Fuzzy M-solid subvarieties

International Journal of Algebra, Vol. 5, 2011, no. 24, 1195-1205 Fuzzy M-Solid Subvarieties Bundit Pibaljommee Department of Mathematics, Faculty of Science Khon kaen University, Khon kaen 40002, Thailand banpib@kku.ac.th Abstract In this paper, we define the concept of fuzzy M-solid subvarieties of a given M-solid variety and show that the lattice of all fuzzy M-solid subvarieties is a complete sublattice of the lattice of all fuzzy subvarieties of the same M-solid variety, where M is a submonoid of hypersubstitutions. Moreover, for submonoids M 1,M 2 of hypersubstitutions with M 1 M 2, we show that the lattice of all fuzzy M 2 -solid subvarieties is a complete sublattice of the lattice of all fuzzy M 1 -solid subvarieties. Mathematics Subject Classification: 08A72, 08B15 Keywords: Varieties, Fuzzy subalgebras, Fuzzy subvarieties, M-solid varieties, Fuzzy M-solid subvarieties 1 Introduction The concept of fuzzy sets of a given set is established by L.A. Zadeh in 1965 which is defined as a function from a nonempty set X to the unit real interval [0, 1]. In [10], V. Murali defined fuzzy subalgebras of a given algebra in universal algebra sense, studied its lattice and showed that the lattice of all fuzzy subalgebras forms a complete lattice. In [13], B. Šešelja defined the notion of P -fuzzy subvariety of a given variety, where P is a partially ordered set. In [11], B. Pibaljommee defined the concept of fuzzy subvarieties of a variety V in universal algebra sense and found that the lattice of all subvarieties of the variety can be embedded into the lattice of all fuzzy subvarieties of the variety as complete lattices. In universal algebra many authors investigated structure of solid, M-solid varieties (see e.g., [3, 4, 5, 6, 7, 8, 9]) which play important role in computer science. It is natural to consider the structure of fuzzy M-solid subvarieties. So, the purpose of this work is to define the concept of fuzzy M-solid subvarieties of a given M-solid variety extending from

1196 B. Pibaljommee the concept of fuzzy subvarieties and show that the lattice of all fuzzy M-solid subvarieties of an M-solid variety is a complete sublattice of the lattice of all fuzzy subvarieties of the same M-solid variety. Moreover, for submonoids M 1,M 2 of hypersubstitutions with M 1 M 2, we show that the lattice of all fuzzy M 2 -solid subvarieties is a complete sublattice of the lattice of all fuzzy M 1 -solid subvarieties. 2 Preliminaries Let τ =(n i ) i I be a type of algebras with operation symbols (f i ) i I where f i is an n i -ary operation. An algebra of type τ is an ordered pair A := (A;(fi A) i I), where A is a non-empty set and (fi A) i I is a sequence of operations on A indexed by a non-empty index set I such that to each n i ary operation symbol f i there is a corresponding n i ary operation fi A on A. The set A is called the universe of A and the sequence (fi A) i I is called the sequence of fundamental operations of A. We often write A instead of A := (A;(fi A ) i I ). We denote by Alg(τ) the class of all algebras of type τ. A class V of algebras of type τ is called a variety if it is closed under taking of homomorphic images (H), subalgebras (S) and direct products (P). Let K Alg(τ). We denote by H(K) the class of all homomorphic images of algebras in K, bys(k) the class of all subalgebras of algebras in K and by P(K) the class of all direct products of algebras in K. Instead of H({A}), S({A}) and P({A}), we write H(A), S(A) and P(A), respectively for every algebra A in Alg(τ). Clearly, Alg(τ) and the class of all one-element algebras T (τ) are varieties which are called trivial varieties. Let X n := {x 1,...,x n } be a finite set of variables, W τ (X n ) be the set of all n-ary terms of type τ, and let W τ (X) := W τ (X n ), X := {x 1,...,x n,...} be the set of all terms of type τ. Then we denote by F τ (X) the absolutely free algebra; F τ (X) :=(W τ (X n ); ( f i ) i I ) with f i :(t 1,...,t ni ) f i (t 1,...,t ni ). Now we present the basic knowledge about varieties, identities and their properties which is given in [2, 6]. Let τ be a type. An equation of type τ is a pair (s, t) from W τ (X); such pairs are commonly written as s t. An equation s t is an identity of an algebra A, denoted by A = s t if s A = t A, where s A and t A are the term operations induced by terms s and t on A. A class K of algebras satisfies an equation s t, denoted by K = s t, if for every A K, A = s t. Let Σ be a set of equations of type τ. A class K of algebras of type τ is said to satisfy Σ, denoted by K = Σ,ifK = s t for every s t Σ. Clearly, =:= {(A,s t) A Alg(τ),s t W τ (X) 2 } Alg(τ) n=1

