SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES

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1 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES ERKKO LEHTONEN Abstract. Certain quasi-orders of k-valued logic functions defined by the clones that contain all unary operations on the k-element set are investigated. The quasi-ordering is based on the composition of functions from inside with the members of the clone. It is determined whether the induced partial orders satisfy the descending chain condition. The widths of these posets are also determined. 1. Introduction Generalizing and unifying the various notions of minors and subfunctions presented by Pippenger [7], Wang [12], Zverovich [14], and others, we define for any class C of operations on a fixed nonempty base set A, the C-subfunction relation as follows: a function f is a C-subfunction of a function g, if f = g(h 1,..., h n ) for some h 1,..., h n C. This relation is a preorder (i.e., a reflexive and transitive relation) on the set of all operations on A if and only if C is a clone (i.e., C contains all projections and is closed under functional composition). Previously, we have investigated the C-subfunction relations between Boolean functions in [4], as well as the C-subfunction relations defined by the clones of unary, linear and monotone functions on finite base sets in [5]. In particular, we determined whether the partial order induced by each of these C-subfunction relations satisfies the descending chain condition. We also determined the widths of these posets. We now continue our study of C-subfunctions, focusing on the C-subfunction relations defined by the clones that contain all unary operations on a finite set A. It was shown by Burle [1] that for A = k 2, these clones constitute a (k + 1)- element chain B 0 B 1 B k. We will show that the partial order induced by the B i -subfunction relation does not satisfy the descending chain condition if and only if 2 i k 2, and it contains infinite antichains if and only if i k 2 and k 3 or i 1 and k = 2. This paper is organized as follows. We present our basic definitions and notation and we formulate our current problem precisely in Section 2. We analyze the subfunction relations defined by the smallest and largest Burle s clones B 0, B k, B k 1 in Section 3. In Section 4, we construct an infinite descending chain of B p -subfunctions for 2 p k 2. Section 5 is devoted to unique representability of quasilinear functions and C-decompositions, concepts which we will need in our analysis of B 1 - subfunctions. In Section 6, we show that there is no infinite descending chain of B 1 -subfunctions. We present infinite antichains of B k 2 -incomparable functions in Section 7. Our results are summarised in Section 8. Date: March 12,

2 2 ERKKO LEHTONEN 2. Definitions and notation 2.1. General notation and concepts. We denote vectors by bold face letters and their components by normal math fonts, e.g., a = (a 1,..., a n ). We also denote the ith component of a vector a by a(i), especially when the vector symbols involve subscripts. For an integer n 1, we denote n = {0,..., n 1}. For a collection C of disjoint sets, a transversal is a set containing exactly one member of each of them. A partial transversal of C is a subset of a transversal of C. The characteristic function of a subset S A is the mapping χ S : A 2 defined as { 1, if x S, χ S (x) = 0, if x / S Functions and clones. Let A be a fixed nonempty base set. A function on A is a finitary operation on A, i.e., a mapping f : A n A for some positive integer n, called the arity of f. The set of all functions on A is denoted by O A. For a fixed arity n, and for 1 i n, the n-ary ith projection, denoted by x n i, is the function (a 1,..., a n ) a i. We denote by J A the set of all projections on A. The n-ary constant function having value a A everywhere is denoted by â n. Whenever the arity is clear from the context, we may omit the superscripts indicating arity. The range, or image, of f is the set Im f = {f(a) : a A n }. The kernel of f is the equivalence relation KER f = {(a, b) : f(a) = f(b)} on the domain of f. For 1 i n, we say that the ith variable is essential in an n-ary function f, or f depends on the ith variable, if there are points a = (a 1,..., a n ), a = (a 1,..., a n) such that a i a i and a j = a j for all j i and f(a) f(a ). If a variable is not essential in f, then it is inessential in f. The essential arity of f, denoted EAr f, is the number of essential variables in f. The set of essential variables of f is defined as Ess f = {i : the ith variable is essential in f}. If f is an n-ary function and g 1,..., g n are all m-ary functions, then the composition of f with g 1,..., g n, denoted f(g 1,..., g n ) is an m-ary function defined by f(g 1,..., g m )(a) = f(g 1 (a),..., g n (a)). This is equivalent to the composition f g, where the mapping g : A m A n is defined as g(a) = (g 1 (a),..., g n (a)), which we simply denote by g = (g 1,..., g n ). A class is a subset C O A. A clone on A is a class C O A that contains all projections and is closed under functional composition (i.e., if f, g 1,..., g n C, then f(g 1,..., g n ) C whenever the composition is defined). The clones on A constitute an inclusion-ordered lattice, denoted L A, where the lattice operations are the following: meet is the intersection, join is the smallest clone containing the union. We denote by C the clone generated by C. See [11] for a general account on clones. For any class C, we denote by C (n) the n-ary part of C, i.e., C (n) = {f C : f is n-ary}. We denote by C (m,n) the set of mappings of the form (f 1,..., f m ), where each component f i is a member of C (n), i.e., C (m,n) = {(f 1,..., f m ) (A m ) An : f 1,..., f m C (n) }.

