On a product of universal hyperalgebras

Transcription

1 DOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 23(2),2015, On a product of universal hyperalgebras Nitima Chaisansuk and Josef Šlapal Abstract We introduce and study a new operation of product of universal hyperalgebras which lies, with respect to set inclusion, between the cartesian product of the hyperalgebras and the cartesian product of their idempotent hulls. We give sufficient conditions for the validity of the first exponential law and a weak form of the second exponential law for the direct power of universal hyperalgebras with respect to the product introduced. 1 Introduction In his pioneering paper [1], G. Birkhoff introduced the cardinal (i.e., direct) arithmetic of partially ordered sets and showed that it behaves analogously to the arithmetic of natural numbers. Several authors then extended the cardinal arithmetic to relational systems - see e.g. [6-7] and [9-10]. Conversely, the cardinal arithmetic has been restricted from relational systems to universal algebras in [11], to partial algebras in [14] and to hyperalgebras in [13]. In the present paper, we continue the study of direct arithmetic of hyperalgebras from [13]. In [7], M. Novotný and J. Šlapal introduced and studied a new operation of product of relational systems which is a combination of the direct sum and the direct product of the systems. In the present note, we restrict the operation of product discussed in [7] from relational systems to hyperalgebras. We will Key Words: Universal hyperalgebra, combined product, diagonality, mediality, direct power Mathematics Subject Classification: Primary 08A99, 08B25; Secondary 08A65. Received: October, Revised: December, Accepted: December,

2 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 72 study the behavior of the product restricted and will find conditions under which the direct power of hyperalgebras satisfies the first exponential law and a weak form of the second exponential law with respect to the product. Hyperalgebras proved to be useful for many application in various branches of mathematics and computer science (automata theory). This lead to a rapid development of the theory of hyperalgebras since the beginning of 90 s of the last century - see e.g. [2]. The aim of this paper is to contribute to the development by introducing and studying a new operation of product of hyperalgebras. 2 Combined product of hyperalgebras Given sets G and H, we denote by G H the set of all mappings of H into G. The bijection ϕ : (G H ) K G H K (where denotes the cartesian product) given by ϕ(h)(x, y) = h(y)(x) whenever h (G H ) K, x H and y K will be called canonical. Throughout the paper, maps f : G H (G, H sets) will often be denoted as indexed sets (f i ; i G) where f i means f(i) for every i G. Let Ω be a nonempty set. A family τ = (K λ ; λ Ω) of sets will be called a type. By a universal hyperalgebra (briefly, a hyperalgebra) of type τ we understand a pair G = G, (p λ ; λ Ω) where G is a nonempty set, the socalled carrier of G, and p λ : G K λ exp G { } is a map, the so called K λ -ary hyperoperation on G, for every λ Ω. Of course, if K λ =, then p λ is nothing but a nonempty subset of G. To avoid some nonwanted singularities, all hyperalgebras G = G, (p λ ; λ Ω) are supposed to have a type (K λ ; λ Ω) with K λ for every λ Ω. A hyperalgebra G = G, (p λ ; λ Ω) with the property card p λ (x i ; i K λ ) = 1 for every λ Ω and every (x i ; i K λ ) G K λ is called a (universal) algebra. If τ = (K) where K is a finite set with card K = n, then hyperalgebras of type τ are called n-ary hyperalgebras. In the case card K = 2, hyperalgebras of type τ are called hypergroupoids. Given a hyperalgebra G = G, (p λ ; λ Ω) of type τ = (K λ ; λ Ω), λ Ω and a family X i, i K λ, of subsets of G, we put p λ (X i ; i K λ ) = {p λ (x i ; i K λ ); x i X i for every i K λ }. Let G = G, (p λ ; λ Ω) and H = H, (q λ ; λ Ω) be a pair of hyperalgebras of type τ. Then G is called a subhyperalgebra of H provided that G H and p λ (x i ; i K λ ) = q λ (x i ; i K λ ) whenever λ Ω and x i H for each i K λ. We will write G H if G = H and p λ (x i ; i K λ ) q λ (x i ; i K λ ) whenever λ Ω and (x i ; i K λ ) H K λ. A map f : H G is called a homomorphism of H into G if, for each λ Ω, f(q λ (x i ; i K λ )) p λ (f(x i ); i K λ ). The set of all homomorphisms of H into G will be denoted by Hom(H, G). If f is a bijection of H onto G and f(q λ (x i ; i K λ )) = p λ (f(x i ); i K λ )

