EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5835 ISSN 1307-5543 – ejpam.com Published by New York Business Global Multisorted Algebras of Trees of a Weakly Fixed Variable Thodsaporn Kumduang1, Khwancheewa Wattanatripop2,∗ 1 Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Rattanakosin, Nakhon Pathom 73170, Thailand 2 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna, Chiang Mai 50200, Thailand Abstract. For any algebra of type τ , this paper introduces a novel class of terms (or terms) of a weakly fixed variable of type τ . Algebraic structures in the sence of multisorted algebras are studied. In fact, it is shown that the set of such terms and the multisort operations forms a multisorted algebra that satisfies some axioms from the theory of clone. As a tool for classifying arbitrary algebras to subclasses called weakly fixed variable solid varieties, the seminearring of weakly fixed variable hypersubstitutions is proposed. Characterizations for any variety V of algebras of type τ to be a weakly fixed variable solid variety are explored. 2020 Mathematics Subject Classifications: 08A05, 08B05, 08A68, 20N15, 20M07 Key Words and Phrases: Term, operation, multisorted set, endomorphism, semigroup, variety, hyperidentity 1. Introduction and background The concept of multisorted algebras (also called many-sorted algebras or heterogeneous algebras) generalizes the concept of one-sorted algebras. Any module and vector space are basic examples of multisorted algebras. In general, the S-sorted sets A = (As)s∈S are essential. The set S is called a set of sorts. In addition, the sort mapping ϕ : A → B from an S-sorted set A = (As)s∈S to an S-sorted set B = (Bs)s∈S is an S-sorted family ϕ : (ϕs)s∈S of mappings ϕs : As → Bs where s ∈ S. The authors always refer to [1, 4, 19] for more details. Terms or trees in a study of automata and logic can be applied to form multisorted algebras called the multisorted algebra of terms of type τ . To attain this, we recall some important definitions. Let I be a nonempty indexed set and (fi)i∈I be a sequence of operation symbols. To every operation symbol fi, we assign a natural number ni ∈ N := ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5835 Email addresses: thodsaporn.kum@rmutr.ac.th (T. Kumduang), khwancheewa.wat@rmutl.ac.th (K. Wattanatripop) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 2 of 16 {1, 2, . . .}, called the arity of fi. The type is a sequence τ := (ni)i∈I . Let n ≥ 1, we denote by Xn := {x1, . . . , xn} a finite set called an alphabet and each xi in Xn is called a variable. The set of all n-ary terms of type τ is the smallest set which contains Xn denoted by Wτ (Xn) inductively defined by: (1) Xn ⊆ Wτ (Xn) and (2) If t1, . . . , tni ∈ Wτ (Xn) and fi is an operation symbol of the arity ni, then fi(t1, . . . , tni) ∈ Wτ (Xn). For infinitely many variables X, we denote by Wτ (X) := (Wτ (Xn))n∈N, i.e., for the infinite sequence (Wτ (X1),Wτ (X2),Wτ (X3), . . .) the multisorted set of all terms of type τ . In this matter, the sorts are the sets of n-ary terms of type τ for all n ∈ N. Recent developments of terms in various directions can be found, for instance, in [8, 10, 12, 15, 16, 18]. It is commonly seen that any term can be represented as a tree diagram. For this, we can visualize a term from the left to the right by treating each operation symbol as a vertex and each variable as a leaf of the tree. For example, the term t = f(g(f(x2, x1)), f(h(x4, x5, x2), g(x2))) can be visualized as the following figure. g f f g x2 f x2 x1 h x5 x2x4 For more backgrounds of terms representative by trees, we refer to [7]. The multisorted algebra of terms belongs to the variety of all abstract clones