3 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 3 For any clone C, C (1) is a transformation monoid on A. Assume that M is an arbitrary transformation monoid on A. The stabilizer of M is the set St M = {f O A : f(g 1,..., g n ) M whenever g 1,..., g n M}. It is easy to see that St M is a clone, and M St M. An interval of the form Int M = [ M ; St M] in L A is called monoidal. It is well-known that C (1) = M if and only if C Int M (see, e.g., [11]). Thus, L A is partitioned in intervals Int M, where M ranges over all submonoids of O (1) A. It was shown by Burle [1] that for a finite base set A with k 2 elements, the monoidal interval Int O (1) A is the (k + 1)-element chain O (1) A = B 0 B 1 B 2 B k 1 B k = O A, where B 1 consists of all essentially at most unary functions and all quasilinear functions, i.e., functions having the form g(h 1 (x 1 ) h n (x n )) with h 1,..., h n : A 2, g : 2 A arbitrary mappings and denoting addition modulo 2; and for 2 p k, B p consists of all essentially at most unary functions and all functions whose range contains at most p elements C-subfunctions. Let C be a class of functions on A. We say that a function f is a C-subfunction of a function g, denoted f C g, if f = g(h 1,..., h m ) for some h 1,..., h m C, i.e., f = g h for some h C (m,n) where m and n are the arities of g and f, respectively. If f and g are C-subfunctions of each other, we say that they are C-equivalent and denote f C g. If f C g but g C f, we say that f is a proper C-subfunction of g and denote f < C g. If both f C g and g C f, we say that f and g are C-incomparable and denote f C g. If the class C is clear from the context, we may omit the subscripts indicating the class. We have now defined families of binary relations C and C on O A, parametrised by the class C. Most of the basic properties of C and C that we proved in [4] in the setting of a two-element base set are straightforwardly generalized for arbitrary base sets; we just state these facts and omit the proofs here. For any class C, the set of C-subfunctions of x 1 is C, and therefore the relations C and K are distinct for C K. Also, for any classes C and K, C is a subrelation of K if and only if C K. For any clones C and K, C is a subrelation of K whenever C K. However, it is possible that C and K are the same relation even if C K. The C-subfunction relation C is reflexive if and only if the class C contains all projections; and C is transitive if and only if C is closed under functional composition. Hence, C is a preorder on O A if and only if C is a clone. If C is a clone, then C is an equivalence relation. The C-equivalence class of f is denoted by [f] C. As for preorders, C induces a partial order C on O A / C. It is clear that Im f Im g for any f C g and any C. Therefore, any C-equivalent functions have the same range. This implies in particular that for any element a A, the constant functions â of all arities form a C-equivalence class for any clone C, and these classes are minimal in the partial order C of O A / C. For a general account on ordered sets, see, e.g., the textbook by Davey and Priestley [2] The current problem and known results. Given the preorder C on O A, defined by a clone C on A, two questions about the induced partial order C arise

4 4 ERKKO LEHTONEN immediately. Does there exist an infinite descending chain of C-subfunctions? How large antichains of C-incomparable functions do there exist? The case of a singleton base set A is trivial, because we only have one clone, namely the clone O A of all functions, and all functions are O A -equivalent. We resolved these questions for all clones on a two-element base set in [4]. Since the structure of the lattice of clones on A is largely unknown when A 3, there is little hope of giving such a complete and definite answer in the more general case. We must focus our analysis on some special clones of interest. In [5], we answered these questions for the C-subfunction relations defined by the clones of unary, linear, and monotone functions on any finite base set. In this paper, we analyze the B i - subfunction relations for each of Burle s clones B 0, B 1,..., B k on a k-element finite base set. In what follows, we assume that the base set A is finite and A = k 3. Because it is immaterial what the elements of the base set are, we assume, without loss of generality, that A = {0, 1,..., k 1} = k. 3. The smallest and largest Burle s clones It can be shown by an easy argument on essential arity that for any transformation monoid M on A, there is no infinite descending chain of M -subfunctions. We also proved in [5] that there is no infinite descending chain of O A -subfunctions nor an infinite antichain on O A -incomparable functions. For the sake of comprehensiveness, we reproduce the proofs here B 0 -subfunctions. Denote by J A the clone of all projections on A. It is clear that every nonconstant function f is J A -equivalent (and hence C-equivalent for any clone C) to the function f ess of arity EAr f, obtained by deleting all inessential variables of f. We can also agree that (â n ) ess = â 1. Lemma 3.1. Let M be a transformation monoid on A. If f M g, then EAr f EAr g. Proof. Let f = g(h 1,..., h m ) for some h 1,..., h m M. Each essential variable of f has to be essential in at least one of the inner functions h i substituted for an essential variable of g. Since the h i s are essentially at most unary, it is clear that EAr f EAr g. Proposition 3.2. For any transformation monoid M on A, there is no infinite descending chain of M -subfunctions. Proof. Suppose, on the contrary, that there is an infinite descending chain f 1 > M f 2 > M f 3 > M. Since each f i is M -equivalent to fi ess, we can assume that all variables are essential in f i. Lemma 3.1 implies that there is an m such that all functions f i with i m have the same arity. We have reached a contradiction, because there are only a finite number of functions of any fixed arity. Theorem 3.3. There is no infinite descending chain of B 0 -subfunctions. Proof. A special case of Proposition 3.2.

5 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES B k -subfunctions. For a class C and a subset S A, we denote C S = {f C : Im f = S}. Proposition 3.4. Suppose that C Int O (1) A. Then, for every nonempty subset S of A, every nonempty C S is a C-equivalence class. Proof. Let ξ be any fixed unary function in C S such that ξ(a) = a for all a S, and let f C S be n-ary. It is clear that ξ(f) = f, so f C ξ. For each b S, let u b f 1 (b). For i = 1,..., n, define the unary function g i as g i (a) = (u ξ(a) ) i. Then f(g 1,..., g n )(a) = f(g 1 (a),..., g n (a)) = f(u ξ(a) ) = ξ(a), so ξ = f(g 1,..., g n ). Since C contains all unary functions, ξ C f. We have shown that all functions in C S are C-equivalent to ξ. By the transitivity of C, the members of C S are pairwise C-equivalent. The fact that C-equivalent functions have the same range now implies that C S is a C-equivalence class. Proposition 3.4 gives a complete characterization of the B k -equivalence classes: f Bk g if and only if Im f = Im g. Since A is finite, there are only a finite number of equivalence classes, and therefore there simply cannot exist an infinite descending chain of B k -subfunctions nor an infinite antichain of B k -incomparable functions. In fact, it is easy to see that (O A / Bk, Bk ) is isomorphic to (P(A) \ { }, ), the power set lattice of A with the bottom element removed. The largest chain of this ( poset has k elements, and by Sperner s theorem [10], the largest antichain has k k/2 ) elements. Theorem 3.5. The poset (O A / Bk, Bk ) is isomorphic to (P(A) \ { }, ). The largest chain of B k -subfunctions has k elements. The largest antichain of B k -incomparable functions has ( k k/2 ) elements B k 1 -subfunctions. Let α, β, γ be distinct elements of A. We say that (α, β, γ) is an essential triple for an n-ary function f if there exist a, b, c A n and 1 i n such that a j = b j for every j i, a i = c i, and f(a) = α, f(b) = β, f(c) = γ. The following lemma is due to Mal tsev [6] who slightly improved earlier results by Yablonski [13] and Salomaa [9] (see also [8]). Lemma 3.6. If f has at least two essential variables and takes on more than two values, then f possesses an essential triple. Conversely, if f possesses an essential triple, then f has at least two essential variables and each value of f appears in an essential triple for f. For 1 i n, we define the projection of a subset B A n onto its ith component by pr i B = {b i : (b 1,..., b n ) B}. Corollary 3.7. Let f be an n-ary function with at least two essential variables and Im f = r 3. Then there is a transversal B of KER f such that pr i B < r for 1 i n. Proof. By Lemma 3.6, there are elements α, β, γ A and points a, b, c A n such that the conditions for an essential triple for f are satisfied. Choose any r 3 points d 4,..., d r such that B = {a, b, c, d 4,..., d r } is a transversal of KER f. It is clear that pr i B r 1 for 1 i n.

