A monoidal interval of clones of selfdual functions

Transcription

1 Journal of Automata, Languages and Combinatorics u v w, x y c Otto-von-Guericke-Universität Magdeburg A monoidal interval of clones of selfdual functions Andrei Krokhin Department of Computer Science, University of Durham Science Laboratories, South Road, Durham DH1 3LE, UK andrei.krokhin@durham.ac.uk and Ivo G. Rosenberg Département de mathématiques et de statistique, Université de Montréal C.P succ. Centre-ville, Montréal QC, H3C 3J7, Canada rosenb@dms.umontreal.ca ABSTRACT Let A be a 2p-element set, p prime, and let π be a fixed point-free permutation on A of order p. We study the interval of clones C on A such that C consists of functions that are selfdual with respect to π and C contains all unary functions with this property. Keywords: clone, monoidal interval, selfdual functions 1. Introduction Clones are composition-closed families of finitary functions on a set. They are actively used in classifying finite algebras see, e.g., [16], since algebras with the same clone of term operations have very much in common. A well-known Galois correspondence see, e.g. [9] links clones with relational structures, and this link has been recently used to classify the complexity of combinatorial problems known as constraint satisfaction problems see [3]. Thus the study of clones is motivated, in particular, by applications in universal algebra and in computer science. The set of all clones on a fixed set can be conveniently partitioned into classes where two clones belong to the same class called monoidal interval - see the definition below if they contain exactly the same unary functions. Again, the study of monoidal intervals originated in universal algebra see [16], and recently some known results on monoidal intervals were applied to investigate the complexity of certain constraint satisfaction problems related to hypergraph colouring [2]. In this paper, we study algebraic properties of certain monoidal intervals of clones of selfdual functions. Let A be a finite set. For a positive integer n, let O n A denote the system of all n-ary functions or operations on A, and let O A = n=1 On A. A set C O A is called a clone on A if

2 2 A. Krokhin, I.G. Rosenberg 1. C contains all projections, i.e. functions of the form e n i x 1,..., x n = x i, and 2. C is closed under permutation and identification or fusion of variables, and 3. C is closed under superposition or composition, i.e. for any n-ary f C and m-ary g C, the function f g belongs to C where f g is defined by f gx 1,..., x n+m 1 = fgx 1,..., x m, x m+1,..., x n+m 1. The clones on A, ordered by inclusion, form a complete lattice, denoted L A. For a two-element set A, a complete description of L A was given in [10]; in particular, L A is countable. If A 3 then L A = 2 ℵ0 [17] and a detailed description of the lattice is believed to be impossible. Let us recall some definitions and notation. An m-ary relation on A is simply a subset of A m. The elements of a relation will usually be written in columns. A function f O n A is said to preserve an m-ary relation ρ on A if, for each m n matrix X with all columns from ρ, the column fx, which is calculated coordinatewise from the columns of X, belongs to ρ. The system of all functions preserving a relation ρ forms a clone, which is denoted by P olρ. If Q is a set of relations on A set P olq = ρ Q P olρ. For a subset F of O A, set F n = F O n A and write P ol nθ instead of P olθ n. The intersection of an arbitrary set of clones on A is a clone, and hence there exists the least clone containing F. Call it the clone generated by F and denote it by F. For a clone C, the set C 1 is a submonoid of the full transformation monoid T A = O 1 A ;. It is well known see e.g.[16] that, for an arbitrary submonoid M of T A, the set {C L A C 1 = M} is an interval of L A. It is denoted by IntM and called monoidal. Monoidal intervals form a partition of L A. There has been a considerable interest in the study of monoidal intervals, in particular, of those containing clones of independent interest. One example of such clones are maximal clones, that is, coatoms of L A. A complete description of maximal clones on any finite set was obtained in [11]. Let us recall some definitions. The graph of a permutation π on A is defined as the binary relation π = {x, πx x A}. A 4-ary relation ρ is called affine if ρ = {x, y, z, x y + z x, y, z A} where A, + is an Abelian group. An h-ary relation ρ, 1 h A 1, is called central if ρ A h and ρ is totally reflexive i.e., it contains all h-tuples with repeated components, totally symmetric i.e., it is invariant under all permutations of components, and the center {a A {a} A h 1 ρ} of ρ is non-empty. For h 3, a family T = {ϑ 1,..., ϑ m } of equivalence relations on A is called h-regular if each ϑ i has exactly h equivalence classes and {B i 1 i m} is non-empty for arbitrary equivalence classes B i of ϑ i, 1 i m. The h-ary relation λ T on A determined by T consists of all h-tuples whose set of components meets at most h 1 classes of each ϑ i 1 i m. Expressed differently, x 1,..., x h λ T if and only if, for some 1 i m, the set {x 1,..., x h } meets every every equivalence class of ϑ i. Relations of the form λ T are called regular. Theorem 1 [11] Let A be finite set, A 2. The maximal clones on A are the clones P olρ where ρ is a relation of one of the following types: O a bounded partial order,