which is a family of N-sorted algebras satisfying the following three identities: (C1) S̃n m(S̃p n(Z̃, Ỹ1, . . . , Ỹp), X̃1, . . . , X̃n) ≈ S̃p m(Z̃, S̃n m(Ỹ1, X̃1, . . . , X̃n), . . . , S̃ n m(Ỹp, X̃1, . . . , X̃n)),m, n, p ∈ N, (C2) S̃n m(λj , X̃1, . . . , X̃n) ≈ X̃j , n,m ∈ N, 1 ≤ j ≤ n, (C3) S̃n n(Ỹ , λ1, . . . , λn) ≈ Ỹ , n ∈ N, where S̃n m, S̃p n, S̃ p m, S̃n n are operation symbols, Z̃, Ỹ1, . . . , Ỹp, X̃1, . . . , X̃n, Ỹ are variables for terms, and λj are symbols for variables. In general, (C1) is said to be the superassociative law since it generalizes the associative law. A class of algebras that satisfies (C1) is called T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 3 of 16 a Menger algebra or a superassociative algebra. In (C1), if n = m = p = 1, then it reduces to the usual associative law. For an overview of clone theory we refer to [2, 3, 9]. The viewpoint taken in Menger algebras can be found in [5, 6, 11, 14]. The multisorted superposition operation defined on (Wτ (Xn))n∈N is a multisorted map- ping Sn m : Wτ (Xn)× (Wτ (Xm))n → Wτ (Xm) defined by: (1) Sn m(xi, t1, . . . , tn) = ti if xi ∈ Xn, (2) Sn m(fi(s1, . . . , sni), t1, . . . , tn) = fi(S n m(s1, t1, . . . , tn), . . . , S n m(sni , t1, . . . , tn)) where n,m ∈ N and t1, . . . , tn ∈ Wτ (Xm). Thus, the multisorted algebra of terms of type τ or the clone of all terms of type τ denoted by clone(τ) = ((Wτ (Xn))n∈N, (S n m)m,n∈N, (xi)i≤n∈N) is formed. Actually, it also satisfies the axioms (C1), (C2) and (C3), thus it is an example of abstract clones. Particularly, if we let A := (A, (fA i )i∈I) be an algebra of type τ and let t be an n-ary term of type τ , then a term t induces an n-ary operation tA on A as follows: (1) If t = xj ∈ Xn, then tA = xAj = prn,Aj where prn,Aj is an n-ary projection mapping on A, (2) if t = fi(t1, . . . , tni) is an n-ary term of type τ and tA1 , . . . , t A ni are the term operations which are induced by t1, . . . , tni , then tA = fA i (tA1 , . . . , t A ni ). Hence, tA is called the term operation induced by the term t on the algebra A. The set of all n-ary term operations on A will be denoted by Wτ (Xn) A. Moreover, let IdA = {s ≈ t ∈ Wτ (X)×Wτ (X) | sA = tA} be the set of all identities satsified in A. The multisorted algebra CloneA = ((Wτ (Xn) A)n∈N, (On,A m )n,m∈N, (pr n,A i )i≤n,n∈N) is constructed. Another example is the quotient algebra clone(V ) = clone(τ)/Id(V ) where Id(V ) is a congruence in the form of the multisorted set (Idn(V ))n∈N of all n-ary identities s ≈ t satisfied in a variety V of algebras of type τ . Recall from [3] that a mapping σ : {fi | i ∈ I} → Wτ (X) such that for each i ∈ I, σ(fi) ∈ Wτ (Xni) is called a hypersubstitution of type τ . Moreover, each σ : {fi | i ∈ I} → Wτ (X) can be uniquely extended to the mapping σ̂ : Wτ (X) → Wτ (X) defined by: T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 4 of 16 (1) σ̂[xi] = xi for every xi ∈ X, (2) σ̂[fi(t1, . . . , tni)] = Sni m (σ(fi), σ̂[t1], . . . , σ̂[tni ]). The setHyp(τ) of all hypersubstitutions of type τ forms a monoid under the associative binary operation defined by: σ ◦h α = σ̂ ◦ α for all σ, α ∈ Hyp(τ) and the hypersubstitution σid : {fi | i ∈ I} → Wτ (X) defined by σid(fi) = fi(x1, . . . , xni) for all i ∈ I which acts as an identity element. In fact, each hypersubstitution can be considered as a multisorted mapping. For n ∈ N, let In ⊆ I be the set of all indexes such that fj with j ∈ In is an n-ary. Let Fn τ = {fj | j ∈ In}. Thus, a hypersubstitution is the sequence (σn)n∈N where σ : Fn τ → Wτ (Xn). Let (Hypn(τ))n∈N be the multisorted set of all hypersubstitutions of type τ . By the definition σ̂n[xi] = xi and σ̂n[fi(t1, . . . , tn)] = Sn n(σn(fi), σ̂n[t1], . . . , σ̂n[tn]) we