Fuzzy M-solid subvarieties 1197 W τ (X) 2 is a relation between the class Alg(τ) and the set W τ (X) 2. Therefore, the relation = gives a Galois connection (Mod, Id) between the class Alg(τ) and the set W τ (X) 2 defined by IdK := {s t W τ (X) 2 K = s t}, ModΣ := {A Alg(τ) A =Σ}. By the properties of Galois connections, we obtain the following theorem. Theorem 2.1 Let K, K 1,K 2 Alg(τ) and Σ, Σ 1, Σ 2 W τ (X) 2. Then (i) K ModIdK and Σ IdModΣ; (ii) K 1 K 2 IdK 2 IdK 1 and Σ 1 Σ 2 ModΣ 2 ModΣ 1 ; (iii) ModId and IdMod are closure operators on Alg(τ) and W τ (X) 2, respectively; (iv) the closed sets under IdMod are exactly the sets of the form IdK, K Alg(τ), and the closed sets under ModId are exactly the sets of the form ModΣ, Σ W τ (X) 2. A class K of algebras of type τ is called an equational class if there is a set Σ W τ (X) 2 such that K = ModΣ. A class Σ of equations of type τ is called an equational theory if there is a class K of algebras of type τ such that Σ = IdK. It is known that IdK is a congruence relation on F τ (X). Let Y be a non-empty set of variables, and K be a class of algebras of type τ. The algebra F K (Y ):=F τ (Y )/IdK is called the K-free algebra over Y or the free algebra with respect to K generated by Y/IdK. Theorem 2.2 ([1]) Let K Alg(τ). equivalent: Then the following conditions are (i) K is a variety of type τ; (ii) K is an equational class, i.e., K = ModΣ for some Σ W τ (X) 2 ; (iii) K = HSP({F K (X n ) n N + })=HSP({F K (X)}). Note that Alg(τ) =Mod{x x} and T (τ) =Mod{x y}. Let V be a variety of type τ. We denote by Sub(V ) the class of all subvarieties of V. It

1198 B. Pibaljommee is known ([2]) that the lattice L(V ):=(Sub(V );, ) is a complete lattice, where {Wi Sub(V ) i I} = and i I {Wi Sub(V ) i I} = {W Sub(V ) W W i } W i = ModId( i I W i ) i I for every family {W i i I} of subvarieties of V. 3 Fuzzy subvarieties In this section, we present the concept of fuzzy subvarieties of a variety V and some results which are shown in [11]. Definition 3.1 Let V be a variety of type τ. A mapping μ : V [0, 1] is called a fuzzy subvariety of V if the following properties are satisfied: (i) B V A H(B),μ(A) μ(b), (ii) B V A S(B),μ(A) μ(b) and, (iii) μ( A j ) inf{μ(a j ) j J}. We denote by FS(V ) the set of all fuzzy subvarieties of V. By the definition above, we note that if A, B V such that A is isomorphic to B, then μ(a) = μ(b). Since every one element algebra of type τ is contained in every variety of type τ and by Theorem 2.2, it is easy to see that (i) every one-element algebra in V has the same image with respect to μ and it is the greatest value, i.e., if A V with A = 1 then μ(a) μ(b) for all B V, and (ii) μ(f V (X)) μ(a) for all A V. A fuzzy subset μ of a variety V of type τ is a function μ : V [0, 1] and for t [0, 1], the set μ t := {A V μ(a) t} is called a level subset of V. Next Proposition is correspondence to the definition of fuzzy subvarieties. Proposition 3.2 Let V be a variety of type τ, μ : V [0, 1] a fuzzy subset of V and t [0, 1]. Then μ is a fuzzy subvariety of V iff for all t [0, 1] the level subset μ t is a subvariety of V.