3 A monoidal interval of clones of selfdual functions 3 E a nontrivial equivalence relation, P the graph of a fixed point free permutation of prime order, A an affine relation determined by an elementary Abelian p-group p-prime, C a central relation, R a regular relation. It was proved in [4, 5] that if ρ is a relation of type E or C, respectively, then the monoidal interval IntP ol 1 ρ has 2 ℵ 0 elements. Monoidal intervals containing a maximal clone of type O were studied in [6, 7] where it was proved that the interval has 2 ℵ 0 elements if the order is not total, and otherwise, at least for A < 5, the interval is finite. Monoidal intervals IntP ol 1 ρ where ρ is either of type P, for A prime, or of type A were described in [15, 14], respectively; these intervals are finite chains. If ρ is the A -ary regular relation λ T determined by the system T consisting of the single equality relation i.e., P olρ is the Slupecki clone then, as proved in [1], the corresponding monoidal interval is an A + 1-element chain. If ρ is the graph of a permutation π then functions from P olρ are called selfdual or autodual, in another terminology with respect to π. Throughout the paper and denote addition and subtraction modulo p, respectively. Let A = Z p, p prime, and let σx = x 1. For M = P ol 1 σ, the interval IntM is the 3-element chain M M, x y z P olπ [15] where M, x y z is the clone generated by M {f} with fx, y, z = x y z. We shall see that in the case A = 2p the structure of the corresponding interval is much less transparent. In this paper, we study monoidal intervals containing a maximal clone P olρ where ρ is of type P and A = 2p, p prime. It will be convenient to represent A as A = Z 2 Z p. From now on, let π be the permutation on A defined by πa, b = a, b 1 and let M = P ol 1 π. Then, as follows from [8], we have IntM = [ M ; P olπ ]. Recall that a variable x i of a function fx 1,..., x n is said to be fictitious if for all a 1,..., a i 1, a i+1,..., a n A, the function fa 1,..., a i 1, x, a i+1,..., a n is constant. Otherwise x i is called essential and f is said to depend essentially on x i. A function is said to be essentially unary if depends essentially on at most one variable. The structure of the paper is as follows. In Subsection 2.1 we prove that every clone in IntM distinct from P olπ is contained in P ol{π, θ}, where θ is the equivalence relation on A whose classes are the sets {i} Z p i = 0, 1, except in the case p = 2 when the interval also contains some clones of quasi-linear functions. In Subsection 2.2 we show that the structure of the interval [ M ; P ol{π, θ}] is largely determined by the structure of two of its subintervals consisting of clones of functions of restricted types. Finally, in Subsection 2.3 we completely describe, for p = 2, the clones of quasi-linear functions contained in IntM.

4 4 A. Krokhin, I.G. Rosenberg 2. Results 2.1. Maximal subclones of P olπ The maximal subclones of P olπ are described in [12]. It follows from this description that they are of the form P ol{π, ρ} where ρ is a relation of one of the following types: 1. a proper unary relation; 2. a binary relation which is either the graph of a permutation distinct from π s, s 1, or a nontrivial equivalence relation; 3. an affine relation determined by an elementary Abelian p-group A, + such that there exists an element c A with πx = x + c for all x A; 4. a central relation; 5. a regular relation determined by a family {ϑ 1,..., ϑ m } of equivalence relations such that θ ϑ 1 ϑ m. Recall that a coatom of a lattice L with a greatest element 1 is an element covered by 1. Proposition 1 1 If p > 2 then P ol{π, θ} is the only coatom of IntM. 2 If p = 2 then P ol{π, θ} and P ol{π, ρ} are the only coatoms of IntM where ρ is the affine relation determined by the group Z 2 Z 2. Proof. It is obvious that M preserves no nontrivial subset of A. It follows from Lemma 3.2 in [12] and Theorem in [9] that every binary relation, which is invariant under M, is either A 2 or a union of some graphs of permutations of the form π s, s 0. It is not hard to check that, among the binary relations described in item 2 in the list above, only θ is of such a form. Namely, θ is the union of all such graphs of permutations. It is well known see, e.g., [16] Prop.2.6 that if ρ is an affine relation determined by an Abelian group A, + then the clone P olρ consists of all functions of the form ri x i + u where u A and the r i s are endomorphisms of the group. Since the group must be an elementary Abelian p-group, we need to consider the clone P ol{π, ρ} only if p = 2, that is, when A = 4. Then the group is isomorphic to Z 2 2. Then ρ does not depend on the choice of the group. To see this, note that if any two of the elements x, y, z coincide then x y + z is the remaining element; otherwise that is, if x, y, z are all distinct x y +z is the unique element in A\{x, y, z}. Thus, assume that ρ is determined by Z 2 2. To show that P ol{π, ρ} is a coatom in IntM, it is enough to prove that M P ol 1 ρ. Let f M be given by f0, 0 = a, b and f1, 0 = c, d. Then fx, y = rx, y + a, b where f and r are given by the following table x, y 0, 0 0, 1 1,, 1 f a, b a, b 1 c, d c, d 1 r 0, 0 0, 1 a c, b d a c, b d 1

5 A monoidal interval of clones of selfdual functions 5 It is easy to verify that r is an endomorphism of Z 2 2. Thus, M P ol 1 ρ. Now let δ be an h-ary central relation and let a belong to the center of δ. For an arbitrary b = b 1,..., b h A n \ δ, we can choose a permutation f M with fa = b 1. We conclude that the relation δ is not invariant under M, since c = a, f 1 b 2,..., f 1 b h δ and fc = b δ. Finally, any equivalence relation ϑ with θ ϑ is either θ or A 2. Since the definition of a regular relation involves only equivalence relations with at least three classes, we conclude that, in our case, no submaximal clone corresponding to a regular relation belongs to IntM Subclones of P ol{π, θ} Let fx 1, y 1,..., x n, y n be a function from P ol{π, θ}. Then, given arguments of f, the first coordinate of the value of f on these arguments depends only on the first values of the arguments because f P olθ. Therefore we can assign to every such f a Boolean function f β x 1,..., x n defined as follows: for all a 1,..., a n Z 2, let a 0 = f β a 1,..., a n if a 0, b 0 = fa 1, b 1,..., a n, b n for some b 0, b 1,..., b n Z p. It is clear that f β is well-defined. Given a 1,..., a n Z 2, we can consider fa 1, y 1,..., a n, y n as an n-ary function to the set {a 0, 0,..., a 0, p 1} where a 0 = f β a 1,..., a n. Therefore we can assign to f and a = a 1,..., a n Z n 2 an n-ary function f a on Z p defined as follows: for all b 1,..., b n Z p, let b 0 = f a b 1,..., b n if a 0, b 0 = fa 1, b 1,..., a n, b n. It is easy to check that, for every a, the function f a belongs to P olσ where σ is the permutation on Z p defined in Section 1. It follows that every n-ary function f P ol{π, θ} can be represented as a pair f β ; {f a a Z n 2 } where f β is a Boolean function and f a P olσ is a function on Z p for every a Z n 2. Throughout this subsection, we shall use this representation and write f = f β ; {f a a Z n 2 }. The following lemma, that can be verified straightforwardly, describes the superposition of functions given by their representations. For f O n A and g i O m A, i = 1,..., n, let f[g1,..., g n ] denote the m-ary function fg 1 x 1,..., x m,..., g n x 1,..., x m. It is easy to check that every clone is closed under such a superposition. Lemma 1 Let f = f β ; {f a a Z n 2 } and g i = g i β ; {gi a a Z m 2 }, i = 1,..., n. If h = f[g 1,..., g n ] then h β = f β [g 1 β,..., gn β ] and, for every a Zm 2, we have h a = f ga [g a 1,..., g a n ] where ga = g 1 a,..., gn a. β We say that an n-ary function f P olπ, θ is quasi-unary if there exists i in the range from 1 to n such that 1 f β x 1,..., x n = x i for all x 1,..., x n Z 2, and 2 for every a Z n 2 and all y 1,..., y n Z n p, f a y 1,..., y n = y i f a 0,..., 0. β

