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On generating sets of the clone of aggregation functions on finite lattices
Information Sciences 476 (2019) 38–47
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On generating sets of the clone of aggregation functions on finite lattices Radomír Halaš a, Radko Mesiar a,b, Jozef Pócs a,c,∗ a
Faculty of Science, Department of Algebra and Geometry, Palacký University Olomouc, 17. listopadu 12, Olomouc 771 46, Czechia Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, Bratislava 1 810 05, Slovakia c Mathematical Institute,Slovak Academy of Sciences, Grešákova 6, Košice 040 01, Slovakia b
a r t i c l e
i n f o
Article history: Received 13 April 2018 Revised 19 August 2018 Accepted 30 September 2018 Available online 3 October 2018 Keywords: (Monotone) clone Monotone function Aggregation function Lattice Capacity Generating set
a b s t r a c t In a recent paper [12] we have shown that aggregation functions on a bounded lattice L form a clone, i.e., the set of functions closed under projections and composition of functions. Moreover, for any finite lattice L we gave a finite set of unary and binary aggregation functions on L from which the aggregation clone is generated. In this paper, a general method for constructing generating sets of the aggregation clone on L is presented. Our approach is based on extending of L-valued capacities leading to so-called full systems of aggregation functions. Several full systems on L are presented (including singleton ones) and their arities are discussed. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Clones acting on non-empty sets belong to basic algebraic structures [5,15] and they capture a lot of properties important in various applications, especially in discrete mathematics and information sciences. For example, many natural problems studied in discrete mathematics and artificial intelligence can be modeled as so-called constraint satisfaction problems (CSP, for short), see e.g., [6]. Although these are usually of a hard complexity, nevertheless at least some of them have been successfully solved by applying a clone theory. In particular, in recent years, clones on bounded chains and on bounded lattices considering monotone functions satisfying boundary conditions, i.e., considering aggregation functions (of an arbitrary arity), are of interest, cf [2,8,17]. So, for example, when considering as the underlying career L = [a, b] a non-trivial interval of reals, then all continuous aggregation functions invariant under transform by means of an arbitrary automorphisms φ : L → L, were characterized by Ovchinnikov and Dukchovny [16] as a polynomial clone of L (for more details on polynomial clones, see Section 2). Note that this clone is isomorphic to the clone of monotone idempotent Boolean functions. Some years later, all continuous aggregation functions on [a, b] invariant under any monotone bijection τ : [a, b] → [a, b] were characterized by Bronevich and Mesiar [4], to be isomorphic to the clone of self-dual monotone idempotent Boolean functions (observe that this clone is generated by the ternary median, see [13]). For some latest studies concerning clones of aggregation functions see e.g., [3,10–12]. ∗ Corresponding author at: Faculty of Science, Department of Algebra and Geometry, Palacký University Olomouc, 17. listopadu 12, Olomouc 771 46, Czechia. E-mail addresses: [email protected] (R. Halaš), [email protected] (R. Mesiar), [email protected] (J. Pócs).
https://doi.org/10.1016/j.ins.2018.09.070 0020-0255/© 2018 Elsevier Inc. All rights reserved.
R. Halaš et al. / Information Sciences 476 (2019) 38–47
39
In [12] we have shown that for lattices with at least three elements, any set of unary aggregation functions together with the lattice operations is not enough to generate the full aggregation clone. Consequently, at least one aggregation function of arity at least two (and different from the lattice operations) must be present in any generating set. The aim of this contribution is a deeper study of particular clones of aggregation functions on finite lattices. We present a general method for constructing generating sets of the aggregation clone. Our approach is based on extending of L-valued capacities, cf [14]., leading to the so-called full systems of aggregation functions. We show that aggregation functions from any full system together with certain unary aggregation functions and the lattice operations already form a generating set of the aggregation clone. In comparison with the result presented in [12], where only one particular type of generating set of the aggregation clone was presented, the method based on full systems allows to find new families of generating sets. This more flexible method also provides a better upper bound for the number of generators of arity at most two than that which has been found in [12]. Moreover, a straightforward modification of the binary case to higher arities is discussed as well. The paper is organized as follows. More details concerning algebraic preliminaries are given in the next section, bringing also several interesting examples. Section 3 contains our main results concerning the generating sets of the aggregation clone, focusing on aggregation functions with arity n = 2. In Section 4, higher arities of aggregation functions in generating sets are considered. Finally, some concluding remarks are added. 2. Algebraic preliminaries Recall that a lattice L is a partially ordered set (a poset, in brief) (L, ≤ ), where for every pair of elements a, b ∈ L there exists their supremum a∨b and infimum a∧b (with respect to the partial order ≤ ). Equivalently, any lattice can be viewed as an algebra (L; ∨, ∧) with two binary operations ∨ and ∧ representing suprema and infima. A lattice L is bounded whenever it has the least element 0 and the greatest element 1, and distributive if it fulfills the distributive identity a ∧ (b ∨ c ) = (a ∧ b) ∨ (a ∧ c ) (or, equivalently, its dual identity), see e.g., [9]. By a sublattice of L is meant any subset M⊆L which is closed under suprema and infima, i.e., a∨b, a∧b ∈ M for all a, b ∈ M. As the intersection of sublattices is again a sublattice, for any subset X of L there is a least sublattice containing X, the so-called sublattice generated by X. Recall that for any non-void subset X of L, the sublattice of L generated by X consists of elements of the form q(x1 , . . . , xn ) for any lattice polynomial q and x1 , . . . , xn ∈ X. A clone on a non-void set A is a set of (finitary) operations on A which contains all the projection operations on A and that is closed with respect to the composition. Projections and composition of functions are formally defined as follows: let A = ∅ be a set and n ∈ N be a positive integer. For any i ≤ n, the ith n-ary projection is for all (x1 , . . . , xn ) ∈ An defined by
pni (x1 , . . . , xn ) := xi . Composition forms from one k-ary operation f: Ak → A and k n-ary operations g1 , . . . , gk : An → A, an n-ary operation f (g1 , . . . , gk ) : An → A defined by
f g1 , . . . , gk (x1 , . . . , xn ) := f g1 (x1 , . . . , xn ), . . . , gk (x1 , . . . , xn ) ,
(1)
for all (x1 , . . . , xn ) ∈ Obviously, for a given set A = ∅, the set consisting of all the projections (of all finite arities) forms a clone on A, which is contained in any other clone. Thus this is the least clone with respect to set inclusion. The greatest one, called the full clone on A, contains all the operations on A. Further, it can be easily seen that the system of all clones on A forms an intersection closed family, i.e., for any indexed system Ci , i ∈ I (I a nonempty index set) of clones on A, their intersection i∈I Ci is again a clone. Hence for any set F of operations defined on A there exist the least clone [F] containing the set F. Let us recall that [F ] = {C : C clone on A, F ⊆ C }. If C is a clone and C = [F ] for some set of operations F, we say that C is generated by F. Consequently, the system of all clones on A forms a complete lattice with respect to set inclusion, where the infimum operation is the intersection, while the supremum of any family of clones is equal to the clone generated by the union of the respective family. Recall that an aggregation function on a bounded lattice L is a function A: Ln → L that An .
(i) is nondecreasing (in each variable), i.e., for any x, y ∈ Ln :
A(x ) ≤ A(y ) whenever x ≤ y, (ii) fulfills the boundary conditions
A ( 0, . . . , 0 ) = 0
and
A ( 1, . . . , 1 ) = 1.