3 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 73 whenever x i H for each i K λ, then f is called an isomorphism of H onto G. In other words, an isomorphism of H onto G is a bijection f : H G such that f is a homomorphism of H into G and f 1 is a homomorphism of G into H. If there is an isomorphism of H onto G, then we write H = G and say that H and G are isomorphic. We say that H may be embedded into G and write H G if there exists a subhyperalgebra G of G such that H = G. The direct product of a nonempty family G i = G i, (p i λ ; λ Ω), i I, of hyperalgebras of type τ = (K λ ; λ Ω) is the hyperalgebra i I G i = i I G i, (q λ ; λ Ω) of type τ where i I G i denotes the cartesian product and, for any λ Ω and any (f k ; k K λ ) ( i I G i) K λ, q λ (f k ; k K λ ) = i I pi λ (f k(i); k K λ ). If the set I is finite, say I = {1,..., m}, then we write G 1... G m instead of i I G i. If G i = G for every i I, then we write G I instead of i I G i. The direct sum of a nonempty family G i = G, (p i λ ; λ Ω), i I, of hyperalgebras of type τ = (K λ ; λ Ω) is the hyperalgebra i I G i = G, (q λ ; λ Ω) of type τ where, for every λ Ω and every (x i ; i K λ ) G K λ, q λ (x i ; i K λ ) = i I pi λ (x i; i K λ ). If the set I is finite, say I = {1,..., m}, then we write G 1... G m instead of i I G i. Let G = G, (p λ ; λ Ω) be a hyperalgebra of type τ = (K λ ; λ Ω) and let λ 0 Ω. An element x G is called an idempotent with respect to the hyperoperation p λ0 if x p λ0 (x i ; i K λ0 ) whenever x i = x for every i K λ0. If every element of G is an idempotent with respect to each hyperoperation p λ, λ Ω, then the hyperalgebra G is said to be idempotent. Let G = G, (p λ ; λ Ω) be a hyperalgebra of type τ = (K λ ; λ Ω). For every λ Ω and every (x i ; i K λ ) G K λ we put p λ (x i ; i K λ ) {x} if there is x G with x i = x for all p λ (x i ; i K λ ) = i K λ, p λ (x i ; i K λ ) if there are i, j K λ with x i x j. The hyperalgebra G, ( p λ ; λ Ω) is called the idempotent hull of G and is denoted by Ḡ. Let G i = G i, (p iλ ; λ Ω), i I, be a nonempty family of hyperalgebras of type τ = (K λ ; λ Ω). The combined product of the family G i, i I, is the hyperalgebra i I G i i I G i = i I j I G ij where G ij = = i I G i, (r λ ; λ Ω) of type τ given by { Ḡj if i = j, G j if i j. Thus, for any λ Ω and any ((x i k ; i I); k K λ) ( i I G i) K λ, we have (x i ; i I) r λ ((x i k ; i I); k K λ) if and only if there exists a subset J I,