6 6 A. Krokhin, I.G. Rosenberg Let K 1 be the set of all quasi-unary functions. Denote by K 2 the set of all f P ol{π, θ} with constant f β. In other words, K 2 consists of all functions f P ol{π, θ} whose image of is either {0} Z p or {1} Z p. Lemma 2 The sets K 1 M, K 2 M, and K 1 K 2 M are clones in IntM. Proof. Straightforward. Denote by d the decomposition function defined by the following rule dx 1, y 1, x 2, y 2 = x 1, y 2. It is easy to see that d P ol{π, θ}. Theorem 2 Let C IntM be a subclone of P ol{π, θ} such that C M K 1 K 2. Then d C. Proof. We need the next three lemmas. Lemma 3 The clone C contains a binary function g such that g β x 1, x 2 = x 1 and, for some a Z 2 2, the function g a y 1, y 2 essentially depends on y 2. Proof. The clone C contains an n-ary function f K 1 K 2 M. We distinguish two cases. Case 1. f β is essentially unary. Without loss of generality we may assume that f β x 1,..., x n = x c 1 for some c Z 2 where x 0 = x and x 1 = x. Since f is not quasi-unary, there exists a Z n 2 such that f a y 1,..., y n depends on another variable than y 1, say y 2. Define a unary operation m on Z 2 Z p by setting mx, y = x c, y for all x Z 2 and y Z p. Clearly, m M and so f = m f C. Now f β x 1,..., x n = x 1 and f b = f b for all b Z n 2. As f a depends essentially on its second variable, there are c 1, c 3,..., c n Z p such that hy 2 = f ac 1, y 2, c 3,..., c n is non-constant. Now define g by the rule gx 1, y 1, x 2, y 2 = f x 1, y 1 c 1, x 2, y 2, a 3, y 1 c 3,..., a n, y 1 c n. It is easy to see that g C because it is obtained as a superposition of f and of functions from M. Next, g β x 1, x 2 = f β x 1, x 2, a 3,..., a n = x 1. Moreover, g a1,a 20, y 2 = f ac 1, y 2, c 3,..., c n is non-constant, that is, g a1,a 2 depends essentially on y 2 as required. Case 2. f β x 1,..., x n depends essentially on at least two variables. According to Theorem 2 [13], there exist two variables, for notational simplicity say x 1 and x 2, and b 3,..., b n Z2 n 2 such that the binary Boolean function hx 1, x 2 = f β x 1, x 2, b 3,..., b n essentially depends on both its variables. For every 3 i n, the unary function m i on A taking every x, y to b i, y belongs to M. Therefore the function f x 1, y 1, x 2, y 2 = fx 1, y 1, x 2, y 2, b 3, y 2,..., b n, y 2 belongs to C. Here the function f β depends essentially on both arguments. The function f 0,0 belongs to P olσ, that is, f 0,0 is not constant. By exchanging the variables if necessary we can obtain that f 0,0 depends essentially on y 2. If f β x 1, 0 is non-constant then the function f x 1, y 1, 0, y 2 belongs to C and satisfies the conditions of Case 1, and we are done. Thus suppose f β x 1, 0 is constant. Then, since f β depends essentially on x 2, we get that f β x 1, 1 = x c 1 for some c Z 2.

7 A monoidal interval of clones of selfdual functions 7 If at least one of f 0,1 y 1, y 2 and f 1,1 y 1, y 2 depends essentially on y 2 then again the function f x 1, y 1, 1, y 2 belongs to C and satisfies the conditions of Case 1. It remains to consider the case when both f 0,1 y 1, y 2 and f 1,1 y 1, y 2 depend essentially only on y 1. Now there exists a Z 2 such that f β a, x 2 = x c 2 for some c Z 2. Define a binary function k on Z 2 Z p by setting kx 1, y 1, x 2, y 2 = f a, y 1, x 2, y 2. Clearly k C, k β x 1, x 2 = x c 2 while k 1,1 = f a,1 depends essentially on y 1 reducing this case to Case 1. Lemma 4 The clone C contains a binary function g satisfying g β x 1, x 2 = x 1 and g 0,0 y 1, y 2 = y 2. Proof. By Lemma 3, the clone C contains a binary function f such that f β x 1, x 2 = x 1 and, for some a = a 1, a 2 Z 2 2, the function f a y 1, y 2 essentially depends on y 2. Consider the permutation m M such that m = id if a 1 = 0 and mx, y = x, y for all x, y if a 1 = 1. Consider also the binary operation f x 1, y 1, x 2, y 2 = mfmx 1, y 1, a 2, y 2. Clearly f β x 1, x 2 = x 1 while f 0,0 y 1, y 2 = f a1,a 2 y 1, y 2 depends essentially on y 2. We may assume that from the very beginning f 0,0 y 1, y 2 depends essentially on y 2. Since f 0,0 P olσ, it follows from Theorem 1 [15] that either f 0,0 y 1, y 2 = ay 1 by 2 c for some a, b, c Z p with b 0 and a b = 1, or else P olσ is the clone on Z p generated by {σ, f 0,0 }. Case 1. f 0,0 y 1, y 2 = ay 1 by 2 c for a, b, c Z p with b 0 and a b = 1. Define the functions f n on Z p as follows: f 1 = f 0,0 and, for n > 0, f n+1 x 1, y 1, x 2, y 2 = f n x 1, y 1, fx 1, y 1, x 2, y 2. It is not hard to calculate that f n 0,0 y 1, y 2 = a1 b b n 1 y 1 b n y 2 c1 b b n 1. Consider f p 1. We have b p 1 = 1 and by 1 b b p 2 1 b = 1 b p 1 also 1 b b p 2 = 0 in Z p. Therefore f p 1 0,0 y 1, y 2 = y 2. Since, obviously, x 1, x 2 = x 1, we get the required result. Case 2. P olσ is the clone on Z p generated by {σ, f 0,0 }. The function 2y 1 y 2 belongs to P olσ, and hence it can be expressed by a term t using superpositions, the functions f 0,0, σ, and the projections on Z p. Consider the operation f x 1, y 1, x 2, y 2 on Z 2 Z p that can be expressed using the same term t, but using i fx i, y i, x j, y j instead of f 0,0 y i, y j for all i, j, ii x l, y l 1 instead of y l 1 for all l, and iii using the same projections on A as those on Z p appear in t. Using Lemma 1 it is not hard to check that f β x 1, x 2 = x 1 or f β x 1, x 2 = x 2, f p 1 β since f β x 1, x 2 is built from projections. Now f 0,0 y 1, y 2 = 2y 1 y 2 by construction of t, and we go back to Case 1 permuting variables of f if needed.