The integer n represents the arity of the aggregation function. It is not hard to show that the following sets of functions form a clone on any bounded lattice L: • Bound(L ), the set of all bounds-preserving functions on L, i.e., f (0, . . . , 0 ) = 0 and f (1, . . . , 1 ) = 1, • Mon(L ), the set of all nondecreasing functions on L, i.e., f(x) ≤ f(y) for any x, y ∈ Ln with x ≤ y, • Pol(L ), the set of all polynomial functions on L, i.e., the clone generated by lattice operations ∧, ∨,
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• WPol(L ), the set of all weighted polynomial functions on L, i.e., the clone generated by lattice operations ∧, ∨ and all the constant functions on L, • Agg (L ), the set of all aggregation functions on L, the so-called aggregation clone of L, • Id(L ), the set of all idempotent functions on L, i.e., f (x, . . . , x ) = x for any x ∈ L, • Sug (L ), the set of all Sugeno integrals on a distributive lattice L. Clearly, for any lattice L we have the inclusions Pol(L ) ⊆ Mon(L ) ∩ Bound(L ) = Agg (L ), and Sug (L ) = Bound(L ) ∩ WPol(L ) ∩ Id(L ). Let us note that certain generating sets of the clones Agg (L ) and Id(L ) ∩ Agg (L ) have been described in [12] and [3], respectively. 3. Generating sets of aggregation clones In what follows, we describe a very general method how to generate the aggregation clone on finite lattices. In this section, we focus on generation using the smallest possible arity of generating functions, i.e., we use unary and binary aggregation functions as generators only. For a set S, denote by P(S) the power set of S. Lemma 3.1. Let f: A → L be a function and fs : A → L, s ∈ S be a non-void family of functions. Then f = s∈S fs if and only if 1) fs ≤ f for all s ∈ S, 2) there is a mapping ϕ : A → P(S) such that s∈ϕ (x ) fs (x ) = f (x ) for all x ∈ A. Dually, f = s∈S fs if and only if 3) fs ≥ f for all s ∈ S, 4) there is a mapping ϕ : A → P(S) such that s∈ϕ (x ) fs (x ) = f (x ) for all x ∈ A. Proof. Obviously, if f = s∈S fs , then fs ≤ f for all s ∈ S. Further, it suffices to put ϕ (x ) = S for each x ∈ A. Then s∈ϕ (x ) fs (x ) = s∈S f s ( x ) = f ( x ). Conversely, assume that the conditions 1) and 2) are fulfilled. Since ϕ (x)⊆S, for each x ∈ A we obtain
f (x ) =
s∈ϕ ( x )
f s (x ) ≤
f s ( x ) ≤ f ( x ).
s∈S
The dual statement can be proved analogously.
In the following statement we describe certain aggregation functions which fit in the realm of the previous lemma. f
Corollary 3.2. Let f: Ln → L be an aggregation function and for each a ∈ Ln let ha : Ln → L, be a function given by
haf
(x ) =
1, if x = (1, . . . , 1 ); f (a ), if x ≥ a, x = (1, . . . , 1 ); 0, if x a.
(2)
f
Then ha is an aggregation function for all a ∈ Ln and
f =
haf .
(3)
a∈Ln
f
Proof. Let a ∈ Ln be an arbitrary element. It can be easily seen that ha is nondecreasing in Ln . Further, for a = (0, . . . , 0 ) it f f follows from (2) that ha (0, . . . , 0 ) = 0, while for a = (0, . . . , 0 ) we have ha (0, . . . , 0 ) = f (a ) = 0 as well, since f is an aggref f gation function. Also the condition ha (1, . . . , 1 ) = 1 follows directly from (2), hence ha satisfies the boundary conditions and it is an aggregation function. f It can be easily seen that ha ≤ f for all a ∈ Ln . Further, let us define ϕ : Ln → P(Ln ) by ϕ (a ) = {a} for each a ∈ Ln . Then the f n system of functions ha , a ∈ L , satisfies the condition 2) of Lemma 3.1 (note that in this case, using notation as in Lemma 3.1, f A = S = Ln ). Hence we obtain f = a∈Ln ha . Dually, the similar arguments show that f = each a ∈ Ln
a∈Ln
f
f
ga , where ga , a ∈ Ln is a system of aggregation functions such that for
R. Halaš et al. / Information Sciences 476 (2019) 38–47
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gaf
0, if x = (0, . . . , 0 ); f (a ), if x ≤ a, x = (0, . . . , 0 ); 1, if x a.
(x ) =
(4)
Let n ≥ 1 be a positive integer. Recall that a set function c : P({1, . . . , n} ) → [0, 1] (here [0,1] is the standard unit interval of reals) is said to be a capacity if c is nondecreasing and fulfills the boundary conditions, i.e., E1 ⊆ E2 implies c(E1 ) ≤ c(E2 ) and c (∅ ) = 0, c ({1, . . . , n} ) = 1, see [7]. Evidently, each capacity c can be seen as an order homomorphism, and it is a generalization of discrete probability measures, where the additivity is relaxed into the monotonicity. A further generalization yields an L-valued capacity (L bounded lattice) c : P({1, . . . , n} ) → L, which is a monotone L-valued set function satisfying the boundary conditions c (∅ ) = 0 (0 being the bottom element of L) and c ({1, . . . , n} ) = 1 (1 being the top element of L). Clearly, the concept of L-valued capacities and O,1-preserving order homomorphisms between lattices P({1, . . . , n} ) and L coincides. We prefer to keep the terminology for c as L-valued capacity due to the links between aggregation functions and capacities. Observe that the notion of n-ary aggregation function (with domain Ln ) can be seen as a “fuzzy” generalization of the notion of capacity. Hence, concerning the binary case, we say that a binary aggregation function f: L2 → L extends a binary capacity c: P({1, 2}) → L, provided f (1, 0 ) = c ({1} ) and f (0, 1 ) = c ({2} ). As one of the fundamental types of generating functions, beside the lattice operations, we use the following unary aggregation functions. For any element a ∈ L we define χ a : L → L by
1, if x ≥ a, x = 0; 0, otherwise.
χa (x ) =
(5)
Dually, define a function μa : L → L by
0, if x ≤ a, x = 1; 1, otherwise.
μa ( x ) =
(6)
The above functions χ a as well as μa , a ∈ L, capture the order structure of the lattice L. For example, for a = 0 the function χ a represents the characteristic function of the principal filter generated by the element a. f
f ga
In the following two lemmas and subsequent remarks we describe one of the possible ways how to obtain functions ha , given by (2) and (4), respectively.
Lemma 3.3. Let g: L2 → L extends the binary capacity c: P({1, 2}) → L with c ({1} ) = a and c ({2} ) = b. Then for any positive integer n ∈ N and x = (x1 , . . . , xn ) ∈ Ln we obtain
Hg1
n
χ0 (xi ),
(x ) = g
n
i=1
i=1
n
n
χ1 (xi ),
i=1
(7)
a, otherwise.
and
Hg2 (x ) = g
0, if x = (0, . . . , 0 ); χ1 (xi ) = 1, if x = (1, . . . , 1 );
0, if x = (0, . . . , 0 ); χ0 (xi ) = 1, if x = (1, . . . , 1 );
(8)
b, otherwise.
i=1
Proof. Obviously, ni=1 χ0 (xi ) = 0 if and only if χ0 (xi ) = 0 for all i ∈ {1, . . . , n}, which holds if and only if xi = 0 for all i ∈ {1, . . . , n}, i.e., x = (0, . . . , 0 ). Similarly, ni=1 χ1 (xi ) = 1 if and only if x = (1, . . . , 1 ). Hence for any x ∈ Ln the pair n
χ0 (xi ),
i=1
n
χ1 (xi )
i=1
gains the three possible values, (0,0), (1,1) or (1,0). The first two possibilities occur only when x is equal to (0, . . . , 0 ) or (1, . . . , 1 ), respectively. From this it can be easily seen that Hg1 (x ) satisfies (7) for each x ∈ Ln . The equality (8) for Hg2 can be proved analogously. Remark 3.4. Composing the functions μa , a ∈ L, given by (6) and g we obtain the functions G1g , G2g : Ln → L where
G1g
n
(x ) = g
i=1
and
μ0 ( x i ) ,
n
i=1
0, if x = (0, . . . , 0 ); μ1 (xi ) = 1, if x = (1, . . . , 1 );
a, otherwise.