6 6 ERKKO LEHTONEN Now we can characterize the B k 1 -equivalence classes. The classes contained in B k 1 are given by Proposition 3.4, and we will show that O A \ B k 1 is a B k 1 - equivalence class. For this purpose, we only have to show that f Bk 1 g for any f, g O A \ B k 1. Proposition 3.8. If f, g O A \ B k 1, then f Bk 1 g. Proof. Let f be an n-ary function and g an m-ary function, both in O A \ B k 1. Hence f and g are essentially at least binary and Im f = Im g = A. By Corollary 3.7, there is a k-element set B = {d 0,..., d k 1 } A m such that g(d a ) = a for every a A and pr i B k 1 for 1 i n. Let h : A n A m be defined as h(a) = d f(a). We clearly have that f = g h and h B (m,n) k 1. Thus f B k 1 g. Similarly, we can show that g Bk 1 f. There are only a finite number of B k 1 -equivalence classes, and therefore Bk 1 contains no infinite chains or antichains. The structure of the poset (O A / Bk 1, Bk 1 ) can easily be described in more detail. We denote by P Q the linear sum of posets P and Q, and we denote by 1 the one-element chain. (See [2] for more details.) Then (O A / Bk 1, Bk 1 ) is isomorphic to (P(A) \ { }, ) 1, the power set lattice of A with the bottom element removed and a new top element added. Theorem 3.9. The poset (O A / Bk 1, Bk 1 ) is isomorphic to (P(A)\{ }, ) 1. The largest chain of B k 1 -subfunctions has k + 1 elements. The largest antichain of B k 1 -incomparable functions has ( k k/2 ) elements. 4. B p -subfunctions for 2 p k 2 In this section, we assume that A = k 4. For n 3, define the (n + 1)-ary function f n on A as a, if a 1 = = a n = a and a n+1 = 0, f n (a) = 1, if a {u n, v n, w n }, 0, otherwise, where the (n + 1)-vectors u n, v n, w n are defined recursively as u 3 = 1120, u 4 = 12120, u n = 1v n 1, v 3 = 2120, v 4 = 23210, for n 5 v n = 2u n 1, w 3 = 3210, w 4 = 32120, w n = 3u n 1. Proposition 4.1. For 2 p k 2 and for any n 3, f n+1 < Bp f n. Proof. We first observe that f n+1 = f n (x 2, x 3,..., x n, x n+1, g), where the (n + 2)- ary function g is defined as { 0, if a {u n+1, v n+1, w n+1 } or a 1 = = a n+1, g(a) = 1, otherwise. Since all projections and g are members of B 2, we conclude that f n+1 is a B 2 - subfunction of f n (and hence a C-subfunction for every C B 2 ). We then show that f n Bk 2 f n+1 and hence f n C f n+1 for every C B k 2. Suppose, on the contrary, that f n Bk 2 f n+1. Then there exist (n + 1)-ary

7 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 7 functions h 1,..., h n+2 B k 2 such that f n = f n+1 (h 1,..., h n+2 ). Let us denote h = (h 1,..., h n+2 ), and for a A and n 1, denote by e n a the (n + 1)- vector whose first n components are equal to a and the last component is 0. We clearly have that for a {2,..., k 1}, h(e n a) = e n+1 a and h({e n 1, u n, v n, w n }) {e n+1 1, u n+1, v n+1, w n+1 }. We say that a function f is a projection beyond a A if Ess f = {i} for some i and f(a 1,..., a n ) = a i whenever a i a. Note that all projections are projections beyond every a A. If h(e n 1 ) = u n+1, then Im h i k 1 whenever u n+1 (i) = 1 and so h i is essentially unary and hence a projection beyond 0. Apart from h n+2, each of the other inner functions h i is either essentially at least binary, and therefore 0, 1 / Im h i, or h i is essentially unary such that h i (j,..., j) = j for 2 j k 1 but h i (1,..., 1) 1. Whatever the case may be, it is only possible that h maps both u n and v n to u n+1. However, this is an impossibility, because u n and v n do not have 1 s in common positions. Similarly, we deduce that it is not possible that h(e n 1 ) = v n+1 or h(e n 1 ) = w n+1. We are only left with the case that h(e n 1 ) = e n+1 1. Then for 1 i n + 1, Im h i k 1 and hence h i is essentially unary. Moreover, h i is a projection beyond 0 and n + 1 / Ess h i. We observe that u n (1) = 1, v n (1) = 2, w n (1) = 3 and for 2 i n, {u n (i), v n (i), w n (i)} = 2. Since h 1 is a projection beyond 0, it is only possible that Ess h 1 = {1}, and we can deduce that h(v n ) = v n+1, h(w n ) = w n+1 and either h(u n ) = u n+1 or h(u n ) = e n+1 1. From the recursive definition, we see that at most one component of v n+1 equals 3. If v n+1 contains a 3, then v n does not, and there is no way we could produce a 3 from the vector v n not containing a 3 by a projection beyond 0 with the (n + 1)-th variable inessential. Otherwise, u n+1 contains a 3 but u n does not, and we deduce in a similar way that h(u n ) u n+1, so we have that h(u n ) = e n+1 1. Since u n and v n do not have 1 s at common positions, h(v n ) would be a vector with no 1 s, a contradiction. We conclude that f n Bk 2 f n+1 and hence f n C f n+1 for every C B k 2. Theorem 4.2. For 2 p k 2, there is an infinite descending chain of B p -subfunctions. 5. Quasilinear functions and C-decompositions 5.1. Unique representations of quasilinear functions. Functions of the form f = g(h 1 (x 1 ) h n (x n )), where h 1,..., h n : A 2, g : 2 A and denotes addition modulo 2, are called quasilinear. The mappings h i are in fact characteristic functions of subsets S i A. Then h 1 (x 1 ) h n (x n ) is the characteristic function of the set S 1 S 2 S n A n, where denotes symmetric difference and S i = {(a 1,..., a n ) A n : a i S i } = A i 1 S i A n i A n. The negation h of a mapping h : A n 2 is defined as h(a) = h(a) 1. The inner negation g in of a mapping g : 2 A is defined as g in (b) = g(b 1).