8 8 A. Krokhin, I.G. Rosenberg Lemma 5 Let S be a transformation semigroup on Z p generated by all translations x a and a transformation m with range of size q where 1 < q < p. Then S contains all constant transformations. Proof. Let {s 1,..., s q } be the range of m. For every element a Z p, set U a = {s 1 a,..., s q a}. Choose i so that m 1 s i has the minimal number of elements among all m 1 s j, j = 1,..., q. Then we have m 1 s i < p/q because p is prime. For a Z p, set m a x = mmx a. Clearly m a S and the kernels of m a and m satisfy kerm kerm a. Suppose there exist a Z p and 1 j q such that U a m 1 s j contains two distinct elements u and v. As u a = s r1 and v a = s r2 for some r 1 r 2, clearly m 1 s r1 m 1 s r2 m 1 a s j proving that the kernel of m a has less classes than the kernel of m. So we can replace m by m a and continue. Thus assume that U a m 1 s i 1 for all a Z p and all 1 i q. Suppose to the contrary that U a m 1 s i is a singleton for all a Z p. Then the map φx, s j = x s j is a bijection from m 1 s i {s 1,..., s q } onto Z p. Indeed, this map is injective, since if x 1 s j1 = x 2 s j2 = a for some distinct x 1, x 2 m 1 s i then x 1, x 2 U a m 1 s i contradicting the condition that this set is a singleton. Moreover, this map is also surjective, since, for any a Z p, the element x a U a m 1 s i satisfies x a = s j a for some j, and hence x a s j = a. This and m 1 s i < p/q yield the contradiction because p is prime and 1 < q < p. Thus there exists a Z p such that U a m 1 s i =. Now m 1 a s i is empty and so the image of m a is included into {s 1,..., s i 1, s i+1,..., s q }, and we can replace m by m a. Continuing we obtain that S contains a constant transformation. Then, since S contains all translations, we conclude that S contains all constants. Proof of Theorem 2. To construct d in C, first note that it suffices to find a binary function kx 1, y 1, x 2, y 2 C such that 1k β x 1, x 2 = x 1 and 2 k 0,0 y 1, y 2 = k 1,0 y 1, y 2 = y 2. In fact, it can be straightforwardly checked that then dx 1, y 1, x 2, y 2 = x 1, y 2 = kx 1, y 1, 0, y 2 C. By Lemma 4, we know that C contains a binary function f such that f β x 1, x 2 = x 1 and f 0,0 y 1, y 2 = y 2. We distinguish three cases. Case 1. f 1,0 y 1, y 2 depends essentially only on y 2. We have f 1,0 y 1, y 2 = y 2 a for some a Z p. The function m such that m0, y = 0, y and m1, y = 1, y a for every y Z p belongs to M. Then mfx 1, y 1, x 2, y 2 C satisfies the conditions 1 and 2 above, and we conclude that d C. Case 2. f 1,0 y 1, y 2 depends essentially only on y 1. We have f 1,0 y 1, y 2 = y 1 a for some a Z p. Then the function f x 1, y 1, x 2, y 2 = fx 1, y 1 a, x 2, y 2 belongs to C and satisfies f β x 1, x 2 = x 1, f 0,0 y 1, y 2 = y 2, and f 1,0 y 1, y 2 = y 1.

9 A monoidal interval of clones of selfdual functions 9 Consider hx 1, y 1, x 2, y 2 = mf mx 1, y 1, mx 2, y 2 where m M is given by mx, y = x, y. It is easy to check that h β x 1, x 2 = x 1 and that h 0,0 y 1, y 2 = y 1 and h 1,0 y 1, y 2 = y 2. Consider lx 1, y 1, x 2, y 2 = f hx 1, y 1, x 2, y 2, x 2, y 2. Now it is straightforward to verify that l satisfies the above conditions 1 and 2. Case 3. f 1,0 y 1, y 2 depends essentially on both variables. We may assume that f 0,0 = f 0,1 and f 1,0 = f 1,1, otherwise consider fx 1, y 1, 0, y 2 instead of f. Consider the function my = f 1,0 0, y on Z p. This function is not constant, since otherwise f 1,0 would depend essentially only on y 1. Assume that m is a permutation of Z p and let s be the least common multiple of the cycle lengths of m. Then m s is the identity permutation on Z p. Let the binary function ˆf on Z 2 Z p be defined by setting ˆfx 1, y 1, x 2, y 2 = fx 1, y 1, fx 1, y 1,..., fx 1, y 1, x 2, y 2... with s symbols f on the right-hand side, that is, ˆf is obtained from f by successively replacing the second argument in it by f s 1 times. Clearly ˆf C, ˆf β x 1, x 2 = x 1, ˆf 0,0 y 1, y 2 = y 1 and ˆf 1,0 y 1, y 2 = y 2 due to ˆf 1,0 0, y 2 = m s y 2 = y 2 and ˆf 1,0 P olσ. So in this case we go back to Case 1. Assume now that m is not a permutation. According to Lemma 5, all constant transformations on Z p can be constructed from m and the translations., i.e., there exist n 0 and a 0, a 1,..., a n Z p such that mm mmy a n a n 1 a 0 is a constant transformation. Consider the following recursively defined sequence of functions from C: f 0 x 1, y 1, x 2, y 2 = fx 1, y 1, x 2, y 2 a 0, and for 1 i n f i = f i 1 x 1, y 1, fx 1, y 1, x 2, y 2 a i. Then we have f i β x 1, x 2 = x 1 for all 1 i n. Using the fact that f 0,0 = f 0,1 and f 1,0 = f 1,1, it is easy to show by induction that, for every a Z 2 2 and 1 i n, we have f a i y 1, y 2 = f a i 1 y 1, f a y 1, y 2 a i. Using f 0,0 y 1, y 2 = y 2 and the definition of m we get and f n 0,0 y 1, y 2 = y 2 a n a 0, f n 1,0 0, y 2 = mm mmy 2 a n a n 1 a 0. Since f n 1,0 y 1, y 2 P olσ and f n 1,0 0, y 2 is a constant, it follows that f n 1,0 y 1, y 2 essentially depends only on y 1. Denote by r the unary operation on Z 2 Z p defined by setting r 0 y = y a 0 a n, r 1 y = y, and r β x = x.

