The external associativity condition gives the commutative diagramRemark 1.4. Notice that we have natural transformations η A,0,0,B : A✷ 1 B → A✷ 2 B and η 0,A,B,0 : A✷ 1 B → B✷ 2 A.If we had insisted a 2-fold monoidal category be a monoid in the category of monoidal categories and strictly monoidal functors, this would amount to requiring that η = id. In view of the above, this would imply A✷ 1 B = A✷ 2 B = B✷ 1 A and similarly for morphisms. Thus the nerve of such a category would be a commutative topological monoid and its group completion would be equivalent to a product of abelian Eilenberg-MacLane spaces.Remark 1.5. Recall that a braided monoidal category (also known as braided tensor category) is a category C together with a functor ✷ 1 : C × C → C which is strictly associative, has a strict 2-sided unit object 0 and with a natural commutativity isomorphism c A,B : A✷ 1 B −→ B✷ 1 A satisfying the following properties:Assuming that ✷ 1 = ✷ 2 and that the natural isomorphism η A,B,C,D satisfies η A,B,0,C = η A,0,B,C = id A✷1B✷1C , one proceeds as follows to show that we have a braided monoidal category. In the internal associativity diagram take V = W = 0 and obtain that η U,X,Y,Z = id U ✷ 1 η 0,X,Y,Z . Then take X = Y = 0 and obtain that η U,V,W,Z = η U,V,W,0 ✷ 1 id Z . Combining these two facts, one obtains thatwhere c B,C = η 0,B,C,0 . Then take U = Z = W = 0 in the internal associativity law to get the first associativity law for c, and take U = Z = X = 0 to get the other one. With the additional conditions we have here the external associativity law is superfluous. Conversely given a braided monoidal category, we can define a 2-fold monoidal structure by ( * ).Remark 1.6. Joyal and Street [10] considered a very similar concept to our notion of 2-fold monoidal category. They loosened our requirement that the two operations ✷ 1 and ✷ 2 be strictly associative with a strict unit by only requiring these conditions to hold up to coherent natural isomorphisms. More significantly they required the natural transformation η A,B,C,D to be an isomorphism. They then showed that such a category is naturally equivalent to a braided monoidal category. Briefly given such a category one obtains an equivalent braided monoidal category by discarding one of the two operations, say ✷ 2 , and defining the commutativity isomorphism for the remaining operation ✷ 1 to be the compositeOur requirement that the operations be strictly associative and unital are not significant restrictions and were adopted for convenience and simplicity. One can always replace categories with operations which are associative and unital up to coherent natural isomorphisms by equivalent categories with strictly associative and unital operations.

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The aim of this note is the following theorem. THEOREM 1. Let K be an algebraically closed field, and L = Hom K (K™, K"). Suppose that 1 ^ t ^ min (m, ri). Then (a) the subvariety V = {(f>eL\ r k $ < /} can be defined set-theoretically by mn -t 2 + \ equations, but not by fewer equations; (b) the same holds for the corresponding projective variety P(F) <= P(L).In order to give the most general version of Theorem 1, we introduce the following notation: B is a commutative ring, l a n m x / i matrix of indeterminates, and I t (X) the ideal generated by the /-minors of Z i n the polynomial ring B [X]. (We use Bruns and Vetter [3] as a reference for the theory of determinantal ideals.) For an ideal J in a commutative ring R, we call ara / = min {k: there exist f x ,... ,f k e R such that Rad / = Rad (/ 1 5 ... ,f k )} the arithmetical rank of / .Then Theorem 1 clearly is a consequence of the following. THEOREM 2. With the notation just introduced, for all t such that 1 ^ t < min (m, n). Furthermore, the mn -t 2 +1 elements generating \ t {X) up to radical can be chosen homogeneous.The inequality < in Theorem 2 is a result of the first author; it will be used in proving the converse. Therefore we restate [3, (5.21)]. It can be seen easily from its proof that the mn -t 2 +1 elements generating I^A') up to radical may be chosen homogeneous; furthermore, this has been noted explicitly in Bruns [2, (2.1)]. LEMMA 1. Let B be a commutative ring. Then araI t (-^) ^mn -t 2 +\, and there are mn -t 2 +\ homogeneous elements generating l t (X) up to radical for all t such that 1 < t ^ min (m, n).The precise value of ara l t {X) has been found only in the trivial case t = 1 and the cases (i) t = min (m, n) and (ii) t = 2 in characteristic zero. Case (i) was proved by Hochster (compare the remark following the corollary below). The inequality ^ in (ii)

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