4 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 74 card J 1, such that x i p i λ (xi k ; k K λ) for every i I J and x i k = xi for every k K λ and every i J. If the set I is finite, say I = {1,..., m}, we write G 1... G m instead of i I G i. We then clearly have G 1... G m = (Ḡ1 G 2... G m ) (G 1 G 2... G m )... (G 1 G 2... Ḡm). In particular, if I = {1, 2}, then, for every λ Ω and every ((x k, y k ); k K λ ) (G 1 G 2 ) K λ, (x, y) r λ ((x k, y k ); k K λ ) if and only if one of the following three conditions is satisfied: (i) x p 1 λ (x k; k K λ ) and y p 2 λ (y k; k K λ ), (ii) x = x k for every k K λ and y p 2 λ (y k; k K λ ), (iii) x p 1 λ (x k; k K λ ) and y = y k for every k K λ. If I is a singleton, say I = {j}, then i I G i = Ḡj. Remark 2.1. Let G i, i I, be a family of hyperalgebras of the same type. We clearly have i I G i i I G i i I Ḡi = i I Ḡi and i I G i i I Ḡi. If G i is idempotent for every i I, then all the previous inequalities become equalities. If the hyperalgebras G i, i I, are idempotent with the exception of at most one of them, then i I G i is idempotent. Theorem 2.2. Let G i, i I, be a nonempty family of hyperalgebras of the same type τ and let H be a hyperalgebra of type τ. Then i I H G i = H i I G i. Proof. Let G i = G i, (p i λ ; λ Ω) for all i I, H = H, (q λ; λ Ω) and let τ = (K λ ; λ Ω). Let i I G i = G, (s λ ; λ Ω), H G i = H G, (rλ i ; λ Ω) for each i I, H i I G i = H G, (u λ ; λ Ω) and i I H G i = H G, (v λ ; λ Ω). We will show that u λ ((x k, y k ); k K λ ) = v λ ((x k, y k ); k K λ ) for every ((x k, y k ); k K λ ) (H G) K λ. It is easy to see that the following five conditions are equivalent: (a) (x, y) u λ ((x k, y k ); k K λ ); (b) one of the following three cases occurs: (i) x q λ (x k ; k K λ ) and y s λ (y k ; k K λ ); (ii) x = x k for every k K λ and y s λ (y k ; k K λ ); (iii) x q λ (x k ; k K λ ) and y = y k for every k K λ ; (c) one of the following three cases occurs: (i) x q λ (x k ; k K λ ) and y p i λ (y k; k K λ ) for some i I;

5 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 75 (ii) x = x k for every k K λ and y p i λ (y k; k K λ ) for some i I; (iii) x q λ (x k ; k K λ ) and y = y k for every k K λ ; (d) (x, y) r i λ ((x k, y k ); k K λ ) for some i I; (e) (x, y) v λ ((x k, y k ); k K λ ). This proves the statement. 3 Direct power of hyperalgebras In the sequel, we will work with K L-matrices over G where K, L, G are nonempty sets. These matrices are nothing but the maps x : K L G, i.e., the indexed sets (x ij ; i K, j L) denoted briefly by (x ij ). Definition 3.1. A hyperalgebra G, (p λ ; λ Ω) of type τ = (K λ ; λ Ω) is called medial if, for every µ, ν Ω and every K µ K ν -matrix (a ij ) over G, from x i p ν (a ij ; j K ν ) for each i K µ and y j p µ (a ij ; i K µ ) for each j K ν it follows that p ν (x i ; i K µ ) = p µ (y j ; j K ν ). The medial universal algebras are often called commutative or the algebras satisfying the interchange law and they were dealt with e.g. in [4], [5] and [11]. The medial n-ary algebras are studied in [12] and the medial groupoids in [3]. Example 3.2. (1) Let (X, ) be a partially ordered set with a least element 0 and let A be the set of all atoms of (X, ). Put 0 = {0} and, for any x X with x 0, put x = {y X; y < x and y A {0}}. Further, for any pair x, y X put x y = x y. Then (X,, ) is a medial hyperalgebra of type (1, 2). Indeed, it can easily be seen that, for any x X, we have a = {0} = b whenever a x and b x. Further, for any a, b, c, d X, we have x y = f g whenever x a b, y c d, f a c and g b d. Finally, for any a, b X, we have x = y z whenever x a b, y a and z b. (2) Every unary algebra G consisting of two-element cycles only is medial and, moreover, the unary hyperalgebra Ḡ is medial, too. Lemma 3.3. Let H and G = G, (p λ ; λ Ω) be hyperalgebras of the same type τ = (K λ ; λ Ω), let λ Ω and let f i Hom(H, G) for each i K λ. Let f : H G be a map such that f(x) p λ (f i (x); i K λ ) for every x H. If G is medial, then f is a homomorphism from H into G. Proof. Let H = H, (q λ ; λ Ω), µ Ω and x i H for each i K µ. We will show that f(q µ (x i ; i K µ )) p µ (f(x i ); i K µ ). Let y f(q µ (x i ; i K µ )). Then there exists x q µ (x i ; i K µ ) such that y = f(x), so f(x) f(q µ (x i ; i K µ )). Since f j Hom(H, G) for each j K λ and x q µ (x i ; i K µ ), we