10 10 A. Krokhin, I.G. Rosenberg Clearly r M and so r f n C. It is easy to see that r f n satisfies the assumption of Case 2. This proves Theorem 2. Now we show that, in some sense, in order to understand the structure of the interval [ M ; P olπ, θ] it is sufficient to investigate the intervals [ M ; M K 1 ] and [ M ; M K 2 ] of the lattice L A. It is obvious that every clone from [ M ; M K 1 K 2 ] is uniquely determined by its intersections with M K 1 and M K 2. Let C be a subclone of P ol{π, θ} such that M C M K 1 K 2. It follows from Theorem 2 that C contains the decomposition function d. Let f, g C n, f = f β ; {f a a Z n 2 }, g = g β ; {g a a Z n 2 }. Consider the function h = d[f, g] C. We have h = f β ; {g a a Z n 2 }. This means that C n is determined by the sets Y n C = {f β f C n } and Z n C = {{f a a Z n 2 } f C n }. Set Y C = n=1 Y n C and Z C = n=1 Zn C. It follows from Lemma 1 that Y C is a clone on {0, 1}. The clone Y C contains all unary functions because M C. It is well known [10] that in this case Y C is one of the three clones on {0, 1}: the clone of all essentially unary functions, the clone of all linear functions, and the clone of all functions. Set C = C M K 2. Since Y C contains all constant Boolean functions, it is easy to see that Z C = Z C. Now it follows that each clone C [ M ; P olπ, θ] such that d C is determined by a Boolean clone Y C and by the clone C M K The case p = 2: clones of quasi-linear functions In this section we consider the case p = 2 and describe the interval [ M, P olπ, ρ] where ρ is the affine relation determined by the group Z 2 2. It will be convenient to write the elements of Z 2 a x 2 as vertical pairs. We shall denote the pairs,, b y and x i y i by a, x, and x i, respectively. The functions from the clone P olρ are called quasi-linear. It is well known that an n-ary function f is quasi-linear if and only if n fx 1,..., x n = T i x i + a where T i are 2 2 matrices over Z 2 and a is an element from Z 2 2. For convenience, x x we choose notation so that π =. Note that we denote addition in Z 2 2 y y simply by +. It is always clear from the context which addition is meant. It can be easily checked that 1 a 12 M = { x + a a 12, a 22 Z 2, a Z 2 2}. 0 a 22 Lemma 6 We have

11 A monoidal interval of clones of selfdual functions 11 P ol{π, ρ} = { n T i x i + a n N, T i = a i 11 ai 12 a i 21 ai 22 and n ai 11 = 1, n ai 21 = 0, a Z2 2}. Proof. Every function f from P ol{π, ρ} must be of the above form because fx,..., x belongs to M = P ol 1 {π, ρ}. Now it is easy to verify that all such functions indeed belong to P olπ. Consider the following sets of functions from P ol{π, ρ}: C 1 = M { n a i 11 ai 12 C 2 = M { n a i 11 ai 12 x i + a n N, {i : a i 11 = 1} = 1, a Z2 2}, x i + a n N, n ai 11 = 1, a Z2 2}, C 3 = M { n T 1 c i x i + a n N, some T i = with c, d Z 2 and 0 d T j = for all j i, a Z 2 2}, C 4 = { n C 5 = { n C 6 = { n C 7 = { n a i 11 ai 12 0 a i x i + a n N, {i : a i 11 = 1} = 1 and 22 a i 11 ai 12 0 a i 22 a i 11 ai 12 0 a i 22 {i : a i 22 = 1} 1, a Z2 2}, x i + a n N, {i : a i 11 = 1} = 1, a Z2 2}, x i + a n N, n ai 11 = 1 and a i 11 ai 12 0 a i x i + a n N, n ai 22 C 8 = M, f where fx 1, x 2 = x 1 + {i : a i 22 = 1} 1, a Z2 2}, 11 = 1, a Z2 2}, It is easy to see that the functions from C 8 \ M are exactly the functions fx 1,..., x n of one of the following three types: 1 1 x i + x j +a, x i + x j +a, x i + x j +a, where 1 i, j n and, a Z 2 2. So, the clone C 8 consists of essentially at most binary functions. x 2.