obtain the extension of each σn in (σn)n∈N. Recently, in [17], the multisorted set (W fv τ (Xn))n∈N of terms of a fixed variable is introduced. For instance, let us consider the type τ = (2) with a binary operation symbol f . Then, x1, x2, f(x1, x1), f(x2, x2), f(f(x1, x1), x1), f(x2, f(x2, x2)) ∈ W fv τ (X2), x1, x2, x3, f(x1, x1), f(x3, x3), f(f(x3, x3), f(x3, x3)) ∈ W fv τ (X3). Conversely, f(x1, x2), f(x3, x1), f(x2, f(x1, x2)) ∈ Wτ (X3) \W fv τ (X3). By the formal definition, if t is a term, then the set var(t) consisting of all variables of X that appear in t is called the set of all variables for t. Thus, an n-ary term of a fixed variable of type τ is inductively defined by: (1) Every xi ∈ Xn is an n-ary term of a fixed variable of type τ , (2) if t1, . . . , tni are n-ary terms of a fixed variable of type τ , and if var(tj) = var(tk) for all 1 ≤ j < k ≤ ni, then fi(t1, . . . , tni) is an n-ary term of a fixed variable of type τ , (3) the set W fv τ (Xn) is the smallest set which is closed under finite application of (2). This concept always plays a key role in a study of the variety of bands, i.e. all algebras of type (2) satisfying f(x, x) ≈ x. Closed identities of a fixed variable and closed varieties of a fixed variable are also investigated based on multisorted hypersubstitutions of a fixed variable. The main purpose of this paper is to generalize the multisorted set of terms of a fixed variable by naturally reducing certain conditions and constructing the multisorted algebras under the superposition operation. The paper is organized as follows. Section 2 T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 5 of 16 introduces the concept of terms of a weakly fixed variable of type τ and provides some algebraic properties. In Section 3, we present essential tools for defining a novel class of algebras that satisfy identities generated by terms of a weakly fixed variable. Additionally, the multisorted algebras consisting of the multisorted set of mapping whose images are terms of a weakly fixed variable and two associative binary operations are constructed. Applications of the multisorted algebras of terms of a weakly fixed variable are given in Section 4. Finally, we provide some concluding remarks in Section 5. 2. Terms of a weakly fixed variable This section begins with the definition of terms of a weakly fixed variable. We also con- struct the multisorted algebra of such terms under the multisorted superposition operation and projections. Definition 1. For a natural number n, an n-ary term of a weakly fixed variable of type τ is inductively defined by the following: (1) Every variable xi in an alphabet Xn is an n-ary term of a weakly fixed variable of type τ . (2) If t1, . . . , tni are n-ary terms of a weakly fixed variable of type τ and var(tl) = var(tp) for some 1 ≤ l < p ≤ ni, then fi(t1, . . . , tni) is an n-ary term of a weakly fixed variable of type τ . (3) The set Wwfv τ (Xn) of all n-ary terms of a weakly fixed variable of type τ is the smallest set closed under finite application of (2). Some concrete examples are given. Example 1. Consider a type (3, 2) with a ternary operation symbol ⊞ and a binary op- eration symbol ⊟ and an alphabet X4. Then quaternary terms of type (3, 2) which are quaternary terms of a weakly fixed variable are listed, for example, as follows: x1, x2, x3, x4,⊞(x1, x1, x3),⊞(x2, x2, x4),⊟(x4, x4),⊞(⊟(x3, x3), x3,⊞(x1, x2, x1)). On the other hand, ⊞(x1, x2, x3),⊞(x4, x2, x1),⊟(x2,⊟(x1, x1)),⊞(x3,⊟(x1, x1), x4) are not quaternary terms of a weakly fixed variable of type (3, 2). We note that the set of terms of a weakly fixed variable can be viewed as a general- ization of the set of terms of a