Fuzzy M-solid subvarieties 1199 Let V be a variety of type τ and μ, ν FS(V ). We define the order on FS(V )byμ ν : μ ν means μ(a) ν(a) for all A V. We review the notions of unions and intersections of fuzzy subsets of V. Let μ and ν be fuzzy subsets of V. We define (μ ν)(a) := min{μ(a),ν(a)} (μ ν)(a) := max{μ(a),ν(a)}. For arbitrary intersections and unions, we denote by ( i I μ i )(A) := inf{μ i (A) i I} ( i I μ i )(A) := sup{μ i (A) i I} for a family {μ i i I} of fuzzy subsets of V. It is clear that i I μ i is a fuzzy subvariety for all a family {μ i i I} of fuzzy subvarieties of V. But in general, the union of fuzzy subvarieties of V need not to be a fuzzy subvariety of V (see in [11]). For a fuzzy subset μ of V, we define the fuzzy subvariety generated by μ by μ F := {ν FS(V ) μ ν}. The construction of the fuzzy subvariety of V generated by a fuzzy set of V was described by the following Lemma. Lemma 3.3 ([11]) Let V be a variety of type τ and μ be a fuzzy subset of V. Define a mapping ν : V [0, 1] by for every A V, Then ν = μ F. ν(a) := sup{t [0, 1] A HSP(μ t )}. Next, we consider the lattice of all fuzzy subvarieties of V. Let {μ j j J} be a family of fuzzy subvarieties of V. We define the meet and join on FS(V ) as follow: μ j := j I j I μ j j I μ j := {μ FS(V ) j I μ j μ} Theorem 3.4 ([11]) The lattice of all fuzzy subvarieties of V denoted by FS(V ) := (FS(V );, ) forms a complete lattice which has the least and the greatest elements, say 0, 1, respectively where 0(A) = 0, 1(A) = 1 for all A V.

1200 B. Pibaljommee 4 Fuzzy M - solid subvarieties In this section we assume that V is an M-solid variety of type τ and then we give the notion of fuzzy M-solid subvarieties. First of all we start with the concept of hypersubstitutions introduced first in [8] (see also in [4, 5]). A mapping σ : {f i i I} W τ (X) which maps each n i -ary operation symbol to an n i -ary term is called a hypersubstitution of type τ. Each hypersubstitution σ induces a map ˆσ on the set of all terms which is defined by (i) ˆσ[x] :=x if x X is a variable, (ii) ˆσ[f i (t 1,...,t ni )] := σ(f i )(ˆσ[t 1 ],...,ˆσ[t ni ]) for composite terms f i (t 1,...,t ni ). On the set Hyp(τ) of all hypersubstitutions of type τ, we define a binary operation h : Hyp(τ) Hyp(τ) Hyp(τ) byσ 1 h σ 2 =ˆσ 1 σ 2 where is the usual composition of functions. Then together with the identity element σ id, mapping each f i to the term f i (x 1,...,x ni ), we obtain a monoid (Hyp(τ); h,σ id ). Let A =(A, (f A i ) i I) be an algebra of type τ and σ Hyp(τ). The algebra A := (A, (σ(f i ) A ) i I )) is called derived algebra determined by A and σ. For the class K Alg(τ) and for a submonoid M Hyp(τ) we define χ A M [K] :={σ(a) σ M and A K}. We note that χ A M is a closure operator on Alg(τ). A variety V of type τ is called M-solid if χ A M [V ]=V. Now we give the notion of fuzzy M-solid subvariety of a given M-solid variety. Definition 4.1 Let M be a submonoid of Hyp(τ) and let V be an M-solid variety of type τ. A fuzzy subset μ of V is called a fuzzy M-solid subvariety if (1) μ is a fuzzy subvariety of V, and (2) σ M A V (μ(σ(a)) μ(a)). We denote by FMS(V ) the set of all fuzzy M-solid subvarieties of V. Proposition 4.2 Let V be an M-solid variety of type τ, μ : V [0, 1] be a fuzzy subset of V and t [0, 1]. Then μ is a fuzzy M-solid subvariety of V iff for all t [0, 1] the level subset μ t is an M-solid subvariety of V.

