Hyperidentities in many-sorted algebras

Discussiones Mathematicae General Algebra and Applications 29 (2009 ) 47–74 HYPERIDENTITIES IN MANY-SORTED ALGEBRAS Klaus Denecke and Somsak Lekkoksung Universität Potsdam, Institut of Mathematics Am Neuen Palais, 14415 Potsdam, Germany e-mail: Abstract The theory of hyperidentities generalizes the equational theory of universal algebras and is applicable in several fields of science, especially in computers sciences (see e.g., [2, 1]). The main tool to study hyperidentities is the concept of a hypersubstitution. Hypersubstitutions of many-sorted algebras were studied in [3]. On the basis of hypersubstitutions one defines a pair of closure operators which turns out to be a conjugate pair. The theory of conjugate pairs of additive closure operators can be applied to characterize solid varieties, i.e., varieties in which every identity is satisfied as a hyperidentity (see [4]). The aim of this paper is to apply the theory of conjugate pairs of additive closure operators to many-sorted algebras. Keywords: hypersubstitution; hyperidentity; heterogeneous algebra. 2000 Mathematics Subject Classification: 08A68, 08B15. 1. Preliminaries Hyperidentities in one-based algebras were considered by many authors (for references see e.g., [4, 2]). An identity s ≈ t is satisfied as a hyperidentity in the one-based algebra A = (A; (f iA )i∈I ) of type τ if after any 48 K. Denecke and S. Lekkoksung replacements of the operation symbols occurring in s and t by terms of the same arity the arising equation is satisfied in A. These replacements can be described by hypersubstitutions, i.e., mappings from the set of operation symbols into the set of all terms of type τ . Hypersubstitutions cannot only be applied to terms or equations but also to algebras. This gives a pair of additive closure operators which are related to each other by the socalled conjugate property and which form a conjugate pair of additive closure operators (see [4]). A variety of one-based algebras is called solid if every identity is satisfied as a hyperidentity. Characterizations of solid varieties are based on the theory of conjugate pairs of additive closure operators. For more background see [4]. In this paper we want to apply the theory of conjugate pairs of additive closure operators to many-sorted algebras and identities and want to define hyperidentities and solid varieties of many-sorted algebras. Many-sorted algebras occur in various branches of mathematics. They have found their way into computer science through abstract data type specifications. Many-sorted algebras, varieties and quasivarieties of manysorted algebras are the mathematical fundament of approaches to abstract data types in programming and specification languages. For basic concepts on many-sorted algebras we refer the reader to [5]. The concept of terms in many-sorted algebras was discussed in [5]. First we want to give a slightly different version of the definitions and results from [3]. S Let I be a non-empty set, let N+ := N \ {0} , n ∈ N+ , let I ∗ := n≥1 I n and Σ ⊆ I ∗ × I. Then we define Σn := Σ ∩ I n+1 . For γ ∈ Σ let γ(l) denote the l-th component of γ. Let Kγ be a set of indices with respect to γ. If |Kγ | = 1, we will drop the index. (n) Definition 1.1. Let n ∈ N+ and X (n) := (Xi )i∈I be an I-sorted set (n) of variables, also called an n-element I-sorted alphabet, with X i := {xi1 , . . . , xin }, i ∈ I and let ((fγ )k )k∈Kγ ,γ∈Σ be an indexed set of Σ-sorted operation