fixed variable as follows. Remark 1. For any n ≥ 1, the connection between the set Wwfv τ (Xn) and W fv τ (Xn) is described as follows: (1) If τ = (1, 1, . . . , ) or τ = (2, 2, . . .), then Wwfv τ (Xn) = W fv τ (Xn). T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 6 of 16 (2) If τ = (ni)i∈I and ni > 2 for some i ∈ I, then W fv τ (Xn) ⊂ Wwfv τ (Xn). To guaruntee that the multisorted superposition can be applied to the family of the set of all terms of a weakly fixed variable of type τ , the following lemma is required. Lemma 1. For m,n ∈ N, if s is an n-ary term of a weakly fixed variable of type τ and t1, . . . , tn are m-ary terms of a weakly fixed variable of type τ , then Sn m(s, t1, . . . , tn) ∈ Wwfv τ (Xm). Proof. Suppose that s ∈ Wwfv τ (Xn) and t1, . . . , tn ∈ Wwfv τ (Xm). We prove on the com- plexity of a term s that Sn m(s, t1, . . . , tn) ∈ Wwfv τ (Xm). It is obvious that Sn m(s, t1, . . . , tn) is a term of a weakly fixed variable in Wwfv τ (Xm) if s is a variable xk for all 1 ≤ i ≤ n. We now inductively assume that s = fi(s1, . . . , sni) and var(sl) = var(sk) for some fixed integers 1 ≤ l < k ≤ ni. From the definition of multisorted superposition, we have Sn m(fi(s1, . . . , sni), t1, . . . , tn) = fi(S n m(s1, t1, . . . , tn), . . . , S n m(sni , t1, . . . , tn)), we prove that each Sn m(sj , t1, . . . , tn) is an m-ary term of a weakly fixed variable of type τ for every 1 ≤ j ≤ ni and var(Sn m(sl, t1, . . . , tn)) = var(Sn m(sk, t1, . . . , tn)) for some fixed integers 1 ≤ l < k ≤ ni. Let j ∈ {1, . . . , ni}. If sj = xp for some xp ∈ Xn, then Sn m(sj , t1, . . . , tn) = Sn m(xp, t1, . . . , tn) = tp ∈ Wwfv τ (Xm). Assume that sj = fi(s ′ 1, . . . , s ′ ni ) and Sn m(s′p, t1, . . . , tn) ∈ Wwfv τ (Xm) for all p = 1, . . . , ni. Without loss of generality, suppose that var(s′l) = var(s′k) for some 1 ≤ l < k ≤ ni. Then we obtain that Sn m(fi(s ′ 1, . . . , s ′ ni ), t1, . . . , tn) ∈ Wwfv τ (Xm). Actually, since we know that var(sl) = var(sk) for some fixed integers 1 ≤ l < k ≤ ni, then we have var(Sn m(sl, t1, . . . , tn)) = var(Sn m(sk, t1, . . . , tn)), which completes the proof. To enhance understanding of the computation process for terms with a weakly fixed variable under the multisorted superposition, the following example is provided. Example 2. Consider n = 3 and m = 4 and the multisorted superposition S3 4 . On the sets Wwfv (3,2)(X3) and Wwfv (3,2)(X4), if we put a = ⊞(x1, x3, x1), b = ⊞(x4,⊞(x2, x1, x2), x4), c = ⊟(x4, x4), d = ⊟(x2,⊞(x2, x2, x2)), e = ⊞(⊟(x4, x4),⊟(x4, x4), x3), f = ⊟(x1, x1), T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 7 of 16 then S3 4(a, b, c, c) = S3 4(⊞(x1, x3, x1),⊞(x4,⊞(x2, x1, x2), x4),⊟(x4, x4),⊟(x4, x4)) = ⊞(⊞(x4,⊞(x2, x1, x2), x4),⊟(x4, x4),⊞(x4,⊞(x2, x1, x2), x4)) belongs to Wwfv (3,2)(X4) because the sets of variables in the first and the third positions are equal, i.e., {x1, x2, x4}. Moreover, S3 4(y, z1, z2, z3) ∈ Wwfv (3,2)(X4) if y ∈ {a, d, f} and z1, z2, z3 ∈ {a, b, c, d, e, f} and S4 3(u,w1, w2, w3, w4) ∈ Wwfv (3,2)(X3) if u ∈ {a, b, c, d, e, f} and w1, w2, w3, w4 ∈ {a, d, f}. By Lemma 1, the multisorted superpositions Sn m can be applied to the family (Wwfv τ (Xn))n∈N, which means that we have following multisorted mappings Sn m : Wwfv τ (Xn)× (Wwfv τ (Xm))n → Wwfv τ (Xm) for m,n ∈ N. As a consequence, the multisorted algebra Wwfv τ (X) := ((Wwfv τ (Xn))n∈N, (S n m)n,m∈N, (xi)i≤n,n∈N) is formed. Since we know that (Wwfv τ (Xn))n∈N ⊆ (Wτ (Xn))n∈N, then by Lemma 1 the following result is concluded. Theorem 1. The multisorted algebra Wwfv τ (X) satisfies the axioms (C1), (C2) and (C3). 