(9)
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R. Halaš et al. / Information Sciences 476 (2019) 38–47
G2g
n
(x ) = g
μ1 ( x i ) ,
i=1
n
0, if x = (0, . . . , 0 ); μ0 (xi ) = 1, if x = (1, . . . , 1 );
(10)
b, otherwise.
i=1
For a given a ∈ Ln , a = (0, . . . , 0 ) we put Ja = {i ∈ {1, . . . , n} : ai = 0}. Then
Ha (x ) =
1, if x ≥ a; 0, otherwise.
χai (xi ) =
i∈Ja
(11)
Let us note that the function Ha is considered as an n-ary function, although it need not depend on all the variables. To see (11) it suffices to realize that Ha (x ) = 1 if and only if χai (xi ) = 1 for all i ∈ Ja , i.e., xi ≥ ai for each i ∈ Ja . However ai = 0 for all i ∈ Ja , thus x ≥ a if and only if xi ≥ ai for all i ∈ Ja . Dually, for any a ∈ Ln , a = (1, . . . , 1 ) denoting Jˆa = {1 ≤ i ≤ n : ai = 1}, we obtain
Ga ( x ) =
μa i ( x i ) =
i∈Jˆa
0, if x ≤ a; 1, otherwise.
(12)
Lemma 3.5. Let a ∈ Ln , a = (0, . . . , 0 ), and let f: Ln → L be an aggregation function and g: L2 → L extends a binary capacity c: P({1, 2}) → L with c ({1} ) = f (a ). Then for all x ∈ Ln
haf (x ) = Hg1 (x ) ∧ Ha (x ), where Hg1 is given by (7) and Ha by (11), respectively. If g: L2 → L extends a capacity c with c ({2} ) = f (a ), then
haf (x ) = Hg2 (x ) ∧ Ha (x ), where Hg2 is given by (8). Proof. Assume that g: L2 → L fulfills g(1, 0 ) = c ({1} ) = f (a ). Then, with respect to (7) and (11), we obtain Hg1 (x ) ∧ Ha (x ) = 1 for x = (1, . . . , 1 ), Hg1 (x ) ∧ Ha (x ) = f (a ) for x ≥ a, x = (1, . . . , 1 ), and Hg1 (x ) ∧ Ha (x ) = 0 for xa. This shows that Hg1 (x ) ∧ f
Ha (x ) satisfies (2), i.e., Hg1 (x ) ∧ Ha (x ) = ha (x ).
f
f
Remark 3.6. (i) For a = (0, . . . , 0 ) we obtain either ha (x ) = Hg1 (x ) or ha (x ) = Hg2 (x ), depending on whether f (a ) = g(1, 0 ) = 0 or g(0, 1 ) = 0. (ii) Dually, using the functions G1g , G2g and G defined by (9),(10) and (12), respectively, for any a ∈ Ln , a = (1, . . . , 1 ) we obtain ga (x ) = G1g (x ) ∨ Ga (x ) for g: L2 → L extending a capacity c with c ({1} ) = f (a ) and ga (x ) = G2g (x ) ∨ Ga (x ) for g extending c with c ({2} ) = f (a ). f f (iii) Similarly as in the case (i), for a = (1, . . . , 1 ) we obtain ga (x ) = G1g (x ) or ga (x ) = G2g (x ) depending on whether f (a ) = g(1, 0 ) = 1 or g(0, 1 ) = 1. f
f
As we have shown, the crucial role play binary functions and their values at boolean inputs. In order to generate all aggregation functions, one needs a system of functions attaining a sufficient amount of values at boolean elements. This leads to the following definition of a full system on L. Definition 3.7. Let L be a bounded lattice. We say that a system G = {gi : L2 → L : i ∈ I} of binary aggregation functions is full on L, if the set of elements M = {gi (1, 0 ) : i ∈ I} ∪ {gi (0, 1 ) : i ∈ I} generates L as a lattice, i.e., the smallest sublattice containing M equals to L. Lemma 3.8. Let G = {gi : i ∈ I} be a full system of binary aggregation functions on a bounded lattice L. Then for any a ∈ L there is a binary aggregation function g: L2 → L such that g belongs to the clone generated by the set G ∪ {∨, ∧} and g(1, 0 ) = a. Proof. Denote M = {g(1, 0 ) : g ∈ G } ∪ {g(0, 1 ) : g ∈ G } and let C be the clone generated by G ∪ {∨, ∧}. Since G is full, the set M generates the lattice L. Consequently, each element of a ∈ L can be expressed in the form a = q(a1 , . . . , an ) for some {a1 , . . . , an } ⊆ M and an n-ary lattice polynomial q. For each index j ∈ {1, . . . , n} let gi j ∈ G denotes the function with gi j (1, 0 ) = a j or gi j (0, 1 ) = a j . Further, we put
f j (x1 , x2 ) = gi j (x1 , x2 ) if gi j (1, 0 ) = a j and f j (x1 , x2 ) = gi j ( p22 (x1 , x2 ), p21 (x1 , x2 )) = gi j (x2 , x1 ) provided gi j (0, 1 ) = a j . Note
that f j (1, 0 ) = a j holds for all j ∈ {1, . . . , n}. As the functions fj , j = 1, . . . , n, as well as the function q belong to the clone C, it follows that the composition g = q( f1 , . . . , fn ) given by (1) is a member of C, too. Then for the function g ∈ C the following holds
g(1, 0 ) = q f1 , . . . , fn (1, 0 ) = q f1 (1, 0 ), . . . , fn (1, 0 ) = q(a1 , . . . , an ) = a. Let us mention that using dual arguments, one can show that for any a ∈ L there is a function g ∈ C (C clone generated by the set G ∪ {∨, ∧}) with g (0, 1 ) = a.