6 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 76 have f j (x) f j (q µ (x i ; i K µ )) p µ (f j (x i ); i K µ ) for each j K λ. By the assumptions of the statement, f(x i ) p λ (f j (x i ); j K λ ) for each i K µ and f(x) p λ (f j (x); j K λ ). Since G is medial, p λ (f j (x); j K λ ) = p µ (f(x i ); i K µ ). Thus, y = f(x) p µ (f(x i ); i K µ ). Definition 3.4. Let G = G, (p λ ; λ Ω) and H be hyperalgebras of the same type τ = (K λ ; λ Ω) and let G be medial. The direct power of G and H is the hyperalgebra G H = Hom(H, G), (r λ ; λ Ω) of type τ where, for any λ Ω and any (f i ; i K λ ) (Hom(H, G)) K λ, r λ (f i ; i K λ ) = {f G H ; f(x) p λ (f i (x); i K λ ) for each x H}. It is easy to see that the power G H of hyperalgebras of the same type is idempotent whenever G is idempotent (and medial). Theorem 3.5. Let G, H be hyperalgebras of the same type. If G is medial, then so is G H. Proof. Let G = G, (p λ ; λ Ω) and H = H, (q λ ; λ Ω). Let τ = (K λ ; λ Ω) be the type of G and H and let G H = Hom(H, G), (r λ ; λ Ω). Let µ, ν Ω and let (f ij ) be a K µ K ν -matrix over Hom(H, G). Suppose that h i r ν (f ij ; j K ν ) for all i K µ and g j r µ (f ij ; i K µ ) for all j K ν. For every x H, there holds h i (x) p ν (f ij (x); j K ν ) for all i K µ and g j (x) p µ (f ij (x); i K µ ) for all j K ν. Since G is medial, we have p µ (h i (x); i K µ ) = p ν (g j (x); j K ν ) for every x H. Therefore, r µ (h i ; i K µ ) = r ν (h j ; j K ν ), so that G H is medial. Definition 3.6. A hyperalgebra G, (p λ ; λ Ω) of type τ is called diagonal if, for every λ Ω and every K λ K λ -matrix (a ij ) over G, we have p λ (p λ (a ij ; j K λ ); i K λ )) p λ (p λ (a ij ; i K λ ); j K λ ) p λ (a ii ; i K λ ). For idempotent n-ary algebras, the diagonality introduced coincides with the diagonality studied in [8]. Example 3.7. It may easily be seen that the hyperalgebra from Example 3.2(1) is diagonal. Theorem 3.8. Let G, H be hyperalgebras of the same type. If G is medial and diagonal, then G H is diagonal. Proof. Let G = G, (p λ ; λ Ω) and H = H, (q λ ; λ Ω). Let τ = (K λ ; λ Ω) be the type of G and H and let G H = Hom(H, G), (r λ ; λ Ω). Let λ Ω and let (f ij ) be a K λ K λ -matrix over Hom(H, G). Suppose that f r λ (r λ (f ij ; j K λ ); i K λ ) r λ (r λ (f ij ; i K λ ); j K λ ). Then f(x) p λ (p λ (f ij (x); j K λ ); i K λ ) p λ (p λ (f ij (x); i K λ ); j K λ ) for every x H. Since G is diagonal, we have f(x) p λ (f ii (x); i K λ ) for every x H. Thus, f r λ (f ii ; i K λ ). Hence, G H is diagonal.