12 12 A. Krokhin, I.G. Rosenberg P ol{π, ρ} C 7 C 5 C 6 C4 C 2 C 3 C 3 C 2 C 8 C 1 M Figure 1: The interval [ M ; P ol{π, ρ}]. Theorem 3 The interval [ M, P ol{π, ρ}] consists of the 11 clones: M, P ol{π, ρ}, C i where 1 i 8, and C 2 C 3. Moreover, the coverings are as shown on Fig. 1. Proof. It can be checked by routine calculation that all the sets of operations mentioned in the theorem are pairwise distinct clones with inclusions as shown on Fig. 1, and that they belong to [ M, P ol{π, ρ}]. Notice that the monoid M contains all translations and therefore we need not care about the constant term in functions, as it can be arbitrarily changed at any time. Let us fix the notation for the following matrices. Let P 1 =, P 2 =, P 3 =, P 4 =, P 5 =. Note that the function u i x = P i x belongs to M for each i = 1, 2, 3. We will now show via a sequence of lemmas and propositions that the 11 clones mentioned in the theorem form a complete list of clones in [ M, P ol{π, ρ}]. Lemma 7 For every f C 8 \ M, we have f, M = C 8. Proof. Easily follows from the description of functions in C 8 see above.

13 A monoidal interval of clones of selfdual functions 13 Proposition 2 M, C i, 1 i 7, and C 2 C 3 are the only clones in the interval [ M ; C 7 ], with coverings as shown on Fig. 1. We will use Lemmas 8-14 in the proof of the proposition. Lemma 8 The interval [ M ; C 2 ] is the chain M C 1 C 2. Proof. Take arbitrary f C 1 \ M and show f, M = C 1. We may assume that fx 1,..., x n = P i x 1 + P 4 x x n for some i {1, 2} and n 2. Denote this function by f n. Using P 4 P 1 = we get f k x 1,..., x k 1, u 1 x k 1 = f k 1 x 1,..., x k 1 for all k 2 and thus f, M contains all f 1,..., f n. For all j, l 1 we have f j f l x 1,..., x l, x l+1,..., x j+l 1 = f j+l 1 x 1,..., x j+l 1 due to P 2 i = P i and P i P 4 = P 4. Thus f, M contains all f m. Finally, letting i = 3 i, we get P i x 1 + P 4 x x m = f m u i x 1, x 2,..., x m because P i P i = P i. Thus, f, M = C 1. Now we take arbitrary f C 2 \ C 1 and show f, M = C 2. Using f C 1 and P1 2 = P 2 P 1 = P 1, P 4 P 1 =, after a coordinate exchange we get fu 1 x 1,..., u 1 x n = P 1 x x m = g m x 1,..., x m for some odd m 3. For odd k 3, g k x 1,..., x k 2, x k 1, x k 1 = g k 2 x 1,..., x k 2 and so f, M contains all g 1, g 3,..., g m. For all i, j 1 we have g i g j x 1,..., x j, x j+1,..., x i+j 1 = g i+j 1 x 1,..., x i+j 1, so the clone f, M contains all g 2h+1 with h 0. Let nonnegative r, s and t be such that r + s is odd. Consider the function obtained from g r+s+2t by substituting u 2 x i for each x i such that either r + 1 i r + s or i is odd with i > r + s. In view of P 1 P 2 = P 2 and P 1 + P 2 = P 4, the obtained function is P 1 x x r + P 2 x r x r+s + P 4 x r+s x r+s+t. Thus f, M = C 2. Lemma 9 The interval [ M ; C 5 ] is the chain M C 1 C 3 C 4 C 5. Proof. Similar to Lemma 8.

14 14 A. Krokhin, I.G. Rosenberg Lemma 10 If f C 6 \ C 2 C 3 C 4 then f, M = C 6. Proof. Set N = f, M and let f depend essentially on all its n variables. Claim. The clone N contains the function g 3 = x 1 + P 1 x 2 + x 3. After a suitable permutation of variables, f is of the form a b x 1 +P 1 x 2 + +x k +P 2 x k+1 + +x l +P 4 x l+1 + +x n 1 where a, b Z 2, 2 k l n, and l + a 3 is odd. For a = 0, we replace f by fx 1, x 1, x 2,..., x n 1, so we may assume that a = 1. For b = 1, we replace f by f u 3, so we may assume that a = 1 and b = 0. Using P 2 1 = P 2 P 1 = P 1, P 2 4 = we may reduce f to g 3 proving the claim. For k 1, set f 2k+1 x 1,..., x n = x 1 + P 1 x 2,..., x 2k+1. Clearly, f 3 = g 3 and f 2k+1 f 3 = f 2k+3 and hence N contains all f 2k+1. Now from P 1 P 2 = P 2, replacing some variables x i by u 2 x i, we can get any function of the form x 1 + P 1 x x l + P 2 x l+1 + x 2k+1. Using P 1 + P 2 = P 4, via variable fusion we can get any function of the form 1 with a, b = 1, 0. Using + P 1 =, + P 2 = P 5, + P 4 = P 3 and variable fusion, we get 1 for the pair a, b. Hence N includes C 6 \C 2 C 3 C 4. Clearly, g = f 3 u 1 N. Here gx 1, x 2, x 3 = P 1 x 1 + x 2 + x 3 C 2 \ C 1, and from Lemma 8 we obtain C 2 N. In order to prove C 3 N, we use tx 1,..., x 4 = x 1 + P 1 x 2 +x 3 +P 4 x 4 from N. Now t x 1, x 2 = tx 1, x 2, x 2, x 2 = x 1 +P 4 x 2 belongs to C 3 \ C 1 proving C 3 N by Lemma 9. Finally, hx 1,..., x 4 = x 1 + P 1 x 2 belongs to N C 4 \ C 3. Now Lemma 9 implies that C 4 N. This completes the proof. Lemma 11 If f C 7 \ C 5 C 6 then f, M = C 7. Proof. Set N = f, M and let f depend essentially on all its n variables. Claim. The clone N contains the function f 3 x 1, x 2, x 3 = x 1 + x 2 + P 1 x 3. The function f is of the form n a i 11 ai 12 0 a i x i 2 22

15 A monoidal interval of clones of selfdual functions 15 where s = n a i 11, s = n a i 22 3 satisfy s 3 odd and s > 1 note that, in 3, elements from Z 2 are considered as integers and the sums here are the usual sums of integers. By an appropriate variable exchange and/or fusion, we can achieve that a j 11 = 1 exactly for j = 1, 2, 3 while a 1 22 = a2 22 = 1. Using P 3 2 =, we can get the function 1 a m x 1 + x 2 + x 3 + T i x i 0 b i=4 with a, b Z 2, m 3, and, for i = 4,..., m, T i is one of the matrices P 4, P 5,. Using P 4 P 1 = P 5 P 1 = 1 a P 1 =, 0 b P 1 = P 1. we get f 3 N which proves the claim. For k 1, set f 2k+1 x 1,..., x 2k+1 = x x 2k + P 1 x 2k+1. Let gx 1, x 2, x 3 = f 3 f 3 x 1, x 2, x 3, x 3, x 3 = x 1 + x 2 + x 3. From f 2k+3 = f 2k+1 g for all k 1, we obtain that N contains all f 2k+1. Now consider arbitrary n-ary g C 7 \ C 5 C 6 of the above form 2-3. We may assume that a 1 11 =... = as 11 replace in f n+1 the last variable by f n s to obtain = 1 while as+1 11 =... = a n 11 = 0. For n s, n even, hx 1,..., x 2n s = x x n + P 1 x n x 2n s. 1 a j For j = 1,..., n, replace x j by x j to obtain h and then form 0 b j s h 1 a j n 0 a j x 1,..., x n, x s+1,..., x n = x j + x j. 4 0 b j 0 b j j=1 j=s+1 For n odd, take 4 for n + 1 and choose a n+1 = b n+1 = 0. This proves N C 7 \ C 5 C 6. To see C 5 N, consider gx 1, x 2 = f 3 x 1, x 2, x 2 = x 1 + x 2.

16 16 A. Krokhin, I.G. Rosenberg Clearly, g N C 5 \ C 4 and so by Lemma 9 we get C 5 N. To show C 6 N, consider hx 1, x 2, x 3 = f 3 x 1, u 1 x 2, x 3 = x 1 + P 1 x 2 + x 3. Here h N C 6 \ C 2 C 3 C 4, hence C 6 N by Lemma 10. This proves the lemma. Lemma 12 If D is a clone and D C 2 C 3 then D C 2 or D C 3. Proof. Suppose to the contrary that D C 2 and D C 3. Then there exist f D C 3 \ C 2 and g D C 2 \ C 3. By Lemmas 9 and 8, clearly f, M = C 3 and g, M = C 2, and hence D C 2 C 3, a contradiction. Lemma 13 If D is a clone and D C 2 C 3 C 4 then D C 2 C 3 or D = C 4. Proof. Suppose D C 2 C 3. Then there exists f D \ C 2 C 3. Clearly f C 4 \C 3, and D f, M = C 4 by Lemma 9. Suppose to the contrary that C 4 D. Then there exists g D C 2 C 3 \C 4 D C 2 \C 3. Thus D g, C 3 = C 2 C 3, since C 2 C 3 covers C 3 by Lemma 12. Thus D = C 2 C 3 C 4 = C 2 C 4. Since C 2 C 4 can be easily shown not to be a clone, this contradicts the assumption, and so C = C 4. Lemma 14 If a clone D satisfies D C 5 C 6 then D = C 5 or D C 6. Proof. Suppose that D C 6. From D C 5 C 6 there exists f D C 5 \ C 6 D C 5 \ C 4. Then D f, M = C 5 due to Lemma 9. Suppose to the contrary that C 5 D. Then there exists g D C 6 \ C 5 = D C 6 \ C 4. By Lemmas 10 and 13, C 6 covers C 4, and so D g, M = C 6 contrary to the assumption. Thus D = C 5. Proof of Proposition 2. From Lemmas 11 and 14 we obtain that every proper subclone of C 7 containing M and distinct from C 5 is contained in C 6. Similarly, by Lemmas 10 and 13 each proper subclone of C 6 containing M and distinct from C 4 is included in C 2 C 3. Finally, due to Lemmas 8, 9, and 12, the interval [ M ; C 2 C 3 ] is as shown on Fig. 1. This proves the proposition. Lemma 15 If f P ol{π, ρ} \ C 7 C 8 then f, M = P ol{π, ρ}.

17 A monoidal interval of clones of selfdual functions 17 Proof. Case 1. f is binary. We may assume that 1 c 0 u fx 1, x 2 = x 1 + x 2 1 d 1 v where either c d, or u = 1, or both. Subcase 1.1. c d. If c = 0 then substituting u 3 x 1 for x 1 we get c = 1 and d = 0. Also, substitute u 1 x 2 for x 2 to ensure that u = v = 0. Hence, we may assume that fx 1, x 2 = 1 1 x 1 + x 2. Consider the function g defined as 1 1 g 1 = fffx 1, x 2, u 3 x 2, x 3 = x 1 + x 2 + x We can derive the functions g 2 x 1, x 2, x 3, x 4, x 5 = g 1 x 1, g 1 u 2 x 2, x 3, x 4, x 5 = x 1 + x 2 + x 3 + x 4 + x 5, 1 1 h 3 = g 2 x 1, u 2 x 3, x 2, x 3, x 3 = x 1 + x 2 + x 3, and, further, h n = n x i for any odd n. Substituting g 2 in h n for every variable and renaming the variables in an appropriate way, we can get f n = n for any odd n. x 1 i + n n + x 2 i + x 4 i + n n Now we show how to obtain any function of the form m x 3 i + x 5 i a i 11 ai 12 a i 21 ai 22 x i with m ai 11 = 1 and m ai over Z 2 can be obtained as the sum of some of the coefficients of the function g 2. to 21 = 0 from f n with n > m. Note that any 2 2 matrix For every 1 i m, choose the set of coefficients of g 2 whose sum is equal a i 11 ai 12 a i 21 ai 22 and identify the corresponding variables x 1 i with x i and identify the other variables x 1 i with x m+1. For every i > m, identify all variables x j i, 1 j 5, with x m+1. After performing this procedure we obtain a function of the form m+1 a i 11 ai 12 a i 21 ai 22 x i. Note that we have a m+1 11 = a m+1 21 = 0 because