R. Halaš et al. / Information Sciences 476 (2019) 38–47
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Recall that an element a ∈ L is join-irreducible if a = 0 and a = b ∨ c yields a = b or a = c. The notion of meet-irreducible element is defined dually. For a finite lattice L, J(L) and M(L) denotes the set of all join-irreducible and meet-irreducible elements, respectively. Obviously, for any 0 = a ∈ L, a = X holds for some X⊆J(L). Similarly, b = 1 implies that b = X for some X⊆M(L). Now applying all particular results, we obtain the following statement which describes a relatively wide family of generating sets of the aggregation clone. Theorem 3.9. Let L be a finite bounded lattice and let G be a full system of binary aggregation functions on L. Then both the sets Gχ = G ∪ {χa : a ∈ J(L )} ∪ {∧, ∨} and Gμ = G ∪ {μb : b ∈ M(L )} ∪ {∧, ∨} generate the aggregation clone Agg (L ). Proof. Let C be the clone generated by the set Gχ . We prove that C = Agg (L ). First, assume that b = c ∨ d in L. Then χb (x ) = χc (x ) ∧ χd (x ) for all x ∈ L. Indeed χc (x ) ∧ χd (x ) = 1 if and only χc (x ) = 1 and χd (x ) = 1 which is equivalent to x ≥ c and x ≥ d, i.e., x ≥ c∨d. As every non-zero element is a supremum of join irreducible elements, this shows that χ b ∈ C for all b ∈ L, b = 0. For b = 0 it can be easily seen that χ0 = a∈At(L ) χa , where the supremum is taken over the set of all atoms of L. Hence χ 0 ∈ C as well. f According to Corollary 3.2, any n-ary f ∈ Agg (L ) can be expressed as the finite supremum f = a∈Ln ha , hence it suffices f
to show that ha ∈ C for all a ∈ Ln . Due to Lemma 3.8, for each a ∈ Ln there is a binary aggregation function g ∈ C such that f f g(1, 0 ) = f (a ). Further, from Lemma 3.5 and Remark 3.6 we have ha = Hg1 ∧ Ha for a = (0, . . . , 0 ) and ha (x ) = Hg1 (x ) for 1 n a = (0, . . . , 0 ). However, expressions (7) and (11) yield Hg ∈ C and also Ha ∈ C for all a ∈ L , a = (0, . . . , 0 ). Hence we obtain C = Agg (L ). The fact that Gμ generates the aggregation clone Agg (L ) can be proved by using the dual arguments. Corollary 3.10. Let L be an n-element lattice and let T be a generating set of L with the minimal cardinality. Then Agg (L ) can be generated by the set S consisting of certain unary and binary aggregation functions, where
3n |T | +2≤ + 1. |S| ≤ min |J(L )|, |M(L )| +
2
(13)
2
2 and Gμ = |G | + |M(L )| + 2. Hence it suffices to show that given a generating set T of L, there exists a full system G of T cardinality |2| , where · denotes the ceiling function.
Proof. According to the previous theorem, the sets Gχ and Gμ generate the aggregation clone. Obviously Gχ = |G | + |J(L )| +
For this, let a, b ∈ L be arbitrary elements. We put
ga,b (x1 , x2 ) = (x1 ∧ a ) ∨ (x2 ∧ b) ∨ (x1 ∧ x2 ). Using direct computations, it can be easily seen that ga,b (1, 0 ) = a and ga,b (0, 1 ) = b. Consequently, we can take any two subsets A = {a1 , . . . , am } ⊆ T , B = {b1 , . . . , bm } ⊆ T with m =
|T | 2
, A ∪ B = T and put
G = {gai ,bi : 1 ≤ i ≤ m}. Evidently, G represents a full system, which proves the first inequality of (13). Obviously, |J(L )| ≤ n − 1, |M(L )| ≤ n − 1 and |T| ≤ n for any n-element lattice L. Thus we obtain
n 3n |T | min |J(L )|, |M(L )| + +2≤n−1+ +2= + 1.
2
2
2
Observe that for an n-element chain L we have the equality |J(L )| = |M(L )| = n − 1 and |T | = n, since L can be generated by itself only. Hence in this case, the second inequality of (13) is in fact the equality. Let us remark that the inequality (13) improves the result from [12], where the generating set having 2n + 2 elements was described. Further, for any finite lattice L, the set J(L) eventually together with the bottom element (in the case that L possesses the only atom, otherwise 0 can be omitted) generates L. Similarly, M(L) with the top element generates L. Hence from (13) we obtain that Agg (L ) can be generated by the set S, consisting of functions of arity at most 2, and satisfying
|S| ≤ min
3|J(L )| + 1 , 2
3|M(L )| + 1 2
+2
At the end of this section we provide two examples illustrating the obtained results for particular lattices. Example 3.11. Consider a finite distributive lattice L. Observe that the functions ga, b used in the proof of Corollary 3.10 given by
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R. Halaš et al. / Information Sciences 476 (2019) 38–47
ga,b (x1 , x2 ) =
m (I ) ∧
I⊆{1,2}
xi
i∈I
coincide with binary Sugeno integrals determined by the capacity m: P({1, 2}) → L with m({1} ) = a, m({2} ) = b. Moreover, we have
g0,0 ( x1 , x2 ) = ( x1 ∧ 0 ) ∨ ( x2 ∧ 0 ) ∨ ( x1 ∧ x2 ) = x1 ∧ x2 and
g1,1 ( x1 , x2 ) = ( x1 ∧ 1 ) ∨ ( x2 ∧ 1 ) ∨ ( x1 ∧ x2 ) = x1 ∨ x2 . Hence, as a consequence of Theorem 3.9 we obtain the following corollary. Corollary 3.12. The aggregation clone Agg (L ) on a finite distributive lattice L is generated by the set consisting of the functions χ a , a ∈ J(L) (or functions μb , b ∈ M(L)), and any full system of binary Sugeno integrals (which must contain the two Sugeno integrals g0,0 and g1,1 extending the smallest and the greatest capacity on L, respectively). Observe that the expression (3) of Corollary 3.2 is valid for any aggregation function on an arbitrary lattice. Hence, it can be used to express a given aggregation function by means of a full system also in the case of an infinite lattice, although the infinitary operation of supremum must be involved. To illustrate this, we present an example concerning aggregation functions defined on the unit interval of reals. Example 3.13. Let L = [0, 1] be the unit interval of reals. It is obvious that the only generating set (by means of finitary lattice operations) of L is L itself. One of the full systems on L is given by binary Sugeno integrals, similarly as in Example 3.11. We present another binary full system of this lattice. For each a ∈ [0, 12 ] denote by ga : [0, 1]2 → [0, 1] the weighted arithmetic mean
ga ( x1 , x2 ) = a · x1 + ( 1 − a ) · x2 . Obviously, the system {ga : a ∈ [0, 12 ]} is full on L. Using the presented full system, we provide the formula (3) for the function of binary multiplication. According to Lemma 3.5, for (a1 , a2 ) ∈ [0, 1]2 the function ha1 ·a2 (we omit the lower case (a1 , a2 )) has the expression
⎧
1 ⎪ g ai , if 0 ≤ a1 · a2 ≤ , χ (x ) ∨ χ0 (x2 ), χ1 (x1 ) ∧ χ1 (x2 ) ∧ ⎪ ⎨ a1 ·a2 0 1 2 i∈J(a ,a ) 1 2 ha1 ·a2 (x1 , x2 ) =
1 ⎪ ⎪ ai , if < a1 · a2 ≤ 1. ⎩ga1 ·a2 χ1 (x1 ) ∧ χ1 (x2 ), χ0 (x1 ) ∨ χ0 (x2 ) ∧ i∈J(a
Hence, we obtain
x1 · x2 =
ha1 ·a2 (x1 , x2 ) =
(a1 ,a2 )∈[0,1]2
=
a1 · a2 ·
(a1 ,a2 )∈[0,1]2 a1 ·a2 ≤ 12
∨
ha1 ·a2 (x1 , x2 ) ∨
2
ha1 ·a2 (x1 , x2 )
(a1 ,a2 )∈[0,1]2 1 2
ai χ0 (x1 ) ∨ χ0 (x2 ) + (1 − a1 · a2 ) χ1 (x1 ) ∧ χ1 (x2 ) ∧
(a1 ,a2 )∈[0,1]2 a1 ·a2 ≤ 12
1 ,a2 )
i∈J(a
a1 · a2 ·
1 ,a2 )
ai . χ1 (x1 ) ∧ χ1 (x2 ) + (1 − a1 · a2 ) χ0 (x1 ) ∨ χ0 (x2 ) ∧
(a1 ,a2 )∈[0,1]2 1 2
i∈J(a
1 ,a2 )
4. Generation with higher arities In what follows we extend the results from the previous section. We focus on lowering the number of generating functions of the aggregation clone by using of functions with higher than binary arity. Given a bounded lattice L, for a subset I ⊆ {1, . . . , n} = [n], n ≥ 1, we denote the (L-valued) characteristic function of I by 1I , i.e., 1I ∈ Ln and for all i ∈ [n], 1I (i ) = 1 if i ∈ I and 1I (i ) = 0 for i ∈ I. In accordance with Definition 3.7 we say that a system of aggregation functions G = {g j : Ln j → L : j ∈ J} is full on a bounded lattice L, if the set M = j∈J {g j (1I ) : I ⊆ [n j ]} generates the lattice L. Similarly to the case of generation of the aggregation clone by a binary full system, we show that the same result can be obtained by using of arbitrary full system of aggregation functions. Theorem 4.1. Let L be a finite lattice and G be a full system of aggregation functions on L. Then both the sets Gχ = G ∪ {χa : a ∈ J(L )} ∪ {∧, ∨} and Gμ = G ∪ {μb : b ∈ M(L )} ∪ {∧, ∨} generate the aggregation clone Agg (L ).