7 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 77 We will find sufficient conditions for the validity of the first exponential law and a weak form of the second exponential law for the direct power of hyperalgebras with respect to the combined product, namely, the laws (G H ) K = G H K and G i ) H. i I G H i ( i I Lemma 3.9. Let G, H, K be partial algebras of the same type and let G be medial and diagonal. Then the canonical bijection ϕ : (G H ) K G H K restricted to Hom(K, G H ) is a bijection of Hom(K, G H ) onto Hom(H K, G). Proof. Let G = G, (p λ ; λ Ω), H = H, (q λ ; λ Ω) and K = K, (s λ ; λ Ω). Let τ = (K λ ; λ Ω) be the type of G, H and K. Let G H = Hom(H, G), (r λ ; λ Ω), H K = H K, (v λ ; λ Ω) and let G be diagonal and medial. Let λ Ω, h Hom(K, G H ) and let (y i, z i ) H K for every i K λ. To show that ϕ(h)(v λ ((y i, z i ); i K λ )) p λ (ϕ(h)(y i, z i ); i K λ ), let x ϕ(h)(v λ ((y i, z i ); i K λ )). Then there exists (y, z) v λ ((y i, z i ); i K λ ) such that x = ϕ(h)(y, z). Thus, one of the following three conditions is satisfied: (i) y q λ (y i ; i K λ ) and z s λ (z i ; i K λ ); (ii) y = y i for every i K λ and z s λ (z i ; i K λ ); (iii) y q λ (x i ; i K λ ) and z = z i for every i K λ. If z s λ (z i ; i K λ ), then h(z) h(s λ (z i ; i K λ )) r λ (h(z i ); i K λ ), hence ϕ(h)(y, z) = h(z)(y) p λ (h(z i )(y); i K λ )) = p λ (ϕ(h)(y, z i ); i K λ ) for every y H. Thus, ϕ(h)(y, z) p λ (ϕ(h)(y i, z i ); i K λ ) provided that (ii) is satisfied. If y q λ (y i ; i K λ ), then ϕ(h)(y, z) = h(z)(y) h(z)(q λ (y i ; i K λ )) p λ (h(z)(y i ); i K λ ) = p λ (ϕ(h)(y i, z); i K λ ) for every z K. Thus, ϕ(h)(y, z) p λ (ϕ(h)(y i, z i ); i K λ ) provided that (iii) is satisfied. Suppose that (i) is satisfied. Since z s λ (z i ; i K λ ), by the above considerations we have ϕ(h)(y i, z) p λ (ϕ(h)(y i, z j ); j K λ ) for every i K λ and ϕ(h)(y, z) p λ (ϕ(h)(y, z j ); j K λ ). Since y q λ (y i ; i K λ ), by the above considerations we have ϕ(h)(y, z j ) p λ (ϕ(h)(y i, z j ); i K λ ) for every j K λ and ϕ(h)(y, z) p λ (ϕ(h)(y i, z); i K λ ). The diagonality of G implies ϕ(h)(y, z) p λ (ϕ(h)(y i, z i ); i K λ ). We have shown that ϕ(h) Hom(H K, G). Let g Hom(H K, G) and z i K for every i K λ. To show that ϕ 1 (g)(s λ (z i ; i K λ )) r λ (ϕ 1 (g)(z i ); i K λ ), let x ϕ 1 (g)(s λ (z i ; i

8 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 78 K λ )). Then there exists z s λ (z i ; i K λ ) such that x ϕ 1 (g)(z). Let y H. Then (y, z) v λ ((y, z i ); i K λ ). Since g Hom(H K, G), we have g(y, z) g(v λ ((y, z i ); i K λ )) p λ (g(y, z i ); i K λ ). It follows that ϕ 1 (g)(z)(y) p λ (ϕ 1 (g)(z i )(y); i K λ ), hence ϕ 1 (g)(z) p λ (ϕ 1 (g)(z i ); i K λ ). Therefore, ϕ 1 (s λ (z i ; i K λ ) r λ (ϕ 1 (g)(z i ); i K λ ). Consequently, ϕ 1 (g) Hom(K, G H ). Theorem Let G, H, K be hyperalgebras of the same type. If G is medial and diagonal, then (G H ) K = G H K. Proof. Let G = G, (p λ ; λ Ω), H = H, (q λ ; λ Ω) and K = K, (s λ ; λ Ω). Let τ = (K λ ; λ Ω) be the type of G, H and K. Let G H = Hom(H, G), (r λ ; λ Ω), H K = H K, (v λ ; λ Ω), (G H ) K = Hom(K, G H ), (t λ ; λ Ω), G H K = Hom(H K, G), (u λ ; λ Ω) and let G be diagonal and medial. By Lemma 3.9, we are to show that ϕ(t λ (h i ; i K λ )) = u λ (ϕ(h i ); i K λ ) whenever h i Hom(K, G H ) for every i K λ. Let x ϕ(t λ (h i ; i K λ )). Then there exists h t λ (h i ; i K λ ) such that x = ϕ(h). Since h t λ (h i ; i K λ ), we have h(z) r λ (h i (z); i K λ )) for every z K. Therefore, h(z)(y) p λ (h i (z)(y); i K λ )) for every y H. It follows that ϕ(h)(y, z) p λ (ϕ(h i )(y, z); i K λ ) for every (y, z) H K. Hence, ϕ(h) u λ (ϕ(h i ); i K λ ). Thus, ϕ(t λ (h i ; i K λ )) u λ (ϕ(h i ); i K λ ). Let x u λ (ϕ(h i ); i K λ ). By Lemma 3.9, there exists h Hom(K, G H ) with x = ϕ(h), so that ϕ(h) u λ (ϕ(h i ); i K λ ). Thus, ϕ(h)(y, z) p λ (ϕ(h i ) (y, z); i K λ ) for every (y, z) H K, which implies h(z)(y) p λ (h i (z)(y); i K λ ) for every y H and every z K. It follows that h(z) r λ (h i (z); i K λ ) for every z K. Therefore, h t λ ((h i ); i K λ ), hence ϕ(h) ϕ(t λ ((h i ); i K λ )). Consequently, u λ (ϕ(h i ); i K λ ) ϕ(t λ ((h i ); i K λ )). We have shown that ϕ(t λ (h i ; i K λ )) = u λ (ϕ(h i ); i K λ ). Thus, (G H ) K is isomorphic to G H K. It may easily be seen that the direct product of a family of medial hyperalgebras of the same type is a medial hyperalgebra. But this is not generally true for the combined product of hyperalgebras. With respect to the combined product, we have the following weak form of the second exponential law: Theorem Let G i, i I, be a nonempty family of medial hyperalgebras of type τ = (K λ ; λ Ω) and let H be a hyperalgebra of type τ. If i I G i is medial, then G i ) H. i I G H i ( i I