3. Weakly fixed variable hypersubstitutions This section introduces a mapping whose images are terms of a weakly fixed variable and discusses the multisorted composition. Definition 2. A hypersubstitution σ ∈ Hyp(τ) is called a weakly fixed variable hypersub- stitution of type τ if for all i ∈ I, σ maps each operation symbol fi to an ni-ary term of a weakly fixed variable of type τ . The set of all weakly fixed variable hypersubstitutions of type τ is denoted by Hypwfv(τ), i.e., Hypwfv(τ) = {σ | σ : {fi | i ∈ I} → Wwfv τ (X)}. For instance, let τ = (3, 2) be a type with a ternary operation symbol ⊞ and a bi- nary operation symbol ⊟. A hypersubstitution σ : {⊞,⊟} → Wτ (X) such that σ(⊞) = ⊞(x3,⊟(x2, x2), x3) and σ(⊟) = ⊟(x1, x1). Then we get that σ a weakly fixed variable hypersubstitution of type τ . On the other hand, we let β : {⊞,⊟} → Wτ (X) such that β(⊞) = ⊞(x1,⊟(x2, x2), x3) and β(⊟) = ⊟(x1, x2). Then β is not a weakly fixed variable hypersubstitution of type τ since β(⊞) /∈ Wwfv τ (X) and β(⊟) /∈ Wwfv τ (X). To construct the algebra of weakly fixed variable hypersubstitutions, we need the following result to confirm that each extension of σ takes from the set of terms of a weakly fixed variable into itself. T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 8 of 16 Lemma 2. For any weakly fixed variable hypersubstitution σ, we have σ̂ : Wwfv τ (X) → Wwfv τ (X). Proof. Let σ be a mapping on Hypwfv(τ) and t be an element in Wwfv τ (X). We show that σ̂[t] ∈ Wwfv τ (X). Clearly, σ̂[xi] ∈ Wwfv τ (X) for every variable xi. Suppose now that t = fi(t1, . . . , tni) such that σ̂[tk] ∈ Wwfv τ (X) for all k ∈ {1, . . . , ni}. Without loss of generality, we may assume here that var(tl) = var(tp) for some l, p ∈ {1, . . . , ni} and l ̸= p. Then we obtain var(σ̂[tl]) = var(σ̂[tp]). Since σ(fi) belongs to Wwfv τ (X), then by the hypothesis, i.e., var(tl) = var(tp), we have σ̂[t] = Sni m (σ(fi), σ̂[t1], . . . , σ̂[tni ]) ∈ Wwfv τ (X). This finishes the proof. For n ∈ N, the multisorted mapping σ̂n : Wwfv τ (Xn) → Wwfv τ (Xn) can be defined by (1) σ̂n[xi] = xi for any xi ∈ Xn and (2) σ̂n[fi(t1, . . . , tni)] = Sn(σni(fi), σ̂n[t1], . . . , σ̂n[tni ]) if each σ̂n[tj ] is already known for 1 ≤ j ≤ ni. Then we prove: Theorem 2. The extension σ̂ of each σ in Hypwfv(τ) is an endomorphism on the su- perassociative system Wwfv τ (X). Proof. Let σ ∈ Hypwfv(τ). To prove that σ̂ is an endomorphism on Wwfv τ (X), we aim to show that the equation σ̂m[Sn m(s, t1, . . . , tn)] = Sn m(σ̂n[s], σ̂m[t1], . . . , σ̂m[tn]) (1) holds for any s ∈ Wwfv τ (Xn), t1, . . . , tn ∈ Wwfv τ (Xm). To do this, we give a proof on the complexity of a term s. If s is a variable xj in Xn, then σ̂m[Sn m(xj , t1, . . . , tn)] = σ̂m[tj ] = Sn m(xj , σ̂m[t1], . . . , σ̂m[tn]) = Sn m(σ̂n[xj ], σ̂m[t1], . . . , σ̂m[tn]). Suppose that s = fi(s1, . . . , sni) and inductively assume that the equation (1) is satis- fied for s1, . . . , sni . Without loss of generality we may assume that var(sl) = var(sk) for some 1 ≤ l < k ≤ ni. Then by Theorem 1, we obtain σ̂m[Sn m(fi(s1, . . . , sni), t1, . . . , tn)] = σ̂m[fi(S n m(s1, t1, . . . , tn), . . . , S n m(sni , t1, . . . , tn))] = Sni m (σni(fi), σ̂m[Sn m(s1, t1, . . . , tn)], . . . , σ̂m[Sn m(sni , t1, . . . , tn)]) = Sni m (σni(fi), S n m(σ̂n[s1], σ̂m[t1], . . . , σ̂m[tn]), . . . , S nm(σ̂n[sni ], σ̂m[t1], . . . , σ̂m[tn])) = Sn m (Sni n (σni(fi), σ̂n[s1], . . . , σ̂n[sni ]), σ̂m[t1], . . . , σ̂m[tn]) = Sn m (σ̂n[fi(s1, . . . , sni)], σ̂m[t1], . . . , σ̂m[tn]), which shows that (1) holds for s = fi(s1, . . . , sni). T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 9 of 16 Acutally, weakly fixed variable hypersubstitutions of type τ can be considered as mul- tisorted mappings (σn)n∈N. Let Hypn(τ) be the set of all σn and let (Hypn(τ))n∈N be the multisorted