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45
Proof. We show that for a clone C generated by the set Gχ , the equality C = Agg (L ) holds. Let n ≥ 1 be fixed and b ∈ L be such that b = g(1I ) for some g: Lk → L, g ∈ G and I⊆[k]. For any x = (x1 , . . . , xn ) ∈ Ln we put
HgI (x1 , . . . , xn ) = g y1 (x ), . . . , yk (x ) , where the functions yi : Ln → L, i ∈ [k] are given by stipulation
yi ( x1 , . . . , xn ) =
⎧ n ⎪ ⎪ ⎨ χ0 (xl ), if i ∈ I; l=1
n
⎪ ⎪ ⎩ χ1 (xl ), if i ∈/ I. l=1
It can be easily seen that for any x ∈ Ln , (0, . . . , 0 ) = x = (1, . . . , 1 ), yi (x ) = 1 if i ∈ I and yi (x ) = 0 otherwise. Hence in this case y1 (x ), . . . , yk (x ) = 1I and we obtain
HgI
0, if x = (0, . . . , 0 ); (x1 , . . . , xn ) = g y1 (x ), . . . , yk (x ) = 1, if x = (1, . . . , 1 );
b, otherwise.
Obviously HgI ∈ C as the function HgI is a composition of functions belonging to C. Note that concerning the functions χ 0 and χ 1 respectively, we use the results from the proof of Theorem 3.9. Further, let f: Ln → L be an arbitrary aggregation function. As G is a full system, for any a ∈ Ln there exist a system of m functions g j : Ln j → L, g j ∈ G, j = 1, . . . , m, an m-ary lattice polynomial q with q(a1 , . . . , am ) = f (a ), where for all j = 1, . . . , m, a j = g j (1I j ) for some Ij ⊆[nj ]. Consequently, for x ∈ Ln we obtain
haf (x ) = Ha (x ) ∧ q HgI11 (x ), . . . , HgImm (x ) , I
where the function Ha is given by (11). Since all the functions Ha , q and Hgjj , j = 1, . . . , m, belong to the clone C, it follows f
that ha ∈ C as well. According to Corollary 3.2, this yields f ∈ C, i.e., C = Agg (L ). The fact that the set Gμ generates the clone Agg (L ) can be proved analogously.
Allowing higher than binary arities for functions in a full system, it is natural to ask whether singleton full systems can exist and what are the arities of aggregation functions belonging to such systems. In the following theorem we answer this problem positively, and we also give an upper bound for the corresponding arities. Theorem 4.2. Let L be a finite lattice and T be a generating set of L with a minimal cardinality. Then there exists a singleton full system G = {g} on L, where g is a k-ary aggregation function with
k = min n : |T | ≤
nn .
(14)
2
Proof. At first we show that any L-valued capacity can be extended to an aggregation function. Let m: P([n]) → L be an L-valued capacity defined on the power set of [n] = {1, . . . , n}, n ≥ 1. Consider the following function gm : Ln → L given by
gm ( x1 , . . . , xn ) =
m (I ) ∧
xi
(15)
i∈I
I⊆[n]
for all x1 , . . . , xn ∈ L. We show that the function gm extends the capacity m, i.e., gm (1I ) = m(I ) for all subsets I⊆[n]. Let I⊆[n] be a fixed subset. Then
gm ( 1I ) =
J ⊆[n],J ⊆I
m (J ) ∧
j∈J
1I ( j )
∨
J ⊆[n],J I
m (J ) ∧
j∈J
1I ( j ) =
J ⊆[n],J ⊆I
m (J ) ∨
0 = m ( I ),
J ⊆[n],J I
since 1I ( j ) = 1 for all j ∈ J⊆I and 1I ( j ) = 0 for some j ∈ JI. In what follows we find a capacity m, range of which contains the set T. In the most general case, elements of T can form an antichain in the lattice L. According to Sperner’s theorem, cf. eg [1]., for any family F of subsets of an n-element set such that XY for all distinct X, Y ∈ F (F forms an antichain of subsets of an n-element set), the following inequality holds:
46
R. Halaš et al. / Information Sciences 476 (2019) 38–47
n . |F | ≤ n 2
Note that the family of all n2 -element subsets on an n-set forms an antichain and that for this family the equality is attained. Now let k be a positive integer given by (14). Then it is possible to define a capacity m: P([k]) → L in such a way that for all X⊆[k]
⎧ ⎨= 0, if |X | < 2k , m(X ) ∈ T , if |X | = 2k , ⎩ = 1, if |X | > 2k ,
while T ⊆ {m(X ) : |X | > 2k }. With respect to the first part of the proof, the k-ary function gm given by (15), which extends m, forms a full system on the lattice L. Corollary 4.3. Let L be a finite lattice n-element lattice. Then the aggregation clone Agg (L ) can be generated by the set consisting of the lattice operations, min {|J(L)|, |M(L)|} unary aggregation functions and one k-ary aggregation function, where k satisfies (14). Let us mention that the presented upper bound k for arities applies for arbitrary finite bounded lattices. However, if the order structure of an underlying lattice is known, this upper bound can be lowered. In the following example we illustrate this fact in the case of a finite chain. Note that if L is a finite chain, L is the only generating set of L. Example 4.4. Let L be an n-element chain L = {a1 , . . . , an }, n ≥ 1, where a1 < < an , and k be a positive integer such that 2k ≥ n. Consider the cardinality-lexicographic order of P([k]), i.e., for X, Y⊆[k] we put X < Y provided |X| < |Y| or |X | = |Y | and min (XY) < min {YX}. Note that with respect to this ordering, P([k]) becomes a linearly ordered set. Let X1 , . . . , X2k be the enumeration of P([k]) such that X1 < · · · < X2k . Define a capacity m: P([k]) → L by m(Xi ) = ai , i = 1, . . . , 2k , where ai = an for all i ≥ n. From the definition of < it can be easily seen that X⊆Y implies X ≤ Y, which yields that m is a monotone set function on P([k]). Since a1 , a2k is the bottom and the top element of L respectively, we obtain that m is a capacity. Then the singleton {gm }, gm given by (15), forms a full system on L. Consequently, the chain L can be generated by the set {∨, ∧} ∪ {χai : 2 ≤ i ≤ n} ∪ {gm }, where gm is log2 n-ary aggregation function. In the proof of Theorem 4.2 we have presented aggregation functions (15) extending an arbitrary capacity m. In the following example we give another type of aggregation functions satisfying this property. Consequently, such functions can be included in an appropriate full system. Example 4.5. Let n ≥ 1 be a positive integer and m: P([n]) → L be an L-valued capacity. For x = (x1 , . . . , xn ) ∈ Ln put Ix = {i ∈ [n] : xi = 1}. Further we can define fm : Ln → L by
fm (x ) = m(Ix ) for all x ∈ Ln . It can be easily seen that x1 ≤ x2 implies Ix1 ⊆ Ix2 , which due to the monotonicity of m yields fm (x1 ) ≤ fm (x2 ). Since I(0,...,0 ) = ∅ and I(1,...,1 ) = [n], the function fm fulfills the boundary conditions as well, i.e., it is an aggregation function. Obviously, fm extends the capacity m, as I1X = X for any subset X⊆[n]. Observe that the ranges of fm and m coincide, thus fm is the smallest aggregation function extending the capacity m. Finally, omitting the finiteness requirement, in the following example we provide a singleton full system for the real unit interval. Example 4.6. Consider the lattice L = [0, 1]. Since an n-ary capacity attains finite number of values only, it is obvious that none of the aggregation functions of finite arity itself forms a full system of L. Hence, any singleton singleton full system of L possesses an infinitary function. We show that it is possible to find an aggregation function involving ℵ0 = |N| arguments, such that it forms a full system of L. To see this, recall that the set N of all positive integers and Q[0,1] of all rationals between 0 and 1 have the same cardinality, i.e., there is a bijection between them. Due to this well-known fact it is sufficient to find an appropriate capacity on the power set of Q[0,1] . Define a set function m : P(Q[0,1] ) → [0, 1] in the following way: m(∅ ) = 0 and m({q} ) = q for any q ∈ Q[0,1] . Further, for ∅ = X ⊆ Q[0,1] we put
m (X ) =
q.