9 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 79 Proof. Let τ = (K λ ; λ Ω). Let G i = G i, (p i λ ; λ Ω) for every i I, H = H, (q λ ; λ Ω), i I G i = i I G i, (r λ ; λ Ω), G H i = Hom(H, G i ), (u i λ ; λ Ω) for every i I, i I G i H = i I Hom(H, G i), (s λ ; λ Ω) and ( i I G i) H = Hom(H, i I G i), (t λ ; λ Ω). We define a map α : i I Hom(H, G i) ( i I G i) H by α(f i ; i I)(z) = (f i (z); i I) for each z H. Let (f i ; i I) i I Hom(H, G i) and h α(f i ; i I)(q λ (x j ; j K λ )). Then h = α(f i ; i I)(x) where x q λ (x j ; j K λ ). Thus, h = α(f i ; i I)(x) = (f i (x); i I). Since x q λ (x j ; j K λ ) and f i Hom(H, G i ) for every i I, we have f i (x) p i λ (f i (x j ); j K λ ) for every i I. Thus, h = (f i (x); i I) r λ ((f i (x j ); i I); j K λ ) = r λ (α(f i ; i I)(x j ); j K λ ). Consequently, α(f i ; i I)(q λ (x j ; j K λ )) r λ (α(f i ; i I)(x j ); j K λ ). Therefore, α(f i ; i I) Hom(H, i I G i). We have shown that α maps i I Hom(H, G i) into Hom(H, i I G i). Suppose that α(f i ; i I) = α(g i ; i I) where (f i ; i I), (g i ; i I) i I Hom(H, G i). Then (f i (x); i I) = α(f i ; i I)(x) = α(g i ; i I)(x) = (g i (x); i I) for every x H. Therefore, f i (x) = g i (x) for every i I and every x H. Hence, f i = g i for every i I. Thus, α: i I Hom(H, G i) Hom(H, i I G i) is an injection. Finally, let (fj i; i I) i I Hom(H, G i). We will show that α(s λ ((fj i; i I); j K λ)) = t λ (α(fj i; i I); j K λ). It is easy to see that the following seven conditions are equivalent: (a) f α(s λ ((f i j ; i I); j K λ)); (b) f = α(f i ; i I) where (f i ; i I) s λ ((f i j ; i I); j K λ); (c) f = α(f i ; i I) and there exists J I, card J 1, such that f i u i λ (f i j ; j K λ) for every i I J and every f i = f i j for every j K λ and i J; (d) f(x) = (f i (x); i I) for every x H and there exists J I, card J 1, such that f i (x) p i λ (f i j (x); j K λ) for every i I J and every x H and f i (x) = f i j (x) for every j K λ, every i J and every x H; (e) f(x) = (f i (x); i I) r λ ((f i j (x); i I); j K λ) for every x H; (f) f(x) r λ (α(f i j ; i I)(x); j K λ) for every x H; (g) f t λ (α(f i j ; i I); j K λ). Consequently, α(s λ ((f i j ; i I); j K λ)) = t λ (α(f i j ; i I); j K λ), which yields i I GH i ( i I G i) H.

