Categories of universal algebras in which direct products are tensor products

Annates Mathematicae Silesianae 9. Katowice 1995, 7-10 Prace Naukowe Uniwersytetu Śląskiego nr 1523

C A T E G O R I E S O F U N I V E R S A L A L G E B R A S I N W H I C H D I R E C T P R O D U C T S A R E T E N S O R P R O D U C T S

J O S E F Ś L A P A L

A b s t r a c t . In categories of conviniitative universal algebras'of given types we discover full subcategories in which direct products coincide with tensor products.

A concrete category K of structured sets and structure-compatible maps, i.e. a category K with a faithful (=forgetful) functor j | : K —• Set, will be called a, construct. Given two objects A and B of a construct K (we'write A, B € A'), by Hoin(A, B) we denote the set of all morphisms from A into B in A'. The usual symbols x and ® will be used for denotation of the carte­

sian product and the tensor product, respectively, in a construct. We shall need the following known result (see e.g. [1]): Let K be a semifinally com­

plete construct with a unit object. If for arbitrary objects A, B € K there exists a subobject [A,B] of the cartesian product such that \[A, B]\ = Hom(A, B), then for any object A € K the functor A ® — : K —• K is a left adjoint to the functor [.4, -] (and vice versa).

By a type we mean a family r = (n\; A 6 ft) where SI is a set and is a cardinal for each A 6 il. A universal algebra (briefly an algebra) of type r = (n\; A € ft) is a pair A = (X, (p\; A € ft)) where A' is a set - the so called underlying set of A - and p\ is an łi,\-ary operation on X, i.e. p\ : X^nx -* X, for each A G ft. An algebra (X,(p\; A G ft)) of type T = (n\', A € ft) is called

idempotknt if for any A € ft and any i C X from at,- = x for each i € n\

it follows that p\(xt; i 6 n\) = x, diagonal (c.f. [9]) if there holds

Px(Px{xij', 3 € n^A); i € n^x) = px(xu; i € n^A)

A M S (1992) subject classification: Primary 08C05. Secondary 18D15.

8

whenever A 6 fi and x,j G X for all i^fj G ra^A, commutative (see [7]) if the identity

P\(Pn(xij; j € n^M); t G n^A) = p^M(p^A(zij; t' € n^A); j G n^M) holds for all A,/z € fi and all X j j G A , t G n^A, j G n^M.

Given a type T , by Alg^T we denote the category of all idempotent, diagonal and commutative algebras of type r with homomorphisms as morphisms.

Clearly, for arbitrary type r, A l g^r is a semifinally complete construct where

\A\ is the underlying set of A for any algebra A G Alg^T. It is also clear that Alg^T is closed with respect to cartesian (i.e. direct) products of objects. The idempotency implies that every algebra A G Alg^T with card|/4| = 1 is a unit object of A l g^T. By [7], from the commutativity it follows that for any two algebras A, B G Alg^T there exists a subalgebra [A, B] of the direct product flW such that \[A,B]\ = Horn (A,B). Thus, A® - is a left adjoint to the functor [A, —] : Alg^T —• Alg^T for any type r and any algebra A G Alg^T (see also [2], [8]). Using this fact we prove

THEOREM. For any type j and any algebras A,B G Alg^T there holds ĄxB = A®B.

PROOF. Let r = ( »^A; A G fi) be an arbitrary type. According to the previous considerations it is sufficient to show that Ax - is a left adjoint to [A,—] for each A G Alg^T. In other words, we are to prove that for any two algebras A, B G Alg* there is a homomorphism / G Hom(£f, [A, A x B]) such that for any algebra C G Alg^T and any homomorphism g G H o m ( £ , [A, C]) there exists a Unique homomorphism g* G Hom(j4 x B, C) with the property that g(y) = g* o /(y)for each y G

On that account, let A = (A, (p^A; A G fi)), B = {Y, (c^A; A G ft)) and let / : Y -»• {X x Y)^x be the map given by /(y)(s) = (x,y). Denote AxB = (Xx7, (r^A; A G 12)), [A, AxB] = (Hom(A, AxB), (s^A; A G Q)), and let A G and yVG Y for each i G n^A. Then we have

