1. What is the definition of a clonoid?
A clonoid is a clonoid from set A to set B, defined by the existence of subalgebras R i , S i of A m i , B m i , respectively, for m i N and i in a set I, such that C = iI Pol(R i , S i ). In simpler terms, a clonoid is a relation between two sets that can be represented by a combination of subrelations.
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2. What is the proof for Lemma 2.13 in the context of a general ring R?
The proof for Lemma 2.13 in the context of a general ring R involves several steps. First, we assume that R is commutative and semisimple. By the Wedderburn-Artin Theorem, R is isomorphic to a direct sum of finite fields. Since A is distributive and semisimple, it is a direct sum of pairwise non-isomorphic simple R-modules. This implies that A is isomorphic to the regular R-module. Next, we let R be the subring of F_n x n, where F_n is a finite field of order n, and A be the R-module reduct of A. Since |A| = |R|, A is isomorphic to the regular R-module. Applying Lemma 1.10 for A over the commutative ring R, we obtain that for every M <= A k that embeds into A, there exists some T GL k (R) such that T M <= A x 0 k-1. This proves Lemma 2.13 in full generality for a general ring R. For the case where R is not commutative or semisimple, the proof involves considering the subring K_i of R, which is a field of order |F_i|n_i, and applying Lemma 1.10 for A over the semisimple ring R/J, where J is the nilpotent ideal. This leads to the existence of an invertible matrix T_R k x k such that T_R M <= A x 0 k-1. Since T_R is invertible over R, we can find an invertible matrix S_R k x k such that ST_R = I_k - U_J k x k, where U_J k x k is some nilpotent matrix. Since J is nilpotent, we can conclude that M V and a V are distinct, and Lemma 2.13 holds in this case as well. Overall, the proof for Lemma 2.13 in the context of a general ring R involves considering different cases based on the properties of R and applying relevant lemmas and theorems to establish the desired result.
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3. What are uniformly generated functions?
Uniformly generated functions are k-tuples of l-ary term functions on an algebra A that factor through n-ary term functions. They are defined for an R-module A and involve composing n-ary A, B-minors with term functions of A and B. These functions are uniformly generated by n-ary A, B-minors for k-ary functions f U. For example, for a binary function f F (A, B) (2), the minor f ' (x 1 ) is uniformly generated by the unary A, B-minors of f. This concept is used in Section 3 to study properties of uniformly generated functions between modules.
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4. Is Lemma 2.14 related to uniformly generated n-ary A, B-minors?
Yes, Lemma 2.14 discusses the relationship between uniformly generated n-ary A, B-minors. It states that if a function f' is uniformly generated by n-ary A, B-minors of f, then there exists a set S with finite support such that for all f and x, f'(x) can be expressed as a combination of rR kxl, where rk(r) is less than or equal to n. This lemma also establishes that if f' is uniformly generated by n-ary A, B-minors and f - f' is uniformly generated by its n-ary A, B-minors, then every f is uniformly generated by its n-ary A, B-minors. The proof of Lemma 2.14 is straightforward from the definition and involves the use of sets S and T to demonstrate the uniform generation of f and f' by their respective n-ary A, B-minors.
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