Fuzzy M-solid subvarieties 1201 Proof. ( ) : Assume that μ is a fuzzy M-solid subvariety of V and t [0, 1]. Since V is a variety and by Proposition 3.2, we have that μ t is a subvariety of V. Let σ M and A μ t. By assumption, we obtain μ(σ(a)) μ(a) t. Hence, σ(a) μ t. Altogether, μ t is an M-solid subvariety of V. ( ) : Assume that for every t [0, 1],μ t is an M-solid subvariety of V. Since V is a variety and by Proposition 3.2, we have μ is a fuzzy subvariety of V. Let σ M and A μ t. We choose t = μ(a) and by assumption, we have σ(a) μ t. This means μ(σ(a)) t = μ(a). Therefore, μ is a fuzzy M-solid subvariety of V. The following proposition is a consequence of Proposition 4.2. Proposition 4.3 Let W be a subclass of an M-solid variety V. Then the characteristic function μ W is a fuzzy M-solid subvariety of V iff W is an M-solid subvariety of V. Example 4.4 Let τ = (2) with a binary operation symbol, X 2 := {x 1,x 2 }, W (2) (X 2 ) the set of all binary terms of type τ = (2) and Hyp(2) the monoid of all hypersubstitutions of type τ = (2). Let Pre := {σ t Hyp(2) t W (2) (X 2 ),t x 1,x 2 }, where σ t maps the operation symbols to the term t W (2) (X 2 ). It is easy to show that Pre is a submonoid of Hyp(2) (see in [9]). Consider the Pre-solid variety of commutative semigroups V PC. In [3], it was proved that the complete lattice L(V PC ) of all Pre-solid varieties of commutative semigroups is precisely L(V PC )={P n n N} {V PC }, where P n := Mod{(xy)z x(yz),xy yx, x 2 y 2,x 2 y = xy 2,x 0...x n y 0...y n } for n N and clearly P 0 = T (τ). Now, we define the mapping μ : V PC [0, 1] by μ(a) := 1, if A P 0 = T (τ); 1 n+1, if A P n \ P n 1 ; 0, otherwise, for every A V PC. Then μ is a fuzzy Pre-solid subvariety of V PC. Because every level subset of V PC is a Pre-solid subvariety of V PC. For a fuzzy subset μ of an M-solid variety V, we define the fuzzy M-solid subvariety of V generated by μ by μ F := {ν FMS(V ) μ ν}.

1202 B. Pibaljommee Now, we want to describe the construction of the fuzzy M-solid subvariety of V generated by a fuzzy set of V. We refer a result of E. Gracsyńska in [7] (see also in [9]) that for every subclass K of V, HSP(χ A M [K]) is the M-solid subvariety of V generated by K. Lemma 4.5 Let V be an M-solid variety of type τ and μ be a fuzzy subset of V. Define a mapping ν : V [0, 1] by for every A V. Then ν = μ F. ν(a) := sup{t [0, 1] A HSP(χ A M [μ t])}, Proof. Let A V and let I A = {t [0, 1] A HSP(χ A M [μ t])}. First, we prove that ν is a fuzzy M-solid subvariety of V. It suffices to prove that for all t Im(ν), ν t is an M-solid subvariety of V. Let t Im(ν) and t n = t 1,n N \{0}. Let A ν n t. Then ν(a) t, so ν(a) > t n, n N \{0}. Hence, there exists s I A such that s > t n, since if s I A,s t n, then ν(a) = sup I A t n. This gives a contradiction. Thus, μ s μ tn and so, A HSP(χ A M [μ s]) HSP(χ A M [μ t n ]), for all n N\{0}, since χ A M is a closure operator. Therefore, A HSP(χ A M [μ t n ]). Conversely, n N\{0} if A HSP(χ A M [μ t n ]), then t n I A, n N \{0}. Then t n = t 1 n n N\{0} sup I A = ν(a), n N \{0}. Hence ν(a) t, i.e., A ν t. Then we have ν t = HSP(χ A M [μ t n ]), which means that ν t is an M-solid subvariety of n N\{0} V. Therefore, ν is a fuzzy M-solid subvariety of V. Next step is to show that μ ν. Let A V. Since μ(a) I A, we have ν(a) μ(a). This means μ ν. Finally, we want to prove that for any fuzzy M-solid subvariety δ of V containing μ, we have ν δ. It is clear that for all t [0, 1], we have HSP(χ A M [μ t]) is an M solid subvariety of δ t, since for every A V,A μ t implies t μ(a) δ(a), i.e, A δ t and so, HSP(χ A M [μ t]) is an M-solid subvariety of δ t. Now, we prove that for every A V,ν(A) δ(a). Let A V and t I A. Then A HSP(χ A M [μ t]) δ t implies δ(a) t. Therefore, ν(a) = sup I A δ(a), i.e., ν μ. By Theorem 3.4, we obtain the following lemma. Lemma 4.6 The lattice FMS(V ) := (FMS(V ),, ) of all fuzzy M-solid subvarieties of V is a complete lattice. As the same result in [11], we obtain that the lattice of all M-solid subvarieties of V can be embedded into the lattice FMS(V ).