sets of all weakly fixed variable hypersubstitutions. We see that the composition mapping (F n τ )n∈N (αn)n∈N−−−−−→ (Wwfv τ (Xn))n∈N (σ̂n)n∈N−−−−−→ (Wwfv τ (Xn))n∈N is an element in the multisorted set (Hypwfv n (τ))n∈N. Consequently, as in the one-sorted case, the composition between any two mappings Hypwfv n (τ) is defined by σn ◦hn αn = σ̂n ◦n αn where ◦n is a usual composition on the nth sort. Then we prove: Theorem 3. (Hypwfv n (τ))n∈N is a subsemigroup of (Hypn(τ))n∈N with respect to the multisorted operation (ohn)n∈N. Proof. The proof follows from Lemma 2. Note that there is no an identity element in (Hypwfv n (τ))n∈N bacause the image of the mapping σid given by σid(fi) = fi(x1, x2, . . . , xni) is not a term of a weakly fixed variable. Consequently, (Hypwfv n (τ))n∈N does not form a monoid. For any σn and αn on Hypwfv n (τ), we define the binary operation +n on Hypwfv n (τ) by (σn +n αn)(fi) = Sni n (σn(fi), αn(fi), . . . , αn(fi)). It is obvious that σn +n αn is again a weakly fixed variable hypersubstitution of type τ . From this, we have the following result. Theorem 4. ((Hypwfv n (τ))n∈N, (◦hn)n∈N, (+n)n∈N) forms a left-seminearring. Proof. For each n ∈ N, we first show that the operation +n is associative, i.e., ((σn +n αn) + βn) = (σ1 + (σ2 + σ3)) for all σn, αn, βn ∈ Hypwfv(τ). For this, let fi be an operation symbol. Then by Theorem 1, we obtain ((σn +n αn) + βn)(fi) = Sn n((σn +n αn)(fi), βn(fi), . . . , βn(fi)) = Sn n(S n n(σn(fi), αn(fi), . . . , αn(fi)), βn(fi), . . . , βn(fi)) = Sn n(σn(fi), S n n(αn(fi), βn(fi), . . . , βn(fi)), . . . , S n n(αn(fi), βn(fi), . . . , βn(fi))) = Sn n(σn(fi), (αn +n βn)(fi), . . . , (αn +n βn)(fi)) = (σn +n (αn + βn))(fi). T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 10 of 16 Furthermore, the left distributive law, i.e., σn ◦hn (αn +n βn) = (σn ◦hn αn) +n (σn ◦hn βn) is also obtained. Indeed, from the fact that the extension of each mapping on Hypwfv(τ) preserves the operations on the multisorted algebra Wwfv τ (X) proved in Theorem 2, we have (σn ◦hn (αn +n βn))(fi) = σ̂n[(αn +n βn)(fi)] = σ̂n[S n n(αn(fi), βn(fi), . . . , βn(fi))] = Sn n(σ̂n[αn(fi)], σ̂n[βn(fi)], . . . , σ̂n[βn(fi)]) = Sn n((σn ◦hn αn)(fi), (σn ◦hn βn)(fi), . . . , (σn ◦hn βn)(fi)) = ((σn ◦hn αn) +n (σn ◦hn βn))(fi). Therefore, ((Hypwfv n (τ))n∈N, (◦hn)n∈N, (+n)n∈N) is a left-seminearring. Generally, the right distributivity does not hold, as demonstrated by the following counterexample. Example 3. Let τ = (3) be a type with a ternary operation symbol ⊞. Assume that σ3, α3 and β3 be weakly fixed variable hypersubstitutions of type (3) which are defined by σ3(⊞) = ⊞(x1, x3, x1), α3(⊞) = ⊞(x3, x1, x1), β3(⊞) = ⊞(x2, x3, x3). Consider ((σ3 +3 α3) ◦h3 β3)(⊞) = ̂(σ3 +3 α3)[⊞(x2, x3, x3)] = S3 3((σ3 +3 α3)(⊞), x2, x3, x3) = S3 3(⊞(⊞(x3, x1, x1),⊞(x3, x1, x1),⊞(x3, x1, x1)), x2, x3, x3) = ⊞(⊞(x3, x2, x2),⊞(x3, x2, x2),⊞(x3, x2, x2)) and ((σ3 ◦h3 β3) +3 (α3 ◦h3 β3))(⊞) = S3 3((σ3 ◦h3 β3)(⊞), (α3 ◦h3 β3)(⊞), (α3 ◦h3 β3)(⊞), (α3 ◦h3 β3)(⊞)). Since (σ3 ◦h3 β3)(⊞) = (α3 ◦h3 β3)(⊞) = ⊞(⊞(x2, x3, x3),⊞(x2, x3, x3),⊞(x2, x3, x3)), we conclude that ((σ3 +3 α3) ◦h3 β3)(⊞) ̸= ((σ3 ◦h3 β3) +3 (α3 ◦h3 β3))(⊞), which means that the right distributivity on ((Hypwfv n (τ))n∈N, (◦hn)n∈N, (+n)n∈N) does not hold. T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 11 of 16 4. Weakly fixed variable hyperidentities In this section, we apply the concepts of terms of a weakly fixed variable and wfv- hypersubstitutions to describe classes of algebras. We start with the following definition. Definition 3. Let V be a variety of algebras of type τ . An identity s ≈ t ∈ Id(V ) is said to be a weakly fixed variable identity of V , also called wfv-identity