/ ( ? A ( y n * € » A ) ) ( X ) =(a;,?A(y«; » € » A ) ) = (p^A(z; t G n^A), c^A{yi; i G n^A))

=rx((x,yi); i G n^A) = r^A(/(y;)(x); i G n^A)

=s\{f(ViY, * € n^A)(s) for any x G A'. Hence / G Hom(fl, [A, A x B],

Next^v let C = (Z, (t^A; A G fi)) G AIg^T be an algebra and let g G Hom(B,[A,C]) be a homomorphism. For any (x,y) € X xY put y*(x,y) = g(y)(x). Then we have defined a map g* : X x V -> Let (x,-, yj) G A' x Y

9

for each i G n^A and denote [A,C] = ( Hom(A,C), (u^A; A G Q)). There holds

</*(r^A((a;i,yi); i G n^A)) =5*(p^A(a;,-; * G n^A), g^A(y<; t G n^A))

=g(q\(yi\ * e n^A))(p^A(a;^t-; * G n^A))

=«A(ff(yt); * G n^A)(p^A(a;ⁱ; t G n^A))

= k * G n^A)); j' G n^A)

=t\(t\(g(yj){xi); i G n^A); j G n^A)

=*A(5(y«)(*«); » e n^A)

=*A(ff*(a:.-,yt); * € n^A) . We have shown that g* G Hom(A x B,C).

As the equality g(y) = g* o /(y) for any y G Y' and the uniqueness of g*

are obvious, the proof is complete. • REMARK, (a) A finitely productive construct K having the property that

for any its object A the functor Ax - : K -t K has a right adjoint is called cartesian closed - see [4]. Thus, we have proved that Alg^T is cartesian closed for any type r.

(b) Given a natural number n, by an n-ary algebra we understand a set with one n-ary operation on it. For n-ary algebras the commutativity (more often called the mediality) results from the diagonality. The diagonal and idempotent n-ary algebras are studied in [9]. The medial groupoids are fully dealt with in [6]. One can easily prove the following criterion for the diagonality of n-ary algebras: An n-ary algebra (X,p) is diagonal iff

p{xⁿ

=p(xⁿ,p(x²l,X22,... ,X²ⁿ),X33,.-. ,Xⁿⁿ) = . . .

= p ( x u , X²2 , . . . | S n - l , n - l i J > ( * n l i a f n 2 i ' " >^xn n ) )

holds for any Xij G X, i, j = 1,2,... , n. Consequently, a groupoid (X, •) is diagonal iff it is a semigroup with xyz = xz for each x, y, z G X. For example, any rectangular band (see [3]) is a diagonal (and idempotent) groupoid and hence an object of Alg^T for r = (2).

A criterion for the diagonality of algebras of arbitrary types is given in [10].

REFERENCES

[1] J . Adamek, Theory of Mathematical Structures, D. Reidel PubL Comp., Dordrecht - Boston - Lancaster, 1983.

[2] B . Banaschewski and E . Nelson, Tensor products and, bimorphisms, Canad. Math.

Bull. 1 9 (1976), 385-402.

10

[3] A . H . Clifford and G . B . Preston, The Algebraic Theory of Semigroups, Providence, Rhode Island, 1964.

[4] S. Eilenberg and G . M . Kelly, Closed categories, Proc. Conf. Cat. Alg. La Jolla, 1965, 421-562.

[5] G . Gratzer, Universal Algebra, Second Ed., Springer Verlag, New York - Heidelberg - Berlin, 1979.

[6] J . Jezek and T . Kepka, Medial groupoids, Rozpravy C S A V , Rada Mat. a Pfir. Ved.

93/1, Academia, Prague, 1983.

[7] L . Klukovits, On commutative universal algebras, Acta. Sci. Math. 34 (1973), 171- 174.

[8] F . E . Linton, Autonomous equational categories, J . Math. Mech. 15 (1966), 637-642.

[9] J . Płonka, Diagonal algebras, Fund. Math. 58 (1966), 309-321.

[10] J . Ślapal, A note on diagonal algebras, Math. Nachr. 158 (1992), 195-197.

D E P A R T E M E N T O F M A T H E M A T I C S T E C H N I C A L U N I V E R S I T Y O F B R N O 616 69 B R N O

C Z E C H R E P U B L I C