q∈X
It can be easily seen, that the function m fulfills the boundary conditions and it is monotone, i.e., it is a capacity. Moreover, as any real number c ∈ [0, 1] can be expressed as a supremum of some rationals, it follows that for the range of m the
R. Halaš et al. / Information Sciences 476 (2019) 38–47
47
equality Rng (m ) = [0, 1] holds. Consequently, any (ℵ0 -ary) aggregation function extending m, e.g., the infinitary analogue of the function fm from the previous example, can be seen as a singleton full system of L. 5. Conclusion In this paper, a new general method for searching generating sets of aggregation clones on finite lattices has been presented. As the main tool we developed the so-called full systems of aggregation functions which together with certain unary aggregation functions and lattice operations generate the clone of aggregation functions. We have also discussed the existence and minimal arity of singleton full systems of functions. We believe that our results will be convenient for further analysis of aggregation functions on finite lattices and also for the development of other special classes of aggregation functions. Acknowledgments ˇ The first author was supported by the project of Grant Agency of the Czech Republic (GACR) no. 18-06915S and by the ˇ no. project MSMT Mobility 7AMB17AT054; the second author by the project of Grant Agency of the Czech Republic (GACR) 18-06915S and by the Slovak VEGA Grant 1/0420/15; the third author by the IGA project of the Faculty of Science Palacký University Olomouc PrF2018012 and by the Slovak Research and Development Agency under the contract No. APVV-16-0073. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
I. Anderson, Combinatorics of Finite Sets, Dover Publications, Mineola, 2011. G. Beliakov, A. Pradera, T. Calvo, Studies fuzziness on soft computing, Aggregation Functions: A Guide for Practitioners, 221, Springer, 2007. M. Botur, R. Halaš, R. Mesiar, J. Pócs, On generating of idempotent aggregation functions on finite lattices, Inf. Sci. (NY) 430–431 (2018) 39–45. A. Bronevich, R. Mesiar, Invariant continuous aggregation functions, Int. J. Gen. Syst. 39 (2) (2010) 177–188. S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, 1981. K. Ghédira, B. Dubuisson, Constrained Satisfaction Problems, Wiley, 2013. M. Grabisch, Set Functions, Games and Capacities in Decision Making, Springer Verlag, Berlin, 2016. M. Grabisch, J.L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, Cambridge, 2009. G. Grätzer, Lattice Theory: Foundation, Birkhäuser, Basel, 2011. R. Halaš, R. Mesiar, J. Pócs, Generators of aggregation functions and fuzzy connectives, IEEE Trans. Fuzzy Syst. 24 (6) (2016) 1690–1694. R. Halaš, J. Pócs, On lattices with a smallest set of aggregation functions, Inf. Sci. (NY) 325 (2015) 316–323. R. Halaš, J. Pócs, On the clone of aggregation functions on bounded lattices, Inf. Sci. (NY) 329 (2016) 381–389. S.W. Jablonskii, G.P. Gawrilov, W.B. Kudrjavcev, Boolesche funktionen und postsche klassen, Braunschweig, Vieweg, 1970. E.P. Klement, R. Mesiar, On the expected value of fuzzy events, Int. J. Uncertain. Fuzziness Knowl. Based Syst. 23 (2015) 57–74. D. Lau, Function Algebras on Finite Sets, Springer-Verlag, Berlin, 2006. S. Ovchinnikov, A. Dukchovny, Integral representation of invariant functionals, J. Math. Anal. Appl. 244 (1) (20 0 0) 228–232. V. Torra, Y. Narukawa, Modeling Decisions: Information Fusion and Aggregation Operators, Springer-Verlag, 2007.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
On generating sets of the clone of aggregation functions on finite lattices Radomír Halaš a, Radko Mesiar a,b, Jozef Pócs a,c,∗ a
Faculty of Science, Department of Algebra and Geometry, Palacký University Olomouc, 17. listopadu 12, Olomouc 771 46, Czechia Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, Bratislava 1 810 05, Slovakia c Mathematical Institute,Slovak Academy of Sciences, Grešákova 6, Košice 040 01, Slovakia b
a r t i c l e
i n f o
Article history: Received 13 April 2018 Revised 19 August 2018 Accepted 30 September 2018 Available online 3 October 2018 Keywords: (Monotone) clone Monotone function Aggregation function Lattice Capacity Generating set
a b s t r a c t In a recent paper [12] we have shown that aggregation functions on a bounded lattice L form a clone, i.e., the set of functions closed under projections and composition of functions. Moreover, for any finite lattice L we gave a finite set of unary and binary aggregation functions on L from which the aggregation clone is generated. In this paper, a general method for constructing generating sets of the aggregation clone on L is presented. Our approach is based on extending of L-valued capacities leading to so-called full systems of aggregation functions. Several full systems on L are presented (including singleton ones) and their arities are discussed. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Clones acting on non-empty sets belong to basic algebraic structures [5,15] and they capture a lot of properties important in various applications, especially in discrete mathematics and information sciences. For example, many natural problems studied in discrete mathematics and artificial intelligence can be modeled as so-called constraint satisfaction problems (CSP, for short), see e.g., [6]. Although these are usually of a hard complexity, nevertheless at least some of them have been successfully solved by applying a clone theory. In particular, in recent years, clones on bounded chains and on bounded lattices considering monotone functions satisfying boundary conditions, i.e., considering aggregation functions (of an arbitrary arity), are of interest, cf [2,8,17]. So, for example, when considering as the underlying career L = [a, b] a non-trivial interval of reals, then all continuous aggregation functions invariant under transform by means of an arbitrary automorphisms φ : L → L, were characterized by Ovchinnikov and Dukchovny [16] as a polynomial clone of L (for more details on polynomial clones, see Section 2). Note that this clone is isomorphic to the clone of monotone idempotent Boolean functions. Some years later, all continuous aggregation functions on [a, b] invariant under any monotone bijection τ : [a, b] → [a, b] were characterized by Bronevich and Mesiar [4], to be isomorphic to the clone of self-dual monotone idempotent Boolean functions (observe that this clone is generated by the ternary median, see [13]). For some latest studies concerning clones of aggregation functions see e.g., [3,10–12]. ∗ Corresponding author at: Faculty of Science, Department of Algebra and Geometry, Palacký University Olomouc, 17. listopadu 12, Olomouc 771 46, Czechia. E-mail addresses: [email protected] (R. Halaš), [email protected] (R. Mesiar), [email protected] (J. Pócs).