Fuzzy M-solid subvarieties 1203 Lemma 4.7 Let M be a submonoid of Hyp(τ), V an M-solid variety of type τ and {μ j j J} FMS(V ). Then ( μ j ) t is closed under taking the operator χ A M, i.e., χa M [( μ j ) t ]=( μ j ) t. Proof. Let {μ j j J} FMS(V ). Since χ A M is a closure operator on Alg(τ), χ A M [( μ j ) t ] ( μ j ) t. For another inclusion, let A ( μ j ) t and σ M. We have ( μ j )(A) = sup{μ j (A) j J} t. Since μ j (σ(a)) μ j (A), j J, we have sup{μ j (σ(a)) j J} sup{μ j (A) j J} t. Then ( μ j )(σ(a)) t, σ M, i.e., χ A M [( μ j ) t ] ( μ j ) t. Now we obtain χ A M [( μ j ) t ]=( μ j ) t. Theorem 4.8 The lattice FMS(V ) := (FMS(V ),, ) of all fuzzy M- solid subvarieties of V is a complete sublattice of FS(V ) := (FS(V ),, ) of all fuzzy subvarieties of V. Proof. Obviously, FMS(V ) FS(V ). Let {μ j j J} FMS(V ). It is clear that if μ j FS(V ), then μ j FMS(V ). Now we want to show that if μ j FS(V ), then μ j FMS(V ). Let A V. By Lemma 3.3 and Lemma 4.7, we obtain ( μ j )(A) = sup{t [0, 1] A HSP(( μ j ) t )} = sup{t [0, 1] A HSP(χ A M [( μ j ) t ])} for all A V. By Lemma 4.5, we have μ j FMS(V ). These show that FMS(V ) is a complete sublattice of FS(V ). Let V be an M-solid variety. It is natural to ask for the relationship between the complete lattices FM 1 S(V ) := (FM 1 S(V ),, ) and FM 2 S(V ):= (FM 2 S(V ),, ) when M 1 and M 2 are both submonoids of Hyp(τ) and M 1 is a submonoid of M 2. Theorem 4.9 For any two submonoids M 1 and M 2 of the monoid Hyp(τ) with M 1 M 2 and M 1, M 2 solid variety V, the lattice FM 2 S(V ) is a complete sublattice of the lattice FM 1 S(V ). Proof. First, we show that FM 2 S(V ) FM 1 S(V ). Let μ FM 2 S(V ),σ M 2 and A V. We have μ(σ(a)) μ(a). Since M 1 M 2, we have μ(σ(a)) μ(a) for all σ M 1. Hence, μ FM 1 S. Let {μ j j J}

1204 B. Pibaljommee FM 2 S(V ). It is clear that if μ j FM 1 S(V ), then μ j FM 2 S(V ). Next, we want to show that if μ j FM 1 S(V ), then μ j FM 2 S(V ). By Lemma 4.7 and FM 2 S(V ) FM 1 S(V ), we have χ A M 2 [( μ j ) t ]=( μ j ) t = χ A M 1 [( μ j ) t ]. Let A V. Assume that μ j FM 1 S(V ). By Lemma 4.5, we have ( μ j )(A) = sup{t [0, 1] A HSPχ A M 1 [( μ j ) t ]} = sup{t [0, 1] A HSPχ A M 2 [( μ j ) t ]}. This means ( μ j ) FM 2 S(V ). Therefore, the lattice FM 2 S(V ) is a complete sublattice of the lattice FM 1 S(V ). 5 Conclusion We have defined the notion of fuzzy M-solid subvariety of a given M-solid variety and showed that the lattice of all fuzzy M-solid subvarieties of a given M-solid variety is a complete sublattice of the lattice of all fuzzy subvarieties of the M-solid variety. Moreover, for submonoids M 1,M 2 of hypersubstitutions with M 1 M 2, we show that the lattice of all fuzzy M 2 -solid subvarieties is a complete sublattice of the lattice of all fuzzy M 1 -solid subvarieties. It is known that every M-solid variety can be defined by a set of M-hyperidentities. But the connection between the lattice of all fuzzy M-solid subvarieties of a given M-solid variety and the lattice of all M-hyperidentities of the variety is still open. Acknowledgement This work is supported by the Thailand Research Fund under Grant: MRG5180303. References [1] G. Birkhoff, The structure of abstract algebras, Proc. Cambridge Philosophical Society 31 (1935), 433-454.

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