
ON MANY-SORTED ALGEBRAIC CLOSURE OPERATORS J. CLIMENT VIDAL AND J. SOLIVERES TUR Abstract. A theorem of Birkho®-Frink asserts that every algebraic closure operator on an ordinary set arises, from some algebraic structure on the set, as the operator that constructs the subalgebra generated by a subset. However, for many-sorted sets, i.e., indexed families of sets, such a theorem is not longer true without quali¯cation. We characterize the corresponding many-sorted closure operators as precisely the uniform algebraic operators. Some theorems of ordinary universal algebra can not be automatically general- ized to many-sorted universal algebra, e.g., Matthiessen [5] proves that there exist many-sorted algebraic closure systems that can not be concretely represented as the set of subalgebras of a many-sorted algebra. As is well known, according to a representation theorem of Birkho® and Frink [1], this is not so for the single-sorted algebraic closure systems. In [2] it was obtained a concrete representation for the so-called many-sorted uniform 2-algebraic closure operators. However, as will be proved below, con¯rming a conjecture by A. Blass in his review of [2], the main result in [2] remains true if we delete from the above class of many-sorted operators the condition of 2-algebraicity. Therefore a many-sorted algebraic closure operator will be concretely representable as the set of subalgebras of a many-sorted algebra i® it is uniform. We point out that the proof we o®er follows substantially that in GrÄatzer[4] for the single-sorted case, but di®ers from it, among others things, by the use we have to make, on the one hand, of the concept of uniformity, missing in the single-sorted case, and, on the other hand, of the Axiom of Choice, because of the lack, in the many-sorted case, of a canonical choice in the de¯nition of the many-sorted operations. In what follows we use, for a set of sorts S and an S-sorted signature §, the concept of many-sorted §-algebra and subalgebra in the standard meaning, see e.g., [3]. To begin with, as for ordinary algebras, also the set of subalgebras of a many- sorted algebra is an algebraic closure system. Proposition 1. Let A be a many-sorted §-algebra. Then the set of all subalgebras of A, denoted by Sub(A), is an algebraic closure system on A, i.e., we have (1) A 2 Sub(A). i T i (2) If I is not empty and (X )i2I is a family in Sub(A), then i2I X is also in Sub(A). i (3) If I is not empty and (X )i2I is an upwards directed family in Sub(A), then S i i2I X is also in Sub(A). However, as we will prove later on, in the many-sorted case the many-sorted algebraic closure operator canonically associated to the algebraic closure system of the subalgebras of a many-sorted algebra has an additional and characteristic property, that of being uniform. Date: November 22, 2004. 1991 Mathematics Subject Classi¯cation. Primary: 08A30, 06A15; Secondary: 08A68. Key words and phrases. Many-sorted algebra, support, many-sorted algebraic closure operator, many-sorted uniform algebraic closure operator. 1 2 JUAN CLIMENT AND JUAN SOLIVERES Now we recall the concept of support of a sorted set and that of many-sorted algebraic closure operator on a sorted set, essentials to de¯ne that of many-sorted algebraic uniform closure operator. De¯nition 1. Let A be an S-sorted set. Then the support of A, denoted by supp(A), is the subset f s 2 S j As 6= ? g of S. De¯nition 2. Let A be an S-sorted set. A many-sorted algebraic closure operator on A is an operator J on Sub(A), the set of all S-sorted subsets of A, such that, for every X; Y ⊆ A, satis¯es: (1) X ⊆ J(X), i.e., J is extensive. (2) If X ⊆ Y , then J(X) ⊆ J(Y ), i.e., J is isotone. (3) J(J(X))S = J(X), i.e., J is idempotent. (4) J(X) = J(F ), i.e., J is algebraic, where a part F of X is in F 2Subf(X) Subf(X), the set of ¯nite S-sorted subsets of X, i® supp(F ) is ¯nite and, for every s 2 supp(F ), Fs is ¯nite. A many-sorted algebraic closure operator J on A is uniform i®, for X; Y ⊆ A, from supp(X) = supp(Y ), follows that supp(J(X)) = supp(J(Y )). De¯nition 3. Let A be a many-sorted §-algebra. We denote by SgA the many- sorted algebraic closure operator on A canonically associated to the algebraic closure system Sub(A). If X ⊆ A, SgA(X) is the subalgebra of A generated by X. Next, as for ordinary algebras, we de¯ne for a many-sorted §-algebra A an operator on Sub(A) that will allow us to obtain, for every subset of A, by recursion, an N-ascending chain of subsets of A from which, taking the union, we will obtain an equivalent, but more constructive, description of the subalgebra of A generated by a subset of A. Moreover, we will make use of this alternative description to prove the uniformity of the operator SgA and also in the proof of the representation theorem. De¯nition 4. Let A = (A; F ) be a many-sorted §-algebra. (1) We denote by E the operator on Sub(A) that assigns to an S-sorted subset A ¡ S ¢ X of A,E (X) = X [ F [X ] , where, for s 2 S,§ is A σ2§¢;s σ ar(σ) s2S ¢;s the set of all many-sorted formal operations σ such that the coarity of σ is ? Q s and for ar(σ) = (sj)j2m 2 S , the arity of σ, Xar(σ) = j2m Xsj . n 0 (2) If X ⊆ A, then the family (EA(X))n2N in Sub(A) is such that EA(X) = X n+1 n and EA (X) = EA(EA(X)), for n ¸ 0. ! (3) We denote by EA the operator on Sub(A) that assigns to an S-sorted subset ! S n X of A,EA(X) = n2N EA(X) Proposition 2. Let A be a many-sorted §-algebra and X ⊆ A. Then we have that ! SgA(X) = EA(X). Proof. See [2] ¤ Proposition 3. Let A be a many-sorted §-algebra and X; Y ⊆ A. Then we have that n n (1) If supp(X) = supp(Y ), then, for every n 2 N, supp(EA(X)) = supp(EA(Y )). S n (2) supp(SgA(X)) = n2N supp(EA(X)). (3) If supp(X) = supp(Y ), then supp(SgA(X)) = supp(SgA(Y )). Therefore the many-sorted algebraic closure operator SgA is uniform. Proof. See [2] ¤ Finally we prove the representation theorem for the many-sorted uniform alge- braic closure operators, i.e., we prove that for an S-sorted set A a many-sorted MANY-SORTED BIRKHOFF-FRINK 3 algebraic closure operator J on Sub(A) has the form SgA, for some S-sorted signa- ture § and some many-sorted §-algebra A if J is uniform. Theorem 1. Let J be a many-sorted algebraic closure operator on an S-sorted set A. If J is uniform, then J = SgA for some S-sorted signature § and some many-sorted §-algebra A. Proof. Let § = (§w;s)(w;s)2S?£S be the S-sorted signature de¯ned, for every (w; s) 2 S? £ S, as follows: S §w;s = f (X; b) 2 X2Sub(A)(fXg £ J(X)s) j 8t 2 S (card(Xt) = jwjt) g; where for a sort s 2 S and a word w : jwj / S on S, with jwj the lenght of w, the number of occurrences of s in w, denoted by jwjs, is card(fi 2 jwj j w(i) = sg). ? S We remark that for (w; s) 2 S £ S and (X; b) 2 X2Sub(A)(fXg £ J(X)s) the following conditions are equivalent: (1) (X; b) 2 §w;s, i.e., for every t 2 S, card(Xt) = jwjt. (2) supp(X) = Im(w) and, for every t 2 supp(X), card(X ) = jwj . S t t On the other hand, for the index set ¤ = Y 2Sub(A)(fY g £ supp(Y )) and the ¤-indexed family (Ys)(Y;s)2¤ whose (Y; s)-th coordinate is Ys, precisely the s-th coordinate of the S-sorted set Y of the index (Y; s) 2 ¤, let f be a choice function Q ? for (Ys)(Y;s)2¤, i.e., an element of (Y;s)2¤ Ys. Moreover, for every w 2 S and Q w;a w;a a 2 i2jwj Aw(i), let M = (Ms )s2S be the ¯nite S-sorted subset of A de¯ned w;a ¡1 as Ms = fai j i 2 w [s]g, for every s 2 S. ? Now, for (w; s) 2 S £S and (X; b) 2 §w;s, let FX;b be the many-sorted operation Q Q w;a from i2jwj Aw(i) into As that to an a 2 i2jwj Aw(i) assigns b, if M = X and f(J(M w;a); s), otherwise. We will prove that the many-sorted §-algebra A = (A; F ) is such that J = SgA. But before that it is necessary to verify that the de¯nition of the many- ? sorted operations is sound, i.e., that for every (w; s) 2 S £ S,(X; b) 2 §w;s Q w;a and a 2 i2jwj Aw(i), s 2 supp(J(M )) and for this it is enough to prove that supp(M w;a) = supp(X), because, by hypothesis, J is uniform and, by de¯nition, b 2 J(X)s.
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