https://doi.org/10.1016/j.ins.2018.09.070 0020-0255/© 2018 Elsevier Inc. All rights reserved.
R. Halaš et al. / Information Sciences 476 (2019) 38–47
39
In [12] we have shown that for lattices with at least three elements, any set of unary aggregation functions together with the lattice operations is not enough to generate the full aggregation clone. Consequently, at least one aggregation function of arity at least two (and different from the lattice operations) must be present in any generating set. The aim of this contribution is a deeper study of particular clones of aggregation functions on finite lattices. We present a general method for constructing generating sets of the aggregation clone. Our approach is based on extending of L-valued capacities, cf [14]., leading to the so-called full systems of aggregation functions. We show that aggregation functions from any full system together with certain unary aggregation functions and the lattice operations already form a generating set of the aggregation clone. In comparison with the result presented in [12], where only one particular type of generating set of the aggregation clone was presented, the method based on full systems allows to find new families of generating sets. This more flexible method also provides a better upper bound for the number of generators of arity at most two than that which has been found in [12]. Moreover, a straightforward modification of the binary case to higher arities is discussed as well. The paper is organized as follows. More details concerning algebraic preliminaries are given in the next section, bringing also several interesting examples. Section 3 contains our main results concerning the generating sets of the aggregation clone, focusing on aggregation functions with arity n = 2. In Section 4, higher arities of aggregation functions in generating sets are considered. Finally, some concluding remarks are added. 2. Algebraic preliminaries Recall that a lattice L is a partially ordered set (a poset, in brief) (L, ≤ ), where for every pair of elements a, b ∈ L there exists their supremum a∨b and infimum a∧b (with respect to the partial order ≤ ). Equivalently, any lattice can be viewed as an algebra (L; ∨, ∧) with two binary operations ∨ and ∧ representing suprema and infima. A lattice L is bounded whenever it has the least element 0 and the greatest element 1, and distributive if it fulfills the distributive identity a ∧ (b ∨ c ) = (a ∧ b) ∨ (a ∧ c ) (or, equivalently, its dual identity), see e.g., [9]. By a sublattice of L is meant any subset M⊆L which is closed under suprema and infima, i.e., a∨b, a∧b ∈ M for all a, b ∈ M. As the intersection of sublattices is again a sublattice, for any subset X of L there is a least sublattice containing X, the so-called sublattice generated by X. Recall that for any non-void subset X of L, the sublattice of L generated by X consists of elements of the form q(x1 , . . . , xn ) for any lattice polynomial q and x1 , . . . , xn ∈ X. A clone on a non-void set A is a set of (finitary) operations on A which contains all the projection operations on A and that is closed with respect to the composition. Projections and composition of functions are formally defined as follows: let A = ∅ be a set and n ∈ N be a positive integer. For any i ≤ n, the ith n-ary projection is for all (x1 , . . . , xn ) ∈ An defined by
pni (x1 , . . . , xn ) := xi . Composition forms from one k-ary operation f: Ak → A and k n-ary operations g1 , . . . , gk : An → A, an n-ary operation f (g1 , . . . , gk ) : An → A defined by
f g1 , . . . , gk (x1 , . . . , xn ) := f g1 (x1 , . . . , xn ), . . . , gk (x1 , . . . , xn ) ,
(1)
for all (x1 , . . . , xn ) ∈ Obviously, for a given set A = ∅, the set consisting of all the projections (of all finite arities) forms a clone on A, which is contained in any other clone. Thus this is the least clone with respect to set inclusion. The greatest one, called the full clone on A, contains all the operations on A. Further, it can be easily seen that the system of all clones on A forms an intersection closed family, i.e., for any indexed system Ci , i ∈ I (I a nonempty index set) of clones on A, their intersection i∈I Ci is again a clone. Hence for any set F of operations defined on A there exist the least clone [F] containing the set F. Let us recall that [F ] = {C : C clone on A, F ⊆ C }. If C is a clone and C = [F ] for some set of operations F, we say that C is generated by F. Consequently, the system of all clones on A forms a complete lattice with respect to set inclusion, where the infimum operation is the intersection, while the supremum of any family of clones is equal to the clone generated by the union of the respective family. Recall that an aggregation function on a bounded lattice L is a function A: Ln → L that An .
(i) is nondecreasing (in each variable), i.e., for any x, y ∈ Ln :
A(x ) ≤ A(y ) whenever x ≤ y, (ii) fulfills the boundary conditions
A ( 0, . . . , 0 ) = 0
and
A ( 1, . . . , 1 ) = 1.
The integer n represents the arity of the aggregation function. It is not hard to show that the following sets of functions form a clone on any bounded lattice L: • Bound(L ), the set of all bounds-preserving functions on L, i.e., f (0, . . . , 0 ) = 0 and f (1, . . . , 1 ) = 1, • Mon(L ), the set of all nondecreasing functions on L, i.e., f(x) ≤ f(y) for any x, y ∈ Ln with x ≤ y, • Pol(L ), the set of all polynomial functions on L, i.e., the clone generated by lattice operations ∧, ∨,
40
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• WPol(L ), the set of all weighted polynomial functions on L, i.e., the clone generated by lattice operations ∧, ∨ and all the constant functions on L, • Agg (L ), the set of all aggregation functions on L, the so-called aggregation clone of L, • Id(L ), the set of all idempotent functions on L, i.e., f (x, . . . , x ) = x for any x ∈ L, • Sug (L ), the set of all Sugeno integrals on a distributive lattice L. Clearly, for any lattice L we have the inclusions Pol(L ) ⊆ Mon(L ) ∩ Bound(L ) = Agg (L ), and Sug (L ) = Bound(L ) ∩ WPol(L ) ∩ Id(L ). Let us note that certain generating sets of the clones Agg (L ) and Id(L ) ∩ Agg (L ) have been described in [12] and [3], respectively. 3. Generating sets of aggregation clones In what follows, we describe a very general method how to generate the aggregation clone on finite lattices. In this section, we focus on generation using the smallest possible arity of generating functions, i.e., we use unary and binary aggregation functions as generators only. For a set S, denote by P(S) the power set of S. Lemma 3.1. Let f: A → L be a function and fs : A → L, s ∈ S be a non-void family of functions. Then f = s∈S fs if and only if 1) fs ≤ f for all s ∈ S, 2) there is a mapping ϕ : A → P(S) such that s∈ϕ (x ) fs (x ) = f (x ) for all x ∈ A. Dually, f = s∈S fs if and only if 3) fs ≥ f for all s ∈ S, 4) there is a mapping ϕ : A → P(S) such that s∈ϕ (x ) fs (x ) = f (x ) for all x ∈ A. Proof. Obviously, if f = s∈S fs , then fs ≤ f for all s ∈ S. Further, it suffices to put ϕ (x ) = S for each x ∈ A. Then s∈ϕ (x ) fs (x ) = s∈S f s ( x ) = f ( x ). Conversely, assume that the conditions 1) and 2) are fulfilled. Since ϕ (x)⊆S, for each x ∈ A we obtain
f (x ) =
s∈ϕ ( x )
f s (x ) ≤
f s ( x ) ≤ f ( x ).
s∈S
The dual statement can be proved analogously.
