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Basics of Clone Theory Draft BASICS OF CLONE THEORY DRAFT ERHARD AICHINGER Abstract. Some well known facts on clones are collected (cf. [PK79, Sze86, Maˇs10]). Contents 1. Definition of clones 1 2. Polymorphisms and invariant relations 2 3. The invariant relations encode the functions in a clone 4 4. The clones of the form Pol(R) 5 5. How the functions in a clone encode the invariant relations 6 6. Properties of the lattice of all clones 8 7. The definition of relational clones 10 8. The relational clones of the form Inv(F) 10 9. A Theorem on Groups 13 References 14 1. Definition of clones S An Let A be a nonempty set. Then O(A) := fA j n 2 Ng is the set of finitary operations on A. For C ⊆ O(A) and m 2 N, we let C[m] be the functions in (n) n C with arity m. For n 2 N and j 2 f1; : : : ; ng, the function πj : A ! A is Date: June 21, 2011. Course material for the course \Universal Algebra", JKU Linz, Summer term 2011. 1 2 ERHARD AICHINGER (n) defined by πj (x1; : : : ; xn) := xj for all x1; : : : ; xn 2 A. For n 2 N, a subset R of An is also called a n-ary relation on A, and for any set I, a subset of AI is also called a relation on A indexed by I. Let v 2 AI . Then we will denote v also by hv(i) j i 2 Ii. The expression hv(i) j i 2 Ii can also be seen as a shorthand [n] [m] for f(i; v(i)) j i 2 Ig. For m; n 2 N, f 2 O(A) , and g1; : : : ; gn 2 O(A) , m f(g1; : : : ; gn) denotes the function hf(g1(x); : : : ; gn(x)) j x 2 A i. Definition 1.1 (Clone). Let A be a set, J 6= ;, C ⊆ O(A). C is a clone on A if (n) (1) for all n; j 2 N with j ≤ n, we have πj 2 C; [n] [m] (2) for all n; m 2 N, for all f 2 C and for all g1; : : : ; gn 2 C , we have [m] f(g1; : : : ; gn) 2 C . Proposition 1.2. Let A be a set, and let all Cj (j 2 J) be clones on A. Then T fCj j j 2 Jg is a clone on A. Proof: It can be seen from Definition 1.1 that the properties carry over to arbi- trary intersections. 2. Polymorphisms and invariant relations Definition 2.1 (Preservation of a relation). Let A and I be nonempty sets, let n I f : A ! A, and let R ⊆ A . We say that f preserves R if for all v1; : : : ; vn 2 R, we have hf(v1(i); : : : ; vn(i)) j i 2 Ii 2 R: Then R is invariant under f, and we write f £ R. We also say that f is a polymorphism of the relational structure (A; R) and that f is compatible with R. Using the terminology of universal algebra, we see that an operation f preserves R ⊆ AI if and only if R is a subuniverse of hA; fiI . From a relation that is invariant under f, other invariant relations can be constructed in the following ways. Definition 2.2. Let A; I; J be nonempty sets, let R ⊆ AJ , and let σ be a function from I to J. Then R ∗ σ is a subset of AI defined by R ∗ σ := fv ◦ σ j v 2 Rg: Definition 2.3. Let A be a nonempty set, let I;J be sets, let S ⊆ AJ , and let I σ : J ! I. Then (S : σ)I is the subset of A defiuned by (S : σ)I := fg 2 AI j g ◦ σ 2 Sg: BASICS OF CLONE THEORY 3 Lemma 2.4. Let A; I; J be nonempty sets, let R ⊆ AJ , let σ : I ! J and τ : J ! I. Let f 2 O(A) be such that f £ R. Then f £ R ∗ σ and f £ (R : τ)I . Proof: Let n be the arity of f, and let w1; : : : ; wn 2 R ∗ σ. We have to show hf(w1(i); : : : ; wn(i)) j i 2 Ii 2 R ∗ σ: Let k 2 f1; : : : ; ng. Since wk 2 R ∗ σ, there is vk 2 R such that wk = vk ◦ σ. Now we have to show (2.1) hf(v1(σ(i)); : : : ; vn(σ(i))) j i 2 Ii 2 R ∗ σ: Let g := hf(v1(j); : : : ; vn(j)) j j 2 Ji. Since v1; : : : ; vn 2 R, f £ R implies that g 2 R. Therefore, g ◦ σ 2 R ∗ σ. We have g ◦ σ = hf(v1(σ(i)); : : : ; vn(σ(i))) j i 2 Ii: Thus, since g ◦ σ 2 R ∗ σ, (2.1) holds, which completes the proof of f £ R ∗ σ. For proving f £ (S : τ)I , we let g1; : : : ; gn 2 (S : τ)I . We have to show hf(g1(i); : : : ; gn(i)) j i 2 Ii 2 (S : τ)I : To this end, we show (2.2) hf(g1(i); : : : ; gn(i)) j i 2 Ii ◦ τ 2 S: We have hf(g1(i); : : : ; gn(i)) j i 2 Ii◦τ = hf(g1◦τ(j); : : : ; gn◦τ(j)) j j 2 Ji: Since g1◦τ 2 S; : : : ; gn◦τ 2 S, the fact that f £S implies hf(g1◦τ(j); : : : ; gn◦τ(j)) j j 2 Ji 2 S, which implies (2.2). If I is a finite set, a relation R ⊆ AI can therefore often be replaced with a relation R0 on Am, where m := jIj. For a nonempty set A, we let R(A) := S P(An) be the set of all finitary n2N relations on A that are indexed by an initial section of the natural numbers. We will write n for the set f1; : : : ; ng. As is usual, the set An is understood to be the same set as An. For R ⊆ R(A), we let R[n] := fR 2 R j R ⊆ Ang. We note that the R[n] need not be disjoint, since each of them might contain ;. Definition 2.5. Let m 2 N, let A be a nonempty set, and let F ⊆ O(A). We de- fine Inv[m](F) := fR ⊆ Am j 8f 2 F : f £ Rg; and Inv(F) := SfInv[m](F) j m 2 Ng. Definition 2.6. Let m 2 N, let A; I be nonempty sets, and let R ⊆ AI . Then [m] m Pol (fRg) := ff : A ! A j f £ Rg: If Ij (j 2 J with j 6= ;) are sets, Ij [m] Rj ⊆ A for j 2 J, and R := fRj j j 2 Jg, then we define Pol (R) := T [m] S [m] fPol (fRg) j R 2 Rg. Furthermore, Pol(R) := fPol (R) j m 2 Ng: 4 ERHARD AICHINGER Theorem 2.7. Let A be a set, let R; R1; R2 be sets of finitary relations on A, and let F; F1; F2 be sets of finitary operations on A. Then we have: (1) R1 ⊆ R2 ) Pol(R2) ⊆ Pol(R1). (2) F1 ⊆ F2 ) Inv(F2) ⊆ Inv(F1). (3) F ⊆ Pol(Inv(F)). (4) R ⊆ Inv(Pol(R)). (5) Pol(Inv(Pol(R))) = Pol(R). (6) Inv(Pol(Inv(F))) = Inv(F). Proof: (1) Let f 2 Pol(R2), and let R 2 R1. Then R 2 R2, and since f 2 Pol(R2), we have f £ R. (2) Let R 2 Inv(F2), and let f 2 F1. Then f 2 F2, and since R 2 Inv(F2), we have f £ R. (3) Let f 2 F. To prove that f 2 Pol(Inv(F)), we let R 2 Inv(F). Then since f 2 F, we have f £ R. Hence we have f 2 Pol(Inv(F)). (4) Let R 2 R. To prove that R 2 Inv(Pol(R)), we let f 2 Pol(R). Since R 2 R, we have f £ R. Hence we have R 2 Inv(Pol(R)). (5) By item (4), we have R ⊆ Inv(Pol(R)), and therefore by item (1) the inclusion Pol(Inv(Pol(R))) ⊆ Pol(R) holds. The other inclusion follows from (3) by setting F := Pol(R). (6) By item (3), we have F ⊆ Pol(Inv(F)), and therefore by item (2) the inclusion Inv(Pol(Inv(F))) ⊆ Inv(F) holds. The other inclusion follows from item (4) by setting R := Inv(F). 3. The invariant relations encode the functions in a clone Lemma 3.1. Let C be a clone on the set A, let m 2 N, let f 2 C[m], and let R be the subset of AAn defined by R := C[n]. Then f £ R. [n] [m] Proof: Let g1; : : : ; gm 2 R. Since g1; : : : ; gm 2 C and f 2 C , we have [n] f(g1; : : : ; gm) 2 C , and hence f(g1; : : : ; gm) 2 R. Now f(g1; : : : ; gm) = n hf(g1(i); : : : ; gm(i)) j i 2 A i. Therefore, the last expression lies in R, which completes the proof of f £ R. BASICS OF CLONE THEORY 5 Theorem 3.2. Let C be a clone on the set A, let n 2 N, and let f : An ! A, and let R be the subset of AAn defined by R := C[n]. Then the following are equivalent: (1) f 2 C; (2) f £ R; If A is finite and m := jAjn, then each of these properties is furthermore equivalent to (3) f 2 Pol(Inv[m](C)).
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