8 8 ERKKO LEHTONEN Lemma 5.1. The representation of a nonconstant quasilinear function in the form f = g(h 1 (x 1 ) h n (x n )) is unique up to the negation of some of the functions h i and the inner negation of g if the number of negated h i s is odd. Proof. Let f = g(h 1 (x 1 ) h n (x n )) = g (h 1(x 1 ) h n(x n )), where h 1,..., h n, h 1,..., h n 2 A, g, g A 2, be two representations of the nonconstant quasilinear function f. Assume that h i h i. We observe that h i and h i only depend on the ith variable and the other functions h j, h j (j i) do not depend on the ith variable. Let S i and S i be the subsets of A the characteristic functions of which h i and h i are, respectively; S i S i, by the assumption that h i h i. Suppose that S i S i, and assume without loss of generality that S i \ S i (the other case that S i \ S i is treated similarly). Let b S i S i and c S i \ S i, and let a 1,..., a i 1, a i+1,..., a n A. Denote h(a) = h 1 (a 1 ) h i 1 (a i 1 ) h i (a) h i+1 (a i+1 ) h n (a n ), h (a) = h 1(a 1 ) h i 1(a i 1 ) h i(a) h i+1(a i+1 ) h n(a n ). Since h i (c) = h i (b) = h i (b) h i (c), we also have that h(b) = h(c) and h (b) h (c). However, g (h (b)) = g(h(b)) = g(h(c)) = g (h (c)), but this is possible only if g is a constant function and hence also f is a constant function. We have reached a contradiction. Suppose then that S i S i = and S i S i A. Let b A \ (S i S i ) and c S i, and let a 1,..., a i 1, a i+1,..., a n A. Using the same notation as above, we observe that h i (c) = h i (b) = h i (b) h i (c), and so h(b) = h(c) and h (b) h (c). Using a similar argument, we reach a contradiction also in this case. The only remaining case is that S i S i = and S i S i = A, i.e, S i = A \ S i. Then h i = h i. Since h 1 (x 1 ) h i 1 (x i 1 ) h i (x i ) h i+1 (x i+1 ) h n (x n ) = h 1 (x 1 ) h i 1 (x i 1 ) h i (x i) h i+1 (x i+1 ) h n (x n ) = h 1 (x 1 ) h i 1 (x i 1 ) h i (x i) h i+1 (x i+1) h n (x n ) and g(h) = g in (h), we conclude that the negation of h i can be compensated by the inner negation of g. We note that the negation of the characteristic function of a subset S A is the characteristic function of the complement of S. For any fixed element a A, we can choose all the functions h i such that they are characteristic functions of subsets of A that do not contain a, and this way we achieve unique representations of nonconstant quasilinear functions. We say that the representation f = g(h 1 (x 1 ) h n (x n )) of a nonconstant quasilinear function f is in standard form if h i (0) = 0 for every i = 1,..., n. Standard forms are unique C-decompositions. Let C be a clone. If f = g(φ 1,..., φ m ) for φ 1,..., φ m C, we say that the m + 1-tuple (g, φ 1,..., φ m ) is a C-decomposition of f. We often avoid referring explicitly to the tuple and we simply say that f = g(φ 1,..., φ m ) is a C-decomposition. C-decompositions always exist for all clones C and all functions f, because f = f(x 1,..., x n ) and the projections are members of every clone. We

9 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 9 call a C-decomposition (g, φ 1,..., φ m ) of a nonconstant function f minimal, if the number m of inner functions is the smallest possible among all C-decompositions of f, and we call this smallest number the C-degree of f, denoted deg C f. We agree that the C-degree of a constant function is 0. It is clear that deg C f EAr f for any function f. Lemma 5.2. If f C g then deg C f deg C. Proof. Let g = s(φ 1,..., φ d ) be a minimal C-decomposition. Since f C g, we have that f = g(h 1,..., h m ) for some h 1,..., h m C. Then f = s(φ 1,..., φ d )(h 1,..., h m ) = s(φ 1,..., φ d), where φ i = φ i(h 1,..., h m ) C. Thus, deg C f d. The claim also holds for constant functions, because all C-subfunctions of a constant function are constant. Corollary 5.3. C-equivalent functions have the same C-degree. An m-tuple (m 2) (φ 1,..., φ m ) of n-ary functions is functionally dependent, if there is an (m 1)-ary function g and an i such that φ i = g(φ 1,..., φ i 1, φ i+1,..., φ m ). The tuple (φ 1,..., φ m ) is functionally independent if it is not functionally dependent. We often omit the tuple notation and we simply say that functions φ 1,..., φ m are functionally dependent or independent. Lemma 5.4. In a minimal C-decomposition (g, φ 1,..., φ d ) of f, the inner functions φ 1,..., φ d are functionally independent. Proof. Suppose, on the contrary, that there is a (d 1)-ary function h and an i such that φ i = h(φ 1,..., φ i 1, φ i+1,..., φ d ). Then f = g(φ 1,..., φ d ) = g(x 1,..., x i 1, h, x i,..., x d 1 )(φ 1,..., φ i 1, φ i+1,..., φ d ), a contradiction to the minimality of (g, φ 1,..., φ d ). Any m-tuple (m 2) of functions containing a constant function is clearly functionally dependent, and therefore none of the inner functions of a minimal C- decomposition is a constant function. The following more general statement also holds. Lemma 5.5. If f = s(φ 1,..., φ d ) is a minimal C-decomposition, then for every m 1 and for all subsets S {1,..., d} with S = m i S Ess φ i m. Proof. For the sake of contradiction, assume without loss of generality that m Ess φ i = {1,..., p} i=1 for some p < m. Let q = d m + p, and define the q-ary function s as s = s(φ 1,..., φ m, x p+1,..., x q ),

10 10 ERKKO LEHTONEN where φ i = φ i(x 1,..., x p, ˆ0,..., ˆ0). Then s (x 1,..., x p, φ m+1,..., φ d ) = s(φ 1,..., φ m, x p+1,..., x q )(x 1,..., x p, φ m+1,..., φ d ) = s(φ 1(x 1,..., x p, φ m+1,..., φ d ),..., φ m(x 1,..., x p, φ m+1,..., φ d ), x p+1 (x 1,..., x p, φ m+1,..., φ d ),..., x q (x 1,..., x p, φ m+1,..., φ d )) = s(φ 1,..., φ d ) = f. This contradicts the minimality of the C-decomposition f = s(φ 1,..., φ d ). A minimal C-decomposition (g, φ 1,..., φ d ) of f is called optimal, if the cardinality Im(φ 1,..., φ d ) of the range of the inner functions is the smallest possible among all minimal C-decompositions of f, and this smallest cardinality is called the C-range degree of f, denoted deg r C f. Lemma 5.6. If f C g and deg C f = deg C g, then deg r C f deg r C. Proof. Let deg C f = deg C g = d, deg r C g = r, and let (s, φ 1,..., φ d ) be an optimal C-decomposition of g. We have that f = g(h 1,..., h n ) for some h 1,..., h n C, and so f = s(φ 1,..., φ d )(h 1,..., h n ) = s(φ 1,..., φ d), where φ i = φ i(h 1,..., h n ), and therefore (s, φ 1,..., φ d ) is a minimal C-decomposition of f. Since Im(φ 1,..., φ d ) Im(φ 1,..., φ d ), we have that deg r C f r. Corollary 5.7. C-equivalent functions have the same C-range degree. Lemma 5.8. If (g, φ 1,..., φ m ) is an optimal C-decomposition of f, then for every permutation σ of {1,..., m}, there is a function g such that (g, φ σ(1),..., φ σ(m) ) is an optimal C-decomposition of f. Proof. For any permutation σ of {1,..., d}, s(x σ 1 (1),..., x σ 1 (d))(φ σ(1),..., φ σ(d) ) = s(φ 1,..., φ d ). Thus, if (s, φ 1,..., φ d ) is an optimal C-decomposition of f, then so is (s(x σ 1 (1),..., x σ 1 (d)), φ σ(1),..., φ σ(d) ). 6. B 1 -subfunctions If s = s(ψ 1,..., ψ n ) for some essentially unary functions ψ 1,..., ψ n such that the restriction of (ψ 1,..., ψ n ) to S = Im ψ 1 Im ψ n = Im(ψ 1,..., ψ n ) is the identity function on S, then we say that s retracts to S and we call (ψ 1,..., ψ n ) a retraction map. Lemma 6.1. Assume that f = s(φ 1,..., φ d ) is an optimal B 1 -decomposition. Then there is a function s such that f = s (φ 1,..., φ d ) and s retracts to Im φ 1 Im φ n. Proof. For i = 1,..., d, let ψ i be the essentially unary function defined as { a i, if a i Im φ i, ψ i (a) = φ i (0), if a i / Im φ i, and let s = s(ψ 1,..., ψ d ). Then f = s (φ 1,..., φ d ), s = s (ψ 1,..., ψ d ), Im ψ i = Im φ i for 1 i d, and the restriction of (ψ 1,..., ψ d ) to Im φ 1 Im φ d is the identity function.

11 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 11 In the proof of the next proposition, we will make use of the following theorem, due to Foldes and Lehtonen [3]. Theorem 6.2. Let the columns of a p q matrix M over any field be partitioned into n blocks, M = [M 1,..., M n ] (p n). The following are equivalent. (1) All p p submatrices of M with columns from distinct blocks M i are singular. (2) There is an invertible matrix Q and an integer m 1 such that in QM = [QM 1,..., QM n ], there are m rows which are null in all but at most m 1 blocks QM i. Proposition 6.3. If f = s(φ 1,..., φ d ) is an optimal B 1 -decomposition, then Im(φ 1,..., φ d ) = Im φ 1 Im φ d. Furthermore, if s retracts to Im(φ 1,..., φ d ), then f B1 s. Proof. We call a function f B 1 wide, if it is not quasilinear. A wide function is essentially unary and its range contains at least three elements. Let f = s(φ 1,..., φ d ) be an optimal B 1 -decomposition of an n-ary function f. By Lemma 5.8 we may assume that φ 1,..., φ p are quasilinear and φ p+1,..., φ d are wide. Denote Φ = Im φ 1 Im φ d, w = d p, q = n w = n d + p. Since for any permutation σ of {1,..., n} and for any clone C, f C f(x σ(1),..., x σ(n) ), we may assume that, for i = 1,..., w, Ess φ p+i = {q + i}; let φ p+i = ξ p+i (x q+i ) for a unary function ξ p+i. We may assume that the quasilinear inner functions are in canonical form φ i = h 1 (x 1 ) h n (x n ), where h i (0) = 0 for all i. For, if φ i has the standard form φ i = g(h 1 (x 1 ) h n (x n )), then f = s(φ 1,..., φ i,..., φ d ) = s(φ 1,..., φ i 1, g(h 1 (x 1 ) h n (x n )), φ i+1,..., φ d ) = s (φ 1,..., φ i 1, h 1 (x 1 ) h n (x n ), φ i+1,..., φ d ), where s = s(x 1,..., x i 1, g(x i ), x i+1,..., x d ) and g is any permutation of A whose restriction to {0, 1} coincides with g. Note that s B1 s. Also, for the wide inner functions, we may assume that for p + 1 i d, Im φ i = r i, where r i = Im φ i, and φ i (0) = 0. We can make this condition hold with some suitable permutations σ i of A: f = s(φ 1,..., φ d ) = s(x 1,..., x p, σ 1 p+1 (x p+1),..., σ 1 d (x d))(φ 1,... φ p, σ p+1 φ p+1,..., σ d φ d ) and s B1 s(x 1,..., x p, σp+1 1 (x p+1),..., σ 1 d (x d)). We denote ζ = χ {1}. If s retracts to Φ then we also have that for 1 i p, (1) s(x 1,..., x i 1, ζ(x i ), x i+1,..., x d ) = s. We will now show that Im(φ 1,..., φ d ) = Φ. For i = 1,..., w, let S q+i be a transversal of KER ξ p+i such that 0 S q+i. For i = 1,..., p, we may choose φ i = h i 1(x 1 ) h i n(x n ) such that for j = 1,..., w, h i q+j (a) = 0 for every a S q+j. For, assume that for some 1 i p, 1 j w, a S q+j, we have h i q+j (a) = 1. Denote by [a] the equivalence class of a in KER ξ p+j, denote α = ξ p+j (a), and define h i q+j = hi q+j χ [a], i.e., { h i q+j (x), if x / [a], h i q+j(x) = h i q+j (x) 1, if x [a].

12 12 ERKKO LEHTONEN Then h i q+j (a) = 0. Now let φ i = h i 1(x 1 ) h i q+j 1(x q+j 1 ) h i q+j(x q+j ) = φ i χ [a] (x q+j ), h i q+j+1(x q+j+1 ) h i n(x n ) and let s = s(x 1,..., x i 1, ρ, x i+1,..., x d ) with ρ = ζ(x i ) χ {α} (x p+j ). Then s (φ 1,..., φ i 1, φ i, φ i+1,..., φ d ) = s(x 1,..., x i 1, ρ, x i+1,..., x d )(φ 1,..., φ i 1, φ i, φ i+1,..., φ d ) = s(φ 1,..., φ i 1, ρ(φ 1,..., φ i 1, φ i, φ i+1,..., φ d ), φ i+1,..., φ d ) = f, because ρ(φ 1,..., φ i 1, φ i, φ i+1,..., φ d ) = ζ(φ i) χ {α} (φ p+j ) = ζ(φ i χ [a] (x q+j )) χ {α} (ξ p+j (x q+j )) = φ i χ [a] (x q+j ) χ [a] (x q+j ) = φ i. If s retracts to Φ, then s B1 s, because s (x 1,..., x i 1, ρ, x i+1,..., x d ) = s(x 1,..., x i 1, ρ(x 1,..., x i 1, ρ, x i+1,..., x d ), x i+1,..., x d ) = s, where the last equality holds by Equation (1) and since ρ(x 1,..., x i 1, ρ, x i+1,..., x d ) = ζ(ρ) χ {α} (x p+j ) = ζ(ζ(x i ) χ {α} (x p+j )) χ {α} (x p+j ) = ζ(x i ). Repeating this procedure, we will obtain a B 1 -decomposition f = s( φ 1,..., φ p, φ p+1,..., φ d ), where for every 1 i p, 1 j w, a S q+j, we have h i q+j (a) = 0. In other words, the restrictions of φ 1,..., φ p into A q S q+1 S n do not depend on the variables x q+1,..., x n. Furthermore, the B 1 -decomposition f = s( φ i,..., φ p, φ p+1,..., φ d ) is optimal, and if s retracts to Φ then s B1 s. Thus, we can assume that the quasilinear inner functions have the form described above. We then consider the restrictions of the quasilinear inner functions (φ 1,..., φ p ) to A q S q+1 S n. We may now assume that p i=1 Ess φ i = {1,..., q}. We present a system of p quasilinear functions φ 1,..., φ p in canonical form (φ i = h i 1(x 1 ) h i q(x q )) with p i=1 Ess φ i {1,..., q} as a matrix M over the twoelement field {0, 1} as follows. The rows of M are indexed by {1,..., p}, and the columns are indexed by C = {1,..., q} {1,..., k 1}. We let M(i, (j, a)) = h i j (a). (Note that we are assuming that h i j (0) = 0 for all i and j, so this information need not be encoded in M.) We then partition C into q blocks as Π = {C 1,..., C q }, where C j = {j} {1,..., k 1}. The elementary row operations (permutation of rows, addition of one row to another) correspond to permutation of the φ i s and substitution of φ i φ j for φ i for some i j. The modulo 2 sum of quasilinear functions in canonical form is again a function of this type: φ i φ j = (h i 1 h j 1 )(x 1) (h i n h j n)(x n ) (note that χ S χ S = χ S S, where denotes the symmetric difference). If φ 1,..., φ m are functionally independent then so are also φ 1,..., φ i 1, φ i φ j, φ i+1,..., φ m for i j. Let i j, and define the d-ary function s as s = s(x 1,..., x i 1, τ, x i+1,..., x d ),

13 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 13 where τ = ζ(x i ) ζ(x j ). Note that τ is quasilinear and in canonical form. Then s (φ 1,..., φ i 1, φ i φ j, φ i+1,..., φ d ) = s(φ 1,..., φ d ) = f. If s retracts to Φ, then we also have that s = s (x 1,..., x i 1, τ, x i+1,..., x d ) by Equation (1), so s B1 s. Since φ 1,..., φ p are part of an optimal B 1 -decomposition f = s(φ 1,..., φ d ), it follows by Lemma 5.5 that M does not satisfy condition (2) of Theorem 6.2, for otherwise we would have a B 1 -decomposition f = s (φ 1,..., φ d ), where some m inner functions only depend on m 1 variables, contradicting the optimality of the given B 1 -decomposition. Therefore Theorem 6.2 implies that there is a set D = {(c 1, a 1 ),..., (c p, a p )} of p columns of M that is a partial transversal of Π such that the square submatrix R = M[p, D] is nonsingular. Thus the range of (φ 1,..., φ p ) is the whole of {0, 1} p. Since the wide inner functions φ p+1,..., φ d do not depend on the first q variables, we can now conclude that Im(φ 1,..., φ d ) = Φ. We still have to show that if s retracts to Φ then s B1 f. Assume that s = s(ψ 1,..., ψ d ), where (ψ 1,..., ψ d ) is a retraction map with range Φ. For i = 1,..., q, define the subset S i of A as { {0, a j }, if i = c j, S i = {0}, if i / {c 1,..., c p }. Recall that for i = 1,..., w, S q+i was defined as a transversal of KER ξ p+i. It is not difficult to see that S = S 1 S n is a transversal of KER(φ 1,..., φ d ). Let the p p matrix Q = (q ij ) be the inverse of R. Then the inverse mapping of φ S is δ = (δ 1,..., δ n ), where g i (q j1 x 1 q jp x p ), if i = c j, δ i = ˆ0, if i {1,..., q} \ {c 1,..., c p }, ξ 1 p+j (x q+j), if i = q + j for some 1 j w, where g i is the map 0 0, 1 a j. s = f(δ 1,..., δ n) and δ 1,..., δ n B 1. Letting δ i = δ i(ψ 1,..., ψ p ) we have that Proposition 6.4. If f B1 g, deg B1 f = deg B1 g, deg r B 1 f = deg r B 1 g, then f B1 g. Proof. Let g = s(φ 1,..., φ d ) be an optimal B 1 -decomposition. We may assume that s = s(ψ 1,..., ψ d ) where (ψ 1,..., ψ d ) is a retraction map with Im(ψ 1,..., ψ d ) = Im(φ 1,..., φ d ). Then by Proposition 6.3, g B1 s. We have that f = g(h 1,..., h m ) for some h 1,..., h m B 1, and so f = s(φ 1,..., φ d )(h 1,..., h m ) = s(φ 1,..., φ d ), where φ i = φ i(h 1,..., h m ). This must be an optimal B 1 -decomposition of f with Im(φ 1,..., φ d ) = Im(φ 1,..., φ d ), and again by Proposition 6.3, f B1 s. By the transitivity of B1, we have that f B1 g. Theorem 6.5. There is no infinite descending chain of B 1 -subfunctions. Proof. It follows from Lemma 5.2, Lemma 5.6, and Proposition 6.4 that if f < B1 g, then either deg B1 f < deg B1 g or deg B1 f = deg B1 g and deg r B 1 f < deg r B 1 g. The B 1 -degree and the B 1 -range degree are nonnegative integers, and we cannot have an infinite descent in these parameters.

14 14 ERKKO LEHTONEN 7. Antichains Assume that A = k 4. For n 2, define the n-ary function f n as a 1, if a 1 = = a n k 1, f n (a 1,..., a n ) = k 1, if {i : a i = k 1} = n 1, 0, otherwise. It is clear that Ess f n = {1,..., n}. Proposition 7.1. For n m, f n Bk 2 f m. Proof. We say that a function g B k 2 is narrow if Im g k 2. We say that g B k 2 is wide if Im g > k 2. Note that the wide functions are essentially unary. Assume that n < m. Suppose, on the contrary, that f m Bk 2 f n. Then f m = f n (φ 1,..., φ n ) for some φ 1,..., φ n B k 2. Denote φ = (φ 1,..., φ n ). We must have that for i = 1,..., k 2, φ(i,..., i) = (i,..., i), and so all the inner functions have a range of at least k 2 elements. The range of φ must also contain a vector with exactly n 1 elements equal to k 1. Thus, at least n 1 inner functions have a range of at least k 1 elements and are hence wide. Also, in order to obtain a subfunction of higher essential arity, at least one of the inner functions must be essentially at least binary and hence narrow. We conclude that n 1 inner functions are wide and one is narrow; by symmetry and without loss of generality, we may assume that φ n is the narrow one with Im φ n = {1, 2,..., k 2}. The other inner functions depend on one variable; assume that Ess φ i = {π(i)} for some π : {1,..., n 1} {1,..., m}. Consider the vector v with v π(1) = 1 and v j = k 1 for j π(1). We have that φ(v) = (k 1, k 1,..., k 1, x) for some x k 1, so φ 1(1) = k 1, where φ 1 = φ 1 (x π(1),..., x π(1) ). On the other hand, φ(1,..., 1) = (1,..., 1), so φ 1(1) = 1. We have reached a contradiction. Suppose then, on the contrary, that f n Bk 2 f m. Then f n = f m (φ 1,..., φ m ) for some φ 1,..., φ m B k 2. A similar argument as above shows that at least m 1 inner functions must be wide. Assume without loss of generality that the first m 1 inner functions are wide and Ess φ i = {π(i)} for some π : {1,..., m 1} {1,..., n}. The mapping π must be injective; suppose that π(i) = π(j) for some i j. Let v A m with v π(i) = 1 and v h = k 1 for h π(i). Since φ(1,..., 1) = (1,..., 1), we have that φ(v) is a vector with at least two components equal to 1, but this is a contradiction because φ(v) should be a vector with exactly m 1 components equal to k 1 and hence at most one component equal to 1. If n < m 1, then there is no such injective map π. If n = m 1, then π is a permutation. Denote by e n i the n-vector whose ith component is 1 and the other components are equal to k 1. Since for 1 i k 1, φ(i,..., i) = (i,..., i), we must have that φ(e n π(i) ) = em i. But then also φ m is wide; assume that Ess φ m = {π(m)}. But then φ(e n π(m) ) would be a vector with two components equal to 1, a contradiction. Unfortunately, the previous argument does not apply to B 1 in a three-element base set. We have to treat this case differently. Assume that A = k = 3. For