In the following statement we describe certain aggregation functions which fit in the realm of the previous lemma. f
Corollary 3.2. Let f: Ln → L be an aggregation function and for each a ∈ Ln let ha : Ln → L, be a function given by
haf
(x ) =
1, if x = (1, . . . , 1 ); f (a ), if x ≥ a, x = (1, . . . , 1 ); 0, if x a.
(2)
f
Then ha is an aggregation function for all a ∈ Ln and
f =
haf .
(3)
a∈Ln
f
Proof. Let a ∈ Ln be an arbitrary element. It can be easily seen that ha is nondecreasing in Ln . Further, for a = (0, . . . , 0 ) it f f follows from (2) that ha (0, . . . , 0 ) = 0, while for a = (0, . . . , 0 ) we have ha (0, . . . , 0 ) = f (a ) = 0 as well, since f is an aggref f gation function. Also the condition ha (1, . . . , 1 ) = 1 follows directly from (2), hence ha satisfies the boundary conditions and it is an aggregation function. f It can be easily seen that ha ≤ f for all a ∈ Ln . Further, let us define ϕ : Ln → P(Ln ) by ϕ (a ) = {a} for each a ∈ Ln . Then the f n system of functions ha , a ∈ L , satisfies the condition 2) of Lemma 3.1 (note that in this case, using notation as in Lemma 3.1, f A = S = Ln ). Hence we obtain f = a∈Ln ha . Dually, the similar arguments show that f = each a ∈ Ln
a∈Ln
f
f
ga , where ga , a ∈ Ln is a system of aggregation functions such that for
R. Halaš et al. / Information Sciences 476 (2019) 38–47
41
gaf
0, if x = (0, . . . , 0 ); f (a ), if x ≤ a, x = (0, . . . , 0 ); 1, if x a.
(x ) =
(4)
Let n ≥ 1 be a positive integer. Recall that a set function c : P({1, . . . , n} ) → [0, 1] (here [0,1] is the standard unit interval of reals) is said to be a capacity if c is nondecreasing and fulfills the boundary conditions, i.e., E1 ⊆ E2 implies c(E1 ) ≤ c(E2 ) and c (∅ ) = 0, c ({1, . . . , n} ) = 1, see [7]. Evidently, each capacity c can be seen as an order homomorphism, and it is a generalization of discrete probability measures, where the additivity is relaxed into the monotonicity. A further generalization yields an L-valued capacity (L bounded lattice) c : P({1, . . . , n} ) → L, which is a monotone L-valued set function satisfying the boundary conditions c (∅ ) = 0 (0 being the bottom element of L) and c ({1, . . . , n} ) = 1 (1 being the top element of L). Clearly, the concept of L-valued capacities and O,1-preserving order homomorphisms between lattices P({1, . . . , n} ) and L coincides. We prefer to keep the terminology for c as L-valued capacity due to the links between aggregation functions and capacities. Observe that the notion of n-ary aggregation function (with domain Ln ) can be seen as a “fuzzy” generalization of the notion of capacity. Hence, concerning the binary case, we say that a binary aggregation function f: L2 → L extends a binary capacity c: P({1, 2}) → L, provided f (1, 0 ) = c ({1} ) and f (0, 1 ) = c ({2} ). As one of the fundamental types of generating functions, beside the lattice operations, we use the following unary aggregation functions. For any element a ∈ L we define χ a : L → L by
1, if x ≥ a, x = 0; 0, otherwise.
χa (x ) =
(5)
Dually, define a function μa : L → L by
0, if x ≤ a, x = 1; 1, otherwise.
μa ( x ) =
(6)
The above functions χ a as well as μa , a ∈ L, capture the order structure of the lattice L. For example, for a = 0 the function χ a represents the characteristic function of the principal filter generated by the element a. f
f ga
In the following two lemmas and subsequent remarks we describe one of the possible ways how to obtain functions ha , given by (2) and (4), respectively.
Lemma 3.3. Let g: L2 → L extends the binary capacity c: P({1, 2}) → L with c ({1} ) = a and c ({2} ) = b. Then for any positive integer n ∈ N and x = (x1 , . . . , xn ) ∈ Ln we obtain
Hg1
n
χ0 (xi ),
(x ) = g
n
i=1
i=1
n
n
χ1 (xi ),
i=1
(7)
a, otherwise.
and
Hg2 (x ) = g
0, if x = (0, . . . , 0 ); χ1 (xi ) = 1, if x = (1, . . . , 1 );
0, if x = (0, . . . , 0 ); χ0 (xi ) = 1, if x = (1, . . . , 1 );
(8)
b, otherwise.
i=1
Proof. Obviously, ni=1 χ0 (xi ) = 0 if and only if χ0 (xi ) = 0 for all i ∈ {1, . . . , n}, which holds if and only if xi = 0 for all i ∈ {1, . . . , n}, i.e., x = (0, . . . , 0 ). Similarly, ni=1 χ1 (xi ) = 1 if and only if x = (1, . . . , 1 ). Hence for any x ∈ Ln the pair n
χ0 (xi ),
i=1
n
χ1 (xi )
i=1
gains the three possible values, (0,0), (1,1) or (1,0). The first two possibilities occur only when x is equal to (0, . . . , 0 ) or (1, . . . , 1 ), respectively. From this it can be easily seen that Hg1 (x ) satisfies (7) for each x ∈ Ln . The equality (8) for Hg2 can be proved analogously. Remark 3.4. Composing the functions μa , a ∈ L, given by (6) and g we obtain the functions G1g , G2g : Ln → L where
G1g
n
(x ) = g
i=1
and
μ0 ( x i ) ,
n
i=1
0, if x = (0, . . . , 0 ); μ1 (xi ) = 1, if x = (1, . . . , 1 );
a, otherwise.
(9)
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R. Halaš et al. / Information Sciences 476 (2019) 38–47
G2g
n
(x ) = g
μ1 ( x i ) ,
i=1
n
0, if x = (0, . . . , 0 ); μ0 (xi ) = 1, if x = (1, . . . , 1 );
(10)
b, otherwise.
i=1
For a given a ∈ Ln , a = (0, . . . , 0 ) we put Ja = {i ∈ {1, . . . , n} : ai = 0}. Then
Ha (x ) =
1, if x ≥ a; 0, otherwise.
χai (xi ) =
i∈Ja
(11)
Let us note that the function Ha is considered as an n-ary function, although it need not depend on all the variables. To see (11) it suffices to realize that Ha (x ) = 1 if and only if χai (xi ) = 1 for all i ∈ Ja , i.e., xi ≥ ai for each i ∈ Ja . However ai = 0 for all i ∈ Ja , thus x ≥ a if and only if xi ≥ ai for all i ∈ Ja . Dually, for any a ∈ Ln , a = (1, . . . , 1 ) denoting Jˆa = {1 ≤ i ≤ n : ai = 1}, we obtain
Ga ( