Y. For each λG, we define AλA by

    Aλ={αA:XλFix(α)}.

    Hence Aλ. Further, we define a relation λ on Aλ by, for each α,βAλ,

    αλβif and only ifxλα1=xλβ1for allx(XY)λ1. (5.1)

    Clearly, the relation λ is an equivalence relation on Aλ. Let Aλ/λ be the set of all equivalence classes of Aλ with respect to λ. In addition, denote by the notation 1Aλ the equivalence relation {(α,α):αAλ}. We then obtain the following theorem.

    Theorem 5.4. Let AHidX{idX} be a connection set of Γ in which αAFix(α)=Y. If λG, then N(λ) is one-to-one corresponding to Aλ/λ. Consequently, d(λ)=|Aλ/λ|.

    Proof. Let λG. By Theorem 5.2, we have N(λ). Let αN(λ). Then there exists βA such that λ=αβ, that is, α=λβ1. We first show that βAλ. Suppose that XλFix(β). For each xX, we obtain that x(λβ)=(xλ)β=xλ=x(αβ). Thus λβ=αβ and so λ=α which is impossible since αN(λ). Hence XλFix(β) which implies that βAλ. Define φ:N(λ)Aλ/λ by φ(λβ1)=βλ. For each λβ1,λμ1N(λ), assume that λβ1=λμ1. For each x(XY)λ1, we have x(λβ1)=x(λμ1). Then βλμ which means that βλ=μλ. Hence φ is well-defined. We next show that φ is a bijection. Let λβ1,λμ1N(λ) be such that φ(λβ1)=φ(λμ1). Then βλ=μλ, that is, βλμ. Thus xλβ1=xλμ1 for all x(XY)λ1. Furthermore, for each xYλ1, we have xλY and hence xλβ1=xλ=xλμ1. Therefore, λβ1=λμ1 which yields that φ is injective. For proving that φ is surjective, let βλAλ/λ. Thus βAλ which implies that XλFix(β). Let μ=λβ1. By Lemma 5.3, we obtain that λλβ. Hence μ=λβ1λ. Moreover, λ=μβ and then μN(λ). Clearly, φ(μ)=φ(λβ1)=βλ. Consequently, φ is a bijection. This implies that d(λ)=|N(λ)|=|Aλ/λ|.

    To illustrate more clearly, we present the following example for illustrating the above theorem.

    Example 5.5. Let X={1,2,,8} and Y={1,2,3}. Further, let A={α1,α2,α3,α4} be a connection set of Γ contained in HidX{idX} such that

    α1=(1234567812346758),α2=(1234567812357684),α3=(1234567812385647),α4=(1234567812354678).

    Then Fix(α1)=Y{4,8},Fix(α2)=Y{6},Fix(α3)=Y{5,6},Fix(α4)=Y{6,7,8} and thus αAFix(α)=Y. By Theorem 5.2, we have N(λ) for all λG. Let λG be defined as follows:

    λ=(1234567812311156).

    We obtain that Aλ={αA:XλFix(α)}={α1,α2,α4}. By the equivalence relation λ on Aλ defined in (5.1), we obtain that α1λ={α1} and α2λ={α2,α4}. Let μ1,μ2G be defined as follows:

    μ1=(1234567812311175)andμ2=(1234567812311146).

    It is not hard to verify that the following statements hold in Γ[G]. Let βG.

    (i) βα1=λ if and only if β=μ1.

    (ii) βα2=λ if and only if β=μ2.

    (iii) βα3=λ if and only if β=λ.

    (iv) βα4=λ if and only if β=μ2.

    Thus the subdigraph of Γ[G] induced by N[λ] is shown in Figure 1.

    Figure 1.  Γ[N[λ]].

    Hence we get that N(λ)={μ1,μ2}. Therefore, d(λ)=|N(λ)|=2=|{α1λ,α2λ}|=|Aλ/λ|.

    We now define certain graphical terminologies. The induced subdigraph Γ[G] of Γ is said to be semi-regular if d(α)=d(β) for all α,βG. Moreover, if d(α)=k for all αG, then Γ[G] is said to be k-semi-regular. We now present a characterization of the semi-regularity of Γ[G] as follows.

    Theorem 5.6. Let AHidX{idX} be a connection set of Γ such that Fix(α)=Y for all αA. The following statements are equivalent.

    (i) Γ[G] is semi-regular.

    (ii) For each xXY,xαxβ for all αβA.

    (iii) λ=1A for all λG.

    Proof. (ⅰ)(ⅱ): Assume that Γ[G] is semi-regular. Let zXY be fixed. Suppose to the contrary that there exist two distinct α,βA such that zα=zβ. Let yY. Define λG by

    xλ={xifxY,yifxX(Y{z}),zαifx=z.

    We claim that d(λ)<|A|. If λξ1=λ for some ξA, then λ=λξ. Hence (zα)ξ=(zλ)ξ=zλξ=zλ=zα which implies that zαFix(ξ). Since zXY and α is a permutation, we have zαXY which contradicts the assumption. So we conclude that N(λ)={λξ1:ξA}. Thus d(λ)=|N(λ)||A|. Now, we define μG by

    xμ={xifxY,yifxX(Y{z}),zifx=z.

    Therefore, μα=λ=μβ. Since α,βA and Fix(α)=Y=Fix(β), we have zα=zβz which yields that μλ. Furthermore, we see that λα1=μ=λβ1. This implies that d(λ)<|A|, immediately. Next, we define ηG by

    xη={xifxz,yifx=z.

    We show that d(η)=|A|. Indeed, N(η)={ηξ1:ξA}. Suppose that there exist ξ1,ξ2A such that ηξ11=ηξ12. For each xX{z}, we obtain that xξ11=(xη)ξ11=x(ηξ11)=x(ηξ12)=(xη)ξ12=xξ12. Moreover, we can conclude that zξ11=zξ12 since ξ1,ξ2 are permutations. Consequently, ξ11=ξ12 and hence ξ1=ξ2. Thus all elements in N(η) are distinct which leads to d(η)=|N(η)|=|A|. Now, we see that d(λ)<|A|=d(η) which contradicts the semi-regularity of Γ[G].

    (ⅱ)(ⅲ): Assume that the condition holds. Let λG. Since Fix(α)=Y for all αA, we have XλFix(α) for all αA. This implies that Aλ={αA:XλFix(α)}=A. Clearly, 1A=1Aλλ. Next, let α,βAλ be such that αλβ. Since λG, there exists xλXY for some xXY. Hence x(XY)λ1. From αλβ, we get that xλα1=xλβ1XY. If αβ, then (xλα1)α=xλ(α1α)=xλ=xλ(β1β)=(xλβ1)β contradicts the assumption. Therefore, α=β and thus λ1Aλ=1A. Consequently, λ=1A, as required.

    (ⅲ)(ⅰ): Assume that the condition holds. We prove that Γ[G] is semi-regular. Let λG. By Theorem 5.4 and we have known that Aλ=A, it follows that

    d(λ)=|Aλ/λ|=|A/1A|=|A|.

    Hence each element in G has an in-degree |A|. Consequently, Γ[G] is semi-regular.

    Remark 5.7. By the proof of (iii)(i) in Theorem 5.6, we observe that Γ[G] is |A|-semi-regular.

    Theorem 5.8. Let AHidX{idX} be a connection set of Γ such that αAFix(α)=Y. Then

    |G||A|γt(Γ[G])|G|m,

    where m=min{|Aλ/λ|:λG}. The equality holds if Γ[G] is semi-regular.

    Proof. By Theorem 5.2, the fractional total domination number of Γ[G] exists and then N(λ) for all λG. We first prove the lower bound of γt(Γ[G]). Let f be a fractional total dominating function of Γ[G]. Then f(N(λ))1 for all λG. We obtain that

    |G|λGf(N(λ))=λG(μN(λ)f(μ))=μGf(μ)|N+(μ)|Δ+(Γ[G])μGf(μ).

    By Lemma 4.4, we have Δ+(Γ[G])=|A|. Hence

    |G|Δ+(Γ[G])μGf(μ)=|A|μGf(μ),

    that is, |f|=μGf(μ)|G||A|. Since f is arbitrary, we conclude that

    γt(Γ[G])=min{|f|:fis a fractional total dominating function ofΓ[G]}|G||A|.

    For proving the upper bound of γt(Γ[G]), we let m=min{|Aλ/λ|:λG} and define a function f:G[0,1] by f(λ)=1m for all λG. By Theorem 5.4, we obtain that m is the minimum in-degree of Γ[G]. Therefore, it is clear that f is a fractional total dominating function of Γ[G]. We conclude that

    γt(Γ[G])|f|=λGf(λ)=|G|1m.

    We now assume that Γ[G] is semi-regular. By Remark 5.7, we obtain that m=|A|. Therefore, γt(Γ[G])=|G||A|. The equality is attained.

    In this paper, the concepts of fractional domination and fractional total domination have been investigated for Cayley digraphs Γ of transformation semigroups with fixed sets. The results have been divided into two major parts. The first one is related to the fractional (total) domination numbers of Γ with respect to minimal idempotents, which is presented in Section 3. In this part, we have found that the fractional domination number and the domination number coincide. However, the fractional total domination number of Γ does not exist. For the second part, including Sections 4 and 5, the fractional domination and fractional total domination numbers have been provided with respect to permutations. The results have been presented by considering the induced subdigraphs of Γ, separately. Further, we have introduced an equivalence relation for applying to study the fractional total domination number of certain induced subdigraph of Γ.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank Khon Kaen University and Rajamangala University of Technology Lanna. The first author also thanks Assoc. Prof. Dr. Kittikorn Nakprasit for his corrections and suggestions. This work was financially supported by Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation, Grant No. RGNS 65-050.

    No conflicts of interest exists. We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.



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