18 18 A. Krokhin, I.G. Rosenberg the obtained function belongs to P ol{π, ρ} and we also have m ai 11 = 1 and m ai 21 = 0. It remains to substitute u 1x 1 for x m+1, and we get the required function. Subcase 1.2. c = d and u = 1. Substitute u 1 x 1 for x 1. If v = 0 then substitute u 3 x 2 for x 2. Now we may without loss of generality assume that 1 1 f = x 1 + x Consider the function g defined as g = fx 2, fx 2, x 1. It is easy to check that 1 1 g = x 1 + x 2, 1 0 that is, g satisfies the conditions of the previous subcase. Case 2. f depends essentially on more than two variables. Let f = n b i 11 bi 12 b i 21 bi 22 x i where n 3. We may assume that not all of b 3 11,b3 12,b3 21,b3 22 are equal to 0. We show how to obtain, from f, a function satisfying the conditions of Case 1. As f C 7, we know that at least two of elements b i 21, 1 i n, are equal to 1. We may suppose that b 1 21 = b2 21 = 1. Substitute u 1x i for x i for i = 1, 2. Then we have b 1 12 = b1 22 = b2 12 = b2 22 = 0. Denote n i=3 b i 11 bi 12 b i 21 bi 22 by b 11 b 12 b 21 b 22. Note that b 21 = 0 because of n bi 21 = 0. Subcase 2.1. b 11 = 1. Identify all the variables x i, i 3, with x 3 and then substitute u 1 x 3 for x 3. We get the function g = b 1 1 x 1 + b 2 1 x 2 + with b 1 11 = b2 11. Then it is easy to check that the binary function gx 1, u 3 x 2, x 2 satisfies the conditions of Subcase 1.1 for b 1 11 Subcase 2.2. b 11 = 0. We may assume that b 1 11 = 1 and b2 11 = 0. x 3 = 0 and Subcase 1.2 for b1 11 = 1.

19 A monoidal interval of clones of selfdual functions 19 If at least one of b 12 and b 22 is 1 then identifying all x i, i 3, either with x 1 or with x 2 we obtain a binary function satisfying the conditions of Case 1. Now we may assume that b 12 = b 22 = 0. Suppose that at least one of b 3 11 and b3 21 is 1. Then substitute u 3 x 3 for x 3 and consider the obtained function instead of f. b 11 b 12 Calculating the matrix again, we get that at least one of b 12 and b 22 is 1, b 21 b 22 the case already considered. Suppose now that b 3 11 = b3 21 = 0. Identify x 3 with x 2 if b 3 12 = b3 22 = 1, and with x 1 otherwise. Now it can be checked that if we substitute u 1 x 1 for x i for every i 3 then we once again obtain a binary function satisfying the conditions of Case 1. Lemma 16 Suppose that C [ M, P ol{π, ρ}]. If C C 7 and C C 8 then C = P ol{π, ρ}. Proof. If C C 7 C 8 then the result follows from Lemma 15. We now assume that C C 7 C 8 and get a contradiction. Since C C 8, it follows from Proposition 2 that C 1 C, in particular, C contains the function g = x 1 + x 2. Since C C 7, it follows from Lemma 7 that C 8 C, in particular, C contains the function 1 0 h = x 1 + x Then the function 1 0 hx 1, gx 2, x 3 = x 1 + x 2 + x belongs to C and satisfies the conditions of Lemma 15. Hence C = P ol{π, ρ} which is a contradiction with the assumption that C C 7 C 8. Theorem 3 now follows from Proposition 2 and Lemmas 7 and 16. Acknowledgments This research was done during the first author s one month stay at the University of Montreal in 1999 and was partially supported by the grant OGP 5407 by NSERC Canada. The authors would like to thank an anonymous referee for his remarks.

20 20 A. Krokhin, I.G. Rosenberg References [1] G. A. Burle, Classes of k-valued logic which contain all unary functions. Diskr.Anal. 1967, 3 7 in Russian. [2] L. Haddad, B. Larose, Colourings of hypergraphs, permutation groups, and CSP s. To appear in the Proceedings of the ASL, [3] P. Jeavons, On the algebraic structure of combinatorial problems. Theoretical Computer Science 20998, no. 1-2, [4] A. A. Krokhin, Boolean lattices as intervals in clone lattices. Multiple-Valued Logic , no. 3, [5] A. A. Krokhin, Maximal clones in monoidal intervals, I. Siberian Math. Journal 4999, no. 3, in Russian. [6] A. A. Krokhin, B. Larose, A monoidal interval of isotone clones on a finite chain. Acta Sci. Math. Szeged , no. 1-2, [7] A. A. Krokhin, D. Schweigert, On clones preserving a reflexive binary relation. Acta Sci. Math. Szeged , no. 3-4, [8] A.V.Kuznetsov, On the problems of identity and functional completeness of algebraic systems. In: III All-Union Math. Symposium, Moscow, AN USSR, 1956, in Russian. [9] R. Pöschel, L. A. Kalužnin, Funktionen- und Relationenalgebren. VEB DVW, Berlin, [10] E. L. Post, The two-valued iterative systems of mathematical logic. Annals of Math. Studies, no.5, Princeton University Press, N.J., [11] I. G. Rosenberg, Über die funktionale Vollständigkeit in den mehrwertigen Logiken Struktur der Funktionen von mehreren Veränderlichen auf endlichen Mengen. Rozpravy Českoslov. Akad. Věd., Řada Mat. Přirod., Věd. 8970, [12] I. G. Rosenberg, Á. Szendrei, Submaximal clones with a prime order automorphism. Acta Sci. Math. Szeged , [13] A. Salomaa, On essential variables of function, especially in the algebra of logic. Ann. Acad. Sci. Fenn. Ser AI 339, [14] L. Szabó, Á. Szendrei, S lupecki-type criteria for quasi-linear functions over a finite-dimensional vector space. EIK , [15] Á. Szendrei, Algebras of prime cardinality with a cyclic automorphism. Arch. Math. Basel , [16] Á. Szendrei, Clones in universal algebra. Séminaire Math. Supérieures 99, Les Presses de l Université de Montréal, [17] Yu. I. Yanov, A. A. Muchnik, On existence of k-valued closed classes without finite basis. Dokl. AN USSR , in Russian.