Ratio Mathematica Volume 39, 2020, pp. 229-236 On homomorphism of fuzzy multigroups J. A. Awolola* M. A. Ibrahim† Abstract In this paper, the homomorphism of fuzzy multigroups is briefly de- lineated and some related results are shown. In particular, we consider the corresponding isomorphism theorems of fuzzy multigroups. Keywords: fuzzy multiset, fuzzy multigroup, homomorphism of fuzzy multigroups. 2010 AMS subject classifications: 54A40, 03E72, 20N25, 06D72. 1 *Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373, Makurdi, Nigeria; awolola.johnson@uam.edu.ng. †Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria; ami- brahim@abu.edu.ng. 1Received on November 2nd, 2020. Accepted on December 17rd, 2020. Published on De- cember 31st, 2020. doi: 10.23755/rm.v39i0.553. ISSN: 1592-7415. eISSN: 2282-8214. ©J. A. Awolola and A. M. Ibrahim. This paper is published under the CC-BY licence agreement. 229 J. A. Awolola and A. M. Ibrahim 1 Introduction Since the inception of the theory fuzzy multisets introduced by Yager (1986), the subject has become an interesting area for researchers in algebra. The foun- dation of algebraic structures of fuzzy multisets was laid by Shinoj et al. (2015); Ibrahim and Awolola (2015) discussed further some new results which will bring new openings and development of fuzzy multigroup concept. Some group con- cepts like subgroups, abelian groups, normal subgroups and direct product of groups have been established (Ejegwa, 2018a,b,d, 2019). The idea of homo- morphism of fuzzy multigroups and their alpha-cuts have also been discussed (Ejegwa, 2018c, 2020). In this paper, more results on homomorphism of fuzzy multigroups are estab- lished and the corresponding isomorphism theorems of fuzzy multigroups which analogously exist in group setting are discussed. 2 Preliminaries We recall here some basic definitions and results used in the sequel. We refer the reader to (Miyamoto, 2001; Shinoj et al., 2015; Ibrahim and Awolola, 2015). Definition 2.1. (Miyamoto, 2001) Let X be a nonempty set. A fuzzy multi- set U over X is characterized by count membership function CMU : X → [0, 1] (giving a multiset of the unit interval [0, 1]). An expedient notation for a fuzzy multiset U over X is U = {(CMU (a)/a) | a ∈ X} with CMU (a) = {µ1U (a) ,µ 2 U (a), ...,µ m U (a), ...}, where µ 1 U (a) ,µ 2 U (a), ...,µ m U (x), ... ∈ [0, 1] such that (µ1U (x) ≥ µ 2 U (a) ≥, ...,≥ µ m U (a), ...). If the fuzzy multiset U is finite, then CMU (a) = {µ1U (a) ,µ 2 U (a), ...,µ m U (a)}, where µ1U (a) ,µ 2 U (a), ...,µ m U (a) ∈ [0, 1] such that µ 1 U (a) ≥ µ 2 U (a) ≥, ...,≥ µmU (a). The set of all fuzzy multisets over X is denoted by FMS(X). Throughout this paper fuzzy multisets are considered finite. The usual set operations can be carried over to fuzzy multisets. For instance, let U,V ∈ FMS(X), then U ⊆ V ⇐⇒ CMU (a) ≤ CMV (a),∀ a ∈ X, U ∩V = {CMU (a) ∧CMV (a)/a | a ∈ X}, U ∪V = {CMU (a) ∨CMV (a)/x | a ∈ X}. 230 On homomorphism of fuzzy multigroups Definition 2.2 (Shinoj et al., 2015) Let P and Q be two nonempty sets such that ϕ : P → Q is a mapping. Consider the fuzzy multisets U ∈ FMS(P) and V ∈ FMS(Q). Then, (i) the image of U under ϕ is denoted by ϕ(U) has the count membership function CMϕ(U) (b) = { ∨ ϕ(a)=bCMU (a) , ϕ −1 (b) 6= ∅ 0, ϕ−1 (b) = ∅ (ii) the inverse image of V under ϕ denoted by ϕ−1 (V ) has the count mem- bership function CMϕ−1(V ) (a) = CMV (ϕ (a)). Definition 2.4 (Shinoj et al., 2015) Let X be a group. A fuzzy multiset U over X is called a fuzzy multigroup if (i) CMU (ab) ≥ CMU (a) ∧ CMU (b) , ∀ a,b ∈ X, and (ii) CMU (a−1) = CMU (a) , ∀ a ∈ X. The immediate consequence is that CMU (e) ≥ CMU (a) ∀ a ∈ X, where e is the identity element of X. The set all fuzzy multigroups is denoted by FMG(X). The next definition can be found in Shinoj et al. (2015) . Definition 2.5 Let U ∈ FMG(X). Then U is called an abelian fuzzy multigroup over X if CMU (ab) = CMU (ba) , ∀ a,b ∈ X. The set AFMG (X) is the set of all abelian fuzzy multigroups over X. Definition 2.6 Let U ∈ FMS(X). Then U∗ = {x ∈ X | CU (a) = CU (e)} Remark 2.1 For a fuzzy multigroup over a group X, U∗ is a group, certainly a subgroup of X Shinoj et al. (2015). Proposition 2.1 (Ibrahim and Awolola, 2015) Let U ∈ FMG(X), then xU = yU ⇐⇒ x−1y ∈ U∗. The following propositions are shown in (Ibrahim and Awolola, 2015) . Proposition 2.2 Let U ∈ FMG(X). Then the following assertions are equiva- lent: (i) CMU (ab) = CMU (ba), ∀ a,b ∈ X, 231 J. A. Awolola and A. M. Ibrahim (ii) CMU (aba−1) = CMU (b), ∀ a,b ∈ X, (iii) CMU (aba−1) ≥ CMA(b), ∀ a,b ∈ X, (iv) CMU (aba−1) ≤ CMU (b), ∀ a,b ∈ X. Proposition 2.3 Let U ∈ FMG(X). Then CMU (ab−1) = CMU (e) implies CMU (a) = CMU (b). As to the converse problem whether CMU (a) = CMU (b) implies CMU (ab−1) = CMU (e), we give a counter example. Let X = {1,s,t,r} be a klein’s 4-group and U = {(1, 0.7, 0.6, 0.5, 0.5)/1, (0.6, 0.4, 0.2)/s}. We see that U is an abelian fuzzy multigroup over X. Then, while CMU (t) = CMU (r) = 0, we have CMU (tr −1) = CMU (tr) = CMU (s) = (0.6, 0.4, 0.2) 6= (1, 0.7, 0.6, 0.5, 0.5) = CMU (1). Thus the converse problem above does not hold. 3 Main Results Proposition 3.1 Let X be a group such that ϕ : X → X is an automorphism. If U ∈ FMG(X), then ϕ(U) = U if and only if ϕ−1(U) = U. Proof. Let a ∈ X. Then ϕ(a) = a. Now CMϕ−1(U)(a) = CMU (ϕ(a)) = CMU (a) =⇒ ϕ−1(U) = U Conversely, let ϕ−1(U) = U. Since ϕ is an automorphism, then CMϕ(U)(a) = ∨ {CMU (a ′ ) | a ′ ∈ X, ϕ(a ′ ) = ϕ(a)} = CMU (ϕ(a)) = CU(ϕ−1(U))(a) = CMU (a) Hence, the proof. Proposition 3.2 Let ϕ : X → Y be a homomorphism of groups such that U,V ∈ FMG(Y ). If U is a constant on Kerϕ, then ϕ−1(ϕ(U)) = U. Proof. Let ϕ(a) = b. Then we have CMϕ−1(ϕ(U))(a) = CMϕ(U)ϕ(a) = CMϕ(U)(b) = ∨ {CMU (a) | a ∈ X, ϕ(a) = 232 On homomorphism of fuzzy multigroups b}. Since ϕ(a−1c) = ϕ(a−1)ϕ(c) = (ϕ(a))−1ϕ(c) = b−1b = e′, ∀ c ∈ X, such that ϕ(c) = b, which implies that a−1c ∈ Kerϕ. Moreover, since U is constant on Kerϕ, then CMU (a−1c) = CMU (e). Therefore, CMU (a) = CMU (c). This completes the proof. Proposition 3.3 Let U ∈ AFMG(X) such that a map ϕ : X → X/U is defined by ϕ(a) = aU. Then ϕ is a homomorphism with Kerϕ = {a ∈ X | CMU (a) = CMU (e)}. Proof. Clearly, ϕ is a homomorphism. Also, Kerϕ = {a ∈ X : ϕ(a) = eU} = {a ∈ X : aU = eU} = {a ∈ X : CMU (a−1b) = CMU (b) ∀ b ∈ X} = {a ∈ X : CMU (a−1) = CMU (e)} = {a ∈ X : CMU (a) = CMH (e)} = U∗ Proposition 3.4 Let ϕ : X → Y be an epimorphism of groups and U ∈ AFMG(X), then X/U∗ ∼= Y . proof. Define Ψ : X/U∗ → Y by Ψ(xU∗) = ϕ(a) ∀ a ∈ X. Let aU = bU such that CMU (a−1b) = CMU (e). This implies that a−1b ∈ U∗. It is easy to show that Ψ is well-defined, homomorphism and epimorphism. Moreover, ϕ(a) = ϕ(b) =⇒ ϕ(a)−1ϕ(b) = ϕ(e) =⇒ ϕ(a−1)ϕ(b) = ϕ(a−1b) = ϕ(e) =⇒ a−1b ∈ U∗ =⇒ CMU (a−1b) = CMU (e) =⇒ aU = bU This shows that Ψ is an isomorphism. Proposition 3.5 If U,V ∈ AFMG(X) with CMU (e) = CMV (e), then U∗V∗/V ∼= U∗/U ∩V . 233 J. A. Awolola and A. M. Ibrahim Proof. Clearly, for some x ∈ U∗V∗, a = uv such that u ∈ U∗ and v ∈ V∗. Define ϕ : U∗V∗/V → U∗/U ∩V by ϕ(aV ) = u(U ∩V ). If aV = bV with b = u1v1, u1 ∈ U∗ and v1 ∈ V∗, then CMV (a −1b) = CMV ((uv) −1u1v1) = CMV (v −1u−1u1v1) = CMV (u −1u1v −1v1) = CMV (e). Hence, CMV (u−1u1) = CMV (v−1v1) = CMV (e). Thus, CMU∩V (u −1u1) = CMU (u −1u1) ∧CMV (u−1u1) = CMU (e) ∧CMV (e) = CMU∩V (e) That is, u(U ∩V ) = u1(U ∩V ). Therefore, ϕ is well-defined. If aV,bV ∈ U∗V∗/V , then ab = uvu1v1. Since U ∈ AFMG(X), then CMU (vu1v1) = CMU (u1) =⇒ vu1v1 ∈ U∗. Hence, ϕ(aV bV ) = ϕ(abV ) = u(vu1v1)(U ∩V ) = u(U ∩V )vu1v1(U ∩V ) and CMU∩V (u −1 1 (vu1u1)) ≥ CMU (u −1 1 vu1v1) ∧CMV (u −1 1 vu1v1) = CMU (u −1 1 (vu1v1)) ∧CMV (v(u −1 1 u1v1)) = CMU (e) ∧CMV (e) = CMU∩V (e). Hence, vu1v1(U ∩V ) = u1(U ∩V ) That is, ϕ(aV bV ) = u(U ∩V )u1(U ∩V ) = ϕ(aV )ϕ(bV ), and this shows that ϕ is a homomorphism. Undeniably, it is also epimorphism. Furthermore, if a,b ∈ U∗V∗ with a = uv and b = u1v1, u,u1 ∈ U∗ and v,v1 ∈ V∗ and u(U ∩V ) = u1(U ∩V ), then CMU∩V (u−1u1) = CMU∩V (e) That is, CMU (u−1u1) ∧CMV (u−1u1) = CMU (e) ∧CMV (e). However, CMU (e) = CMV (e) and CMU (u−1u1) = CMU (e) =⇒ CMV (u−1u1) = CMV (e). 234 On homomorphism of fuzzy multigroups Therefore, CMV (a −1b) = CMV ((uv) −1u1v1) = CMV (v −1u−1u1u1) = CMV (u −1u1v −1v1) ≥ CMV (u−1u1) ∧CMV (v−1v1) = CMV (e) ∧CMV (e) = CMV (e) =⇒ CMV (a−1b) = CMV (e) Thus, aV = bV . Hence, U∗V∗/V ∼= U∗/U ∩V . Proposition 3.6 Let U,V ∈ AFMG(X) such that U ⊆ V and CMU (e) = CMV (e). Then X/V ∼= (X/U)/(V∗/U). proof. Define ϕ : X/U → X/V by ϕ(aU) = aV ∀a ∈ X such that CMU (a−1b) = CMU (e) = CMV (e) ∀ aU = bU. Since U ⊆ V , we have CMV (a−1b) ≥ CMU (a −1b) = CMV (e) and thus CMV (a−1b) = CMV (e), that is, aV = bV , which implies that ϕ is well-defined. It is homomorphism and epimorphism too. Moreover, Kerϕ = {aU ∈ X/U : ϕ(aU) = eV} = {aU ∈ X/U : aV = eV} = {aU ∈ X/U : CMV (a) = CMV (e)} = {aU ∈ X/U : a ∈ V∗} = V∗/U. Thus, Kerϕ = V∗/U and so X/V ∼= (X/U)/(V∗/U). References P.A. Ejegwa. On fuzzy multigroups and fuzzy submultigroups. Journal of Fuzzy Mathematics, 26(3):641—-654, 2018a. P.A. Ejegwa. On abelian fuzzy multigroups. Journal of Fuzzy Mathematics, 26 (3):655—-668, 2018b. P.A. Ejegwa. Homomorphism of fuzzy multigroups. Applications and Applied Mathematics, 13(1):114—-129, 2018c. 235 J. A. Awolola and A. M. Ibrahim P.A. Ejegwa. On normal fuzzy submultigroups of a fuzzy multigroup. Theory and Applications of Mathematics and Computer Science, 8(1):64—-92, 2018d. P.A. Ejegwa. 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