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HYPERCLONE LATTICE and EMBEDDINGS 1. Introduction

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HYPERCLONE LATTICE and EMBEDDINGS 1. Introduction Novi Sad J. Math. Vol. 36, No. 1, 2006, 87-95 HYPERCLONE LATTICE AND EMBEDDINGS Jovanka Pantovi¶c1, Biljana Rodi¶c2, Gradimir Vojvodi¶c3 Abstract. In this paper, hyperclone lattice is studied via three kinds of embeddings. One is from the clone lattice on A to the hyperclone lattice on A; the second is from the hyperclone lattice on A to the clone lattice on P (A) n f;g and the third one is from the hyperclone lattice on A0 to the hyperclone lattice on A; for A0 ½ A and the ¯nite set A: The second map has usually been used for the description of hyperclone lattice, but one can see from this paper that hyperclone lattice on A is in some way thin in the clone lattice on P (A) n f;g: However, we show that the ¯rst and the third embeddings are full order embeddings and they are used to lift several properties. AMS Mathematics Subject Classi¯cation (2000): 08A40 Key words and phrases: Clone, hyperoperation, embedding 1. Introduction The basic de¯nitions and claims used in Introduction are from [5], [8], [13] and [14]. Many questions which arise from the process of operations extending are studied there. We will follow that work and give an extension of it. Clones of hyperoperations are also studied in [7], [12], [15], etc. Let A be a nonempty ¯nite set and P (A) the power set of A: For a positive integer n; an n-ary hyperoperation on A is a function f : An ! P (A) n f;g: We will denote the set of all operations on A by OA; and the set of all hyperopera- tions by HA: An operation f # on P (A) n f;g is extended from the hyperoperation f n # on A if for all (X1;X2;:::;Xn) 2 (P (A) n f;g) holds f (X1;:::;Xn) = [ff(x1; : : : ; xn)j xi 2 Xi; 1 · i · ng: For an arbitrary set F of hyperopera- tions, let F # = ff #jf 2 F g: For a positive integer n; an i-th projection on A of arity n; 1 · i · n; n n is an n¡ary operation ¼i : A ! A; (x1; : : : ; xn) 7! xi: For positive in- n (n) (m) n (m) tegers n and m, we de¯ne the composition Sm : OA £ (OA ) ! OA ; m (f; g1; : : : ; gn) 7! f(g1; : : : ; gn); where f(g1; : : : ; gn): A ! A; (x1; : : : ; xm) 7! f(g1(x1; : : : ; xm); : : : ; gn(x1; : : : ; xm)): 1Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovi¶ca6, 21000 Novi Sad, Serbia, e-mail: [email protected] 2Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovi¶ca6, 21000 Novi Sad, Serbia, e-mail: [email protected] 3Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovi¶ca,21000 Novi Sad, Serbia, e-mail: [email protected] 88 J. Pantovi¶c,B. Rodi¶c,G. Vojvodi¶c (n) (m) (n) (n) (n¡1) Let f 2 OA and g 2 OA ; then &f 2 OA ; ¿f 2 OA ; ¢f 2 OA ; (m+n¡1) (n+1) f ± g 2 OA and rf 2 OA are de¯ned by (&f)(x1; x2; : : : ; xn) = f(x2; : : : ; xn; x1); n ¸ 2 (¿f)(x1; x2; x3; : : : ; xn) = f(x2; x1; x3; : : : ; xn); n ¸ 2 (¢f)(x1; : : : ; xn¡1) = f(x1; x1; : : : ; xn¡1); n ¸ 2; (&f) = (¿f) = (¢f) = f; n = 1; (f±g)(x1; : : : ; xm; xm+1; : : : ; xm+n¡1) = f(g(x1; : : : ; xm); xm+1; : : : ; xm+n¡1) (rf)(x1; x2; : : : ; xn+1) = f(x2; : : : ; xn+1); where x1; : : : ; xm+n¡1 2 A: The full algebra of operations is OA = (OA; ±; &; ¿; 2 ¢; ¼1): Each subuniverse of OA is a clone. A set of operations is a clone i® it contains all projections and is closed with respect to composition. Let ½ ⊆ Ah be a h-ary relation and f an n-ary operation on A: We say that f preserves ½ if for all h-tuples (a11; : : : ; a1h);:::; (an1; : : : ; anh) from ½ we have (f(a11; : : : ; an1); : : : ; f(a1h; : : : ; anh)) 2 ½: P olA½ is the set of all operations on A which preserve ½. For a positive integer n; an i-th hyperprojection on A of arity n; 1 · i · n; n n is an n¡ary hyperoperation ei : A ! A; (x1; : : : ; xn) 7! fxig: For posi- n (n) (m) n tive integers n and m, we de¯ne the composition Sm : HA £ (HA ) ! (m) m HA ; (f; g1; : : : ; gn) 7! f(g1; : : : ; gn); where f(g1; : : : ; gn): A ! P (A) n f;g; (x1; : : : ; xm) 7! [ff(y1; : : : ; yn): yi 2 gi(x1; : : : ; xm); 1; · i · ng: Let (n) (m) (n) (n) (n¡1) f 2 HpA and g 2 HpA ; then &f 2 HpA ; ¿f 2 HpA ; ¢f 2 HpA ; (m+n¡1) (n+1) f ± g 2 HpA and rf 2 HpA are de¯ned by (&f)(x1; x2; : : : ; xn) = f(x2; : : : ; xn; x1); n ¸ 2 (¿f)(x1; x2; x3; : : : ; xn) = f(x2; x1; x3; : : : ; xn); n ¸ 2 (¢f)(x1; : : : ; xn¡1) = f(x1; x1; : : : ; xn¡1); n ¸ 2; (&f) = (¿f) = (¢f) = f; n = 1; (f ± g)(x1; : : : ; xm; xm+1; : : : ; xm+n¡1) = [ff(y; xm+1; : : : ; xm+n¡1): y 2 g(x1; : : : ; xm)g (rf)(x1; x2; : : : ; xn+1) = f(x2; : : : ; xn+1); where x1; : : : ; xm+n¡1 2 A: The full algebra of hyperoperations is HpA = (HpA; 2 ±; &; ¿; ¢; e1): Each subuniverse of HpA is a partial hyperclone. A set of partial hyperoperations is a partial hyperclone i® it contains all hyperprojections and is closed with respect to composition. If C is a hyperclone, the set of all n-ary hyperoperations from C will be (n) denoted by C : Let us denote the set of all clones on A by LA and the set of all hyperclones by LHA : Each of these sets forms a complete algebraic lattice. The atoms (dual atoms) are called minimal (maximal) elements. The least element in both lattices, trivial clone, will be denoted by JA: For a set F of hyperoperations, the least hyperclone containing F will be denoted by hF i; and the least clone containing the set F of operations on A will be denoted by hF iA: A hyperclone is minimal if it is not trivial and its only subclone is triv- ial. A hyperoperation of minimal arity in a minimal hyperclone, that is not a projection, is called minimal hyperoperation. Hyperclone lattice and embeddings 89 (3) A ternary majority hyperoperation on A; ma 2 HA ; is a ternary hyperop- eration on A de¯ned by ma(x; x; y) = ma(x; y; x) = ma(y; x; x) = fxg for all x; y 2 A: (3) A ternary minority hyperoperation on A; mi 2 HA ; is a ternary hyperop- eration on A de¯ned by mi(x; x; y) = mi(x; y; x) = mi(y; x; x) = fyg for all x; y 2 A: For n > 2 and 1 · i · n; every n-ary hyperoperation s with s(x1; : : : ; xn) = fxig; jfx1; : : : ; xngj < n is called semi-hyperprojection. It is easy to show that the theorem analogous to Rosenberg's classi¯cation theorem ([15]) holds for minimal hyperoperations. Theorem 1. Every minimal hyperoperation is one of the following types: (1) a unary hyperoperation, (2) a binary idempotent hyperoperation, (3) a ternary majority hyperoperation, (4) a ternary minority hyperoperation, (5) an n-ary semi-hyperprojection, n > 2: 2. From the clone lattice to the hyperclone lattice S (n) 0 Let us de¯ne a map ¸ : LA ! LHA by ¸(C) = ff 2 HA : 9f 2 n¸1 n 0 C8(x1; : : : ; xn) 2 A f(x1; : : : ; xn) = ff (x1; : : : ; xn)gg: It is easy to show that ¸(C) is a hyperclone. Lemma 1. The map ¸ is full-order embedding. Proof. Obviously, ¸ is 1-1 map, and holds C1 · C2 , ¸(C1) · ¸(C2): Let H be an arbitrary hyperclone with the property ¸(JA) ⊆ H ⊆ ¸(OA): From the de¯nition of ¸ immediately follows that there is clone C such that ¸(C) = H: 2 Without loss of generality, we will sometimes identify the hyperclone ¸(C) and the clone C; in order to simplify the presentation. Lemma 2. Let A be a ¯nite set with jAj ¸ 2: Every hyperclone generated by constant hyperoperation on A is minimal. Proof. The proof is trivial. 2 Corollary 1. Let A be a ¯nite set with jAj ¸ 2: There are least 2jAj ¡ 1 minimal clones in the lattice LHA : 90 J. Pantovi¶c,B. Rodi¶c,G. Vojvodi¶c Theorem 2. On any ¯nite set A; with jAj ¸ 2; there are three minimal hy- perclones such that their join contains all hyperoperations. Proof. It is proved in [4] that there are two minimal clones such that their join contains all operations on any ¯nite set A: Romov proved in [12] that OA is maximal hyperclone. Hence, it is enough to choose the third minimal hyperclone from the set of minimal hyperclones that are not minimal in the clone lattice on A: From previous lemma follows that such a set is not empty. 2 (1) Theorem 3. The interval [¸(hOA i); ¸(OA)] is a chain. Proof. It is the chain obtained by Burle in 1967 [11]. He has shown that the (1) interval [hOA i;OA] is (jAj + 1)-element chain (1) hOA i = U1 ½ L ½ U2 ½ ::: ½ Uk = OA: Ui is the set of all operations depending on at most one variable and operations taking at most j values and L is the set of operations depending of one variable and operations f(x1; : : : ; xn) = ¸(Ã1(xi1 ) + :::Ãt(xit )); where ¸ : f0; 1g ! A and Ãj : A ! f0; 1g; j 2 fi1; : : : ; itg; 1 · i1 < : : : < it · n; are arbitrary maps and + is addition modulo 2: 2 Corollary 2.
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