symbols. Then for each i ∈ I a set W n (i) which is called the set of all n-ary Σ-terms of sort i, is inductively defined as follows: (n) (i) W0n (i) := Xi , i ∈ I, Hyperidentities in many-sorted algebras 49 n (i) := W n (i) ∪ {f (t , . . . , t ) | γ = (k , . . . , k ; i) ∈ Σ, t (ii) Wl+1 γ k1 1 n kn kj ∈ l n Wl (kj ), 1 ≤ j ≤ n}, l ∈ N. (Here we inductively assume that the sets Wln (i) are already defined for all sorts i ∈ I). S S n Then Wn (i) := ∞ l=0 Wl (i) and we set W (i) := n∈N+ Wn (i). Let Xi := S (n) and X := (Xi )i∈I . Let WΣ (X) := (W (i))i∈I . The set WΣ (X) n∈N+ Xi is called I-sorted set of all Σ-terms and its elements are called I-sorted Σ-terms. For any n ∈ N+ , i ∈ I we set Λn (i) := {(w; i) ∈ I n+1 | w ∈ I n , ∃ m ∈ S N+ , ∃ α ∈ Σm , ∃ j (1 ≤ j ≤ m)(α(j) = i)}. Let Λ(i) := ∞ n=1 Λn (i) and S we set Λ := i∈I Λ(i). To define many-sorted hypersubstitutions we need the following superposition operation for I-sorted Σ-terms. Definition 1.2. Let t ∈ W (i), tj ∈ W (kj ) where 1 ≤ j ≤ n, n ∈ N. Then the superposition operation Sβ : W (i) × W (k1 ) × · · · × W (kn ) → W (i) for β = (k1 , . . . , kn ; i) ∈ Λ, is defined inductively as follows: 1. If t = xij ∈ Xi , then 1.1 Sβ (xij , t1 , . . . , tn ) := xij for i 6= kj and 1.2 Sβ (xij , t1 , . . . , tn ) := tj for i = kj . 2. If t = fγ (s1 , . . . , sm ) ∈ W (i) for γ = (i1 , . . . , im ; i) ∈ Σ and sq ∈ Wn (iq ), 1 ≤ q ≤ m, m ∈ N, and if we assume that S βq (sq , t1 , . . . , tn ) with βq = (k1 , . . . , kn ; iq ) ∈ Λ are already defined, then Sβ (fγ (s1 , . . . , sm ), t1 , . . . , tn ) := fγ (Sβ1 (s1 , t1 , . . . , tn ), . . . , Sβm (sm , t1 , . . . , tn )). 50 K. Denecke and S. Lekkoksung Definition 1.3. Let i ∈ I and ((fγ )k )k∈Kγ ,γ∈Σ be an indexed set of Σsorted operation symbols. Let Σm (i) := {γ ∈ Σm | γ(m + 1) = i}, m ∈ N+ and let [ Σ(i) := Σm (i). m≥1 Any mapping σi : {(fγ )k | k ∈ Kγ , γ ∈ Σ(i)} → W (i), i ∈ I, which preserves arities, is said to be a Σ-hypersubstitution of sort i. Let Σ(i)Hyp be the set of all Σ-hypersubstitutions of sort i. The I-sorted mapping σ := (σi )i∈I is called an I-sorted Σ-hypersubstitution. Let Σ-Hyp be the set of all I-sorted Σ-hypersubstitutions. Any I-sorted Σ-hypersubstitution σ can inductively be extended to an I-sorted mapping σ̂ := (σ̂ i )i∈I . The I-sorted mapping σ̂ : WΣ (X) → WΣ (X) is defined by the following steps: For each i ∈ I we define (i) σ̂i [xij ] := xij for any variable xij ∈ Xi . (ii) σ̂i [fγ (t1 , . . . , tn )] := Sγ (σi (fγ ), σ̂k1 [t1 ], . . . , σ̂kn [tn ]), where γ = (k1 , . . . , kn ; i) ∈ Σ and tq ∈ W (kq ), 1 ≤ q ≤ n, n ∈ N, assumed that σ̂kq [tq ], are already defined. Using the extension σ̂i , we define (σ1 )i ◦i (σ2 )i := (σ̂1 )i ◦ (σ2 )i . Then we have ((σ1 )i ◦i (σ2 )i )ˆ = (σ̂1 )i ◦ (σ̂2 )i . Together with the identity mapping (σ id )i the set Σ(i)-Hyp forms a monoid (see [3]). Now we want to describe the connection between heterogeneous algebras and Σ-terms. Let A be an I-sorted set. Then A is said to be a Σ-algebra if it has the form  

A
A = A; fγ k k∈Kγ ,γ∈Σ 51 Hyperidentities in many-sorted algebras where ((fγ )k )A : Ak1 × · · · × Akn → Ai if γ = (k1 , . . . , kn ; i) ∈ Σ. Let Alg(Σ) be the collection of all Σ-algebras. To connect Σ-terms with Σ-algebras we need to consider operations on I-sorted sets. Let A be an I-sorted set, n ∈ N+ , (ω; i) ∈ I ∗ × I. Then ω is called input sequence on A and i is called output sort. Definition 1.4. Let A be an I-sorted set, let ω = (k 1 , . . . , kn ) ∈ I n , (...truncated)