15 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 15 n 2, define the n-ary function g n as 1, if a 1 = = a n = 1, g n = 2, if {i : a i = 2} = n 1 and {i : a i = 0} = 1, 0, otherwise. Proposition 7.2. For n m, g n B1 g m. Proof. Let n m and suppose, on the contrary, that g m B1 g n. Then g m = g n (φ 1,..., φ n ) for some φ 1,..., φ n B 1. Denote φ = (φ 1,..., φ n ). Denote by vi n the n-vector whose ith component is equal to 0 and the other component are equal to 2. We have that φ(1,..., 1) = (1,..., 1) and φ(vi m) = vn π(i) for some π : {1,..., m} {1,..., n}. The inner functions φ i fall into two types: the quasilinear and the surjective. The surjective functions are essentially unary. We observe that if φ i = ξ(x j ) for some unary surjective function ξ, then ξ(1) = 1 and so ξ is either the identity function on A or the mapping 0 2, 1 1, 2 0. In other words, the surjective inner functions are either projections x j or negations x j = ξ (x j ), where ξ is the latter unary function described above. No projection occurs twice among the inner functions. For, if φ r = φ s = x t for some t and r s, then φ(vt m ) contains (at least) two 0 s, which is not possible. Also, there is at most one negation among the inner functions. For, if φ r = x t, φ s = x u for some r s, then for any l / {t, u}, φ(vl m ) contains (at least) two 0 s, which is not possible. If there are both a projection and a negation among the inner functions, then they depend on the same variable. For, if φ r = x t, φ s = x u for some t u, then φ(vt m ) contains (at least) two 0 s, which is again not possible. So, if there are surjections among the inner functions, then either they are all projections depending on distinct variables; or there are only one projection and one negation, which depend on the same variable; or there is only one negation and no projections. If there is only one negation and no projections, say φ r = x t, then for any l t, φ(vl m ) = vr n. Thus, for any s r, Im φ s = {1, 2}. Now, φ(vt m ) vs n for any s, because φ r (vr m ) = 2 and 0 / Im φ s for s r. This is not possible. If there are a projection and a negation, then assume without loss of generality that φ 1 = x 1, φ 2 = x 1. Then for any l and 3 r n, φ r (vl m ) = 2, and so Im φ r = {1, 2}. Let φ(2,..., 2) = u = (2, 0, u 3,..., u n ) with u j {1, 2} for 3 j n, not all 2. We note that the value of φ i changes from 1 to 2 (or vice versa) if the value of any variable changes from 1 to 2 (or vice versa) if and only if i {j : v j = 1}. It is now easy to see that φ(0, 0, 0, 2, 2,..., 2) = v1 m, a contradiction. Thus, there is no negation among the inner functions. Assume then, without loss of generality, that φ i = x i for 1 i p < n. Since φ(v1 m ) = v1 n, we have that Im φ i = {1, 2} for p + 1 i n. But then φ(vp+1) m vj n for every j, because φ i (vp+1) m = 2 for 1 i p and φ i (vp+1) m 0 for p + 1 i n. This is also an impossible situation. Thus, either all inner functions are projections, or none of them is a projection. If m < n, then the number of distinct projections is less than the number of inner functions. If m > n, then there must be an essentially at least binary inner function in order to incorporate all m essential variables of g m. We have now established that none of the inner functions is surjective, essentially unary. Consider now the only remaining case where all inner functions are quasilinear. Then there is a j such that φ(vi m) = vn j for all i; say j = 1. Let u = φ(2,..., 2),

16 16 ERKKO LEHTONEN and let V = {i : u i = 1}. We observe that the value of φ i changes if the value of any variable changes from 1 to 2 (or vice versa) if and only if i V. It is now easy to see that φ(0, 0, 0, 2, 2,..., 2) = v1 n, a contradiction. Propositions 7.1 and 7.2 can now be merged into one theorem that covers every A = k 3. Theorem 7.3. For every subclone C of B k 2, there is an infinite antichain of C-incomparable functions. 8. Conclusions We have examined the B i -subfunction relations Bi defined by Burle s clones B 0,..., B k on a k-element base set A (k 3). We have determined for 0 i k whether the induced partial order Bi on O A / Bi satisfies the descending chain condition. We have also determined the widths of the posets (O A / Bi, Bi ). In summary, the descending chain condition is not satisfied if and only if 2 i k 2. The width of (O A / Bi, Bi ) is infinite if i k 2 and it is ( k k/2 ) if i = k 1 or i = k. We have also established a simple characterization of the structure of the posets defined by the two largest Burle s clones. The poset (O A / Bk, Bk ) is isomorphic to (P(A) \ { }, ), and the poset (O A / Bk 1, Bk 1 ) is isomorphic to (P(A) \ { }, ) 1. In the special case where A = 2 (Boolean functions), we have three Burle s clones: the clone B 0 of all projections, negations of projections and constant functions; the clone B 1 of all linear functions; and the clone B 2 of all Boolean functions. We have previously shown in [4] that all the three partial orders Bi (0 i 2) satisfy the descending chain condition, the partial orders B0 and B1 contain infinite antichains, and the largest antichain of B2 has 2 elements. Furthermore, the poset (O A / B2, B2 ) is isomorphic to (P(2) \ { }, ). These earlier results for Boolean functions conform with our current results in respect that for every k 2, the partial orders defined by B 0 and B 1 satisfy the descending chain condition but contain infinite antichains; and B k defines a finite poset which is isomorphic to the power set lattice of k with the bottom element removed. References [1] G. A. Burle, The classes of k-valued logics containing all one-variable functions, Diskretnyi Analiz 10 (1967) 3 7 (in Russian). [2] B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, Second edition, Cambridge University Press, Cambridge, [3] S. Foldes, E. Lehtonen, A row-reduced form for column-partitioned matrices, manuscript, Mar. 2006, [4] E. Lehtonen, Order-theoretical analysis of subfunction relations between Boolean functions, manuscript, Apr. 2005, [5] E. Lehtonen, Descending chains and antichains of the unary, linear, and monotone subfunction relations, manuscript, Nov. 2005, [6] A. I. Mal tsev, A strengthening of the theorems of S lupecki and Yablonski, Algebra Logika 6(3) (1967) (in Russian, English summary). [7] N. Pippenger, Galois theory for minors of finite functions, Discrete Math. 254 (2002) [8] I. G. Rosenberg, Completeness properties of multiple-valued logic algebras, in: D. C. Rine (ed.), Computer Science and Multiple-Valued Logic: Theory and Applications, North- Holland, Amsterdam, 1977, pp Second edition, 1984, pp

17 SUBFUNCTION RELATIONS DEFINED BY BURLE S CLONES 17 [9] A. Salomaa, On essential variables of functions, especially in the algebra of logic, Ann. Acad. Sci. Fenn. Ser. A I. Math. 339 (1963) [10] E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928) [11] Á. Szendrei, Clones in Universal Algebra, Séminaire de mathématiques supérieures 99, Les Presses de l Université de Montréal, Montréal, [12] C. Wang, Boolean minors, Discrete Math. 141 (1991) [13] S. V. Yablonski, Functional constructions in a k-valued logic, Tr. Mat. Inst. Steklova 51 (1958) (in Russian). [14] I. E. Zverovich, Characterizations of closed classes of Boolean functions in terms of forbidden subfunctions and Post classes, Discrete Appl. Math. 149 (2005) Institute of Mathematics, Tampere University of Technology, P.O. Box 553, FI Tampere, Finland address: erkko.lehtonen@tut.fi