of V , if both s and t come from the set Wwfv τ (Xn) for some n ∈ N. For example, the identity f(a, b, a) = a in the variety Reg of regular semigroups is a weakly fixed variable identity of Reg. For the set Idn(V ) of all n-ary identities of the variety V , we let Idwfv n (V ) := {s ≈ t | s ≈ t ∈ Id(V ), s, t ∈ Wwfv τ (Xn)}. Alternatively, we say that Idwfv n (V ) = (Wwfv τ (Xn)) 2 ∩ Idn(V ). Moreover, we consider Idwfv(V ) := (Idwfv n (V ))n∈N. Then we prove: Theorem 5. Let V be a variety of algebras of type τ . Then Idwfv(V ) is a congruence on the multisorted algebra Wwfv τ (X). Proof. It is clear that (Idfvn (V ))n∈N+ is preserved by the constant fundamental opera- tions, i.e., projections, of Wwfv τ (X). Assume now that p ≈ q ∈ Idwfv n (V ) and p1 ≈ q1, . . . , pn ≈ qn ∈ Idwfv m (V ). We aim to show that Sn m(p, p1, . . . , pn) ≈ Sn m(q, q1, . . . , qn) ∈ Idwfv m (V ). From Lemma 1, we have that the terms Sn m(p, p1, . . . , pn) and Sn m(q, q1, . . . , qn) contain in the set Wwfv τ (Xm). Thus Sn m(p, p1, . . . , pn) ≈ Sn m(q, q1, . . . , qn) ∈ Idm(V ). As a result, Sn m(p, p1, . . . , pn) ≈ Sn m(q, q1, . . . , qn) ∈ Idwfv m (V ). Theorem 5 allows us to consider the quotient systems of the following form: (Wwfv τ (Xn))n∈N/(Id wfv n (V ))n∈N, which we will call the quotient algebra of terms of a weakly fixed variable of a variety V . T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 12 of 16 Moreover, the multisorted operations on (Wwfv τ (Xn))n∈N/(Id wfv n (V ))n∈N denoted by S n m : Wwfv τ (Xn)/Id wfv n (V )× (Wwfv τ (Xm)/Idwfv m (V ))n → Wwfv τ (Xm)/Idwfv m (V ) can be naturally defined by S n m([t] Idwfv n (V ) , [t1]Idwfv m (V ), . . . , [tn]Idwfv m (V )) = [s] Idwfv m (V ) . Thus, we denote Qwfv τ (V ) := ((Wwfv τ (Xn))n∈N/(Id wfv n (V ))n∈N, (S n m)n,m∈N). Normally, the natural homomorphism is the multisorted mapping (natwfvIdwfvVn)n∈N : (Wwfv τ (Xn))n∈N → (Wwfv τ (Xn))n∈N/(Id wfv n (V ))n∈N defined by natwfvIdwfvVn(t) = [t] Idwfv n (V ) for all t ∈ Wwfv τ (Xn). Clearly, (natwfvIdwfvVn)n∈N is a homomorphism from Wwfv τ (X) to Qwfv τ (V ). In the study of algebra, the concept of hyperidentities represents an extension of iden- tities to a higher level, see [3, 13]. We now discuss identities that involve terms of a weakly fixed variable. Definition 4. Let V be a variety of algebras of type τ and let (Hypwfv n (τ))n∈N be the multisorted semigroup of weakly fixed variable hypersubstitutions of type τ . A weakly fixed variable identity s ≈ t in V is said to be a weakly fixed variable hyperidentity in V if σ̂n[s] ≈ σ̂n[t] ∈ Idn(V ) for s, t ∈ Wwfv τ (Xn), σn ∈ Hypwfv n (τ) and n ∈ N. Furthermore, we call a variety V a weakly fixed variable solid variety if σ̂n[s] ≈ σ̂n[t] ∈ Idn(V ) for s, t ∈ Wwfv τ (Xn), σn ∈ Hypwfv n (τ) and n ∈ N. From Definition 4, we define HIdwfv n (V ) := {s ≈ t | s, t ∈ Wwfv τ (Xn), σ̂n[s] ≈ σ̂n[t] ∈ Idn(V ), σn ∈ Hypwfv n (τ)}. Then (HIdwfv n (V ))n∈N is a multisorted equivalence on (Wwfv τ (Xn))n∈N. Theorem 6. Let V be a variety of type τ . Then (HIdwfv n (V ))n∈N is a congruence on Wwfv τ (X). T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 13 of 16 Proof. Let p ≈ q ∈ HIdwfv n (V ) and let pj ≈ qj ∈ HIdwfv m (V ) for j = 1, . . . , n. According to the definition of weakly fixed variable hyperidentities in a variety V , we have σ̂n[p] ≈ σ̂n[q] ∈ Idwfv n (V ) and σ̂m[pj ] ≈ σ̂m[qj ] ∈ Idwfv m (V ) for all j = 1, . . . , n and σ ∈ (Hypwfv n (τ))n∈N. Our aim is to show that σ̂m[Sn m(p, p1, . . . , pn)] ≈ σ̂m[Sn m(q, q1, . . . , qn)] ∈ Idwfv m (V ) for all σ ∈ (Hypwfv n (τ))n∈N. For this, we let σ be a weakly fixed variable hypersubstitution on the multisorted semigroup (Hypwfv n (τ))n∈N. From σ̂n[p] ≈ σ̂n[q] ∈ Idwfv n (V ), we have that σ̂n[p] and σ̂n[q] are m-ary terms of a weakly fixed variable of type τ and thus σ̂n[p] ≈ σ̂n[q] is an identity in V , i.e., σ̂n[p] ≈ σ̂n[q] ∈ Idn(V ). Similarly, because for every j = 1, . . . , n, σ̂m[pj ] ≈ σ̂m[qj ] ∈ Idwfv m (V ), then σ̂m[pj ] ≈ σ̂m[qj ] ∈ Idm(V ) for all j = 1, . . . , n, and σ̂m[p1], . . . , σ̂m[pn], σ̂m[q1], . . . , σ̂m[qn] ∈ Wwfv τ (Xm). By the fact that σ̂n[p] ≈ σ̂n[q] ∈ Idn(V ) and σ̂m[pj ] ≈ σ̂m[qj ] ∈ Idm(V ), we obtain Sn m(σ̂n[p], σ̂m[p1], . . . , σ̂m[pn]) ≈ Sn m(σ̂n[q], σ̂m[q1], . . . , σ̂m[qn]) ∈ Idm(V ) because (Idn(V ))n∈N is a congruence on the multisorted algebra of terms. From σ̂n[p], σ̂n[q] ∈ Wwfv τ (Xn) and σ̂m[p1], . . . , σ̂m[pn], σ̂m[q1], . . . , σ̂m[qn] ∈ Wwfv τ (Xm), we also conclude that Sn m(σ̂n[p], σ̂m[p1], . . . , σ̂m[pn]) ≈ Sn m(σ̂n[q], σ̂m[q1], . . . , σ̂m[qn]) ∈ Wwfv τ (Xm). This implies that Sn m(σ̂n[p], σ̂m[p1], . . . , σ̂m[pn]) ≈ Sn m(σ̂n[q], σ̂m[q1], . . . , σ̂m[qn]) ∈ Idwfv m (V ). From the fact that (σ̂n)n∈N is an endomorphism on the multisorted algebra Wwfv τ (X) proved in Theorem 2, we have σ̂m[Sn m(p, p1, . . . , pn)] ≈ σ̂m[Sn m(q, q1, . . . , qn)] ∈ Idwfv m (V ). T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 14 of 16 Therefore, (HIdwfv n (V ))n∈N is a congruence on Wwfv τ (X). A congruence (HIdwfv n (V ))n∈N on Wwfv τ (X) is said to be fully invariant if it is com- patible with all endomorphism σ̂ on Wwfv τ (X). Then we prove the following theorem which gives a necessary condition for any variety V to be weakly fixed variable. Theorem 7. Let V be a variety of type τ . If a congruence (HIdwfv n (V ))n∈N is fully invariant, then V is a weakly fixed variable solid variety. Proof. Suppose first that a congruence (HIdwfv n (V ))n∈N is fully invariant. On a variety V of type τ , we let s ≈ t ∈ HIdwfv n (V ) and (σn)n∈N ∈ (Hypwfv n (τ))n∈N. Applying Theorem 2, we have σ̂n[s] ≈ σ̂n[t] ∈ Idn(V ) for all n ∈ N. Hence, s ≈ t is a weakly fixed variable hyperidentity in V , whence, a variety V is weakly fixed variable. We close this section with the following theorem which gives connection between weakly fixed variable identities and weakly fixed variable hyperidentities in a variety V of algebras of type τ . Theorem 8. Every weakly fixed variable identity in a variety V is a weakly fixed variable hyperidentity in V . Proof. Suppose first that s ≈ t ∈ Idwfv(V ) is an identity in the multisorted quotient algebra Qwfv τ (V ). Let σ ∈ Hypwfv(τ). By Theorem 2 and the property of a natural homomorphism, thus the composition mapping natn,Idwfv(V ) ◦ σ̂n : Wwfv τ (X) → Qwfv τ (V ) is a homomorphism. By the hypothesis, we obtain natn,Idvf (V ) ◦ σ̂n(s) = natn,Idvf (V ) ◦ σ̂n(t). That is, natn,Idwfv(V )(σ̂n[s]) = natn,Idwfv(V )(σ̂n[t]). Again by a natural homomorphism natn,Idwfv(V ), we also get [σ̂n[s]]Idwfv(V ) = [σ̂n[t]]Idwfv(V ), which means that σ̂n[s] ≈ σ̂n[t] ∈ Idwfv(V ). Thus, s ≈ t is a weakly fixed variable hyperidentity in V . T. Kumduang, K. Wattanatripop / Eur. J. Pure Appl. Math, 18 (2) (2025), 5835 15 of 16 5. Concluding remarks As a generalization of terms of fixed variable introduced in [17], this work presents an extension called terms of a weakly fixed variable by relaxing the conditions on inductive construction of any term t from an alphabet Xn. Several multibased structures for such terms are developed, including the superassociative system with respect to the multisorted superposition operation, the seminearring of mappings which images are terms of a weakly fixed variable, the quotient multisorted set obtained by dividing the set of all identities induced by terms of a weakly fixed variable, which acts as a congruence. 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