10 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 80 It may easily be shown that, in Theorem 3.12, we may write = instead of provided that G i is idempotent for every i I. We then obtain the second exponential law for the direct product i I GH i = ( i I G i) H. 4 Conclusion The paper contributes to the development of the arithmetic of universal hyperalgebras. We proved that a naturally defined direct sum of universal hyperalgebras of the same type is compatible with their direct product in the sense that the direct product distributes over it (Theorem 2.2). The main results concern the behavior of the operation of combined product introduced for universal algebras (of the same type). In particular, for the direct power of universal hyperalgebras, we studied the validity of the first and second exponential laws with respect to the combined product. Theorem 3.10 states that, with respect to the combined product, the direct power of the hyperalgebras satisfies the first exponential law (G H ) K = G H K whenever G is medial and diagonal. This is not true when replacing the combined product with the direct product. It follows from Theorem 3.10 that the direct power of hyperalgebras satisfies the first exponential law with respect to the direct product if G is medial and diagonal and, moreover, H and K are idempotent (this result was proved in [13]). Theorem 3.11 says that the direct power of universal hyperalgebras satisfies the weak form of the second exponential law i I GH i ( i I G i) H. It may easily be shown that, in Theorem 3.11, we may write = instead of (thus obtaining the second exponential law with respect to the combined product) provided that G i is idempotent for every i I (in which case the combined product coincides with the direct one). It is an open problem to find a necessary and sufficient condition for the validity of the second exponential law for the direct power of hyperalgebras with respect to the combined product. Acknowledgement. The first author acknowledges support from the Royal Golden Jubilee Ph.D. Program of the Thailand Research Fund., the Graduate School and the Faculty of Science, Chiang Mai University, Thailand. The second author acknowledges support from the Brno University of Technology, Specific Research Project no. FSI-S References [1] G. Birkhoff, Generalized arithmetic, Duke Math. J. 9(1942),

11 ON A PRODUCT OF UNIVERSAL HYPERALGEBRAS 81 [2] Š. Hošková (ed.), Algebraic Hyperstructures and Applications, Proceedings of the 10th International Congress, University of Defence, Brno, [3] J. Ježek and T. Kepka, Medial groupoids, Rozpravy ČSAV, Řada Mat. a Přír. Věd. 93/1, Academia, Prague, [4] L. Klukovits, On commutative universal algebras, Acta Sci. Math. (Szeged) 34(1973), [5] F.E. Linton, Autonomous equational categories, J. Math. Mech. 15(1966), [6] V. Novák, On a power of relational structures, Czech. Math. J. 35(1985), [7] M. Novotný and J. Šlapal, On powers of relational structures, Publ. Math. Debrecen 61(2002), [8] J. Plonka, Diagonal algebras, Fund. Math. 58(1966), [9] J. Šlapal, Direct arithmetic of relational systems, Publ. Math. Debrecen 38(1991), [10] J. Šlapal, Cardinal arithmetic of general relational systems, Czech. Math. J. 43(1993), [11] J. Šlapal, A cartesian closed category of universal algebras, Acta Sci. Math. (Szeged) 58(1993), [12] J. Šlapal, On exponentiation of n-ary algebras, Acta Math. Hungar. 63(1994), [13] J. Šlapal, On exponentiation of universal hyperalgebras, Alg. Universalis 44(2000), [14] J. Šlapal, Exponentiality in categories of partial algebras, Mat. Pannon. 13(2002), Nitima CHAISANSUK, Program of Mathematics, Faculty of Science and Technology, Chiang Rai Rajabhat University, Chiang Rai, Thailand. nitimachaisansuk@gmail.com Josef ŠLAPAL, Department of Mathematics, Brno University of Technology, Technicka 2, Brno, Czech Republic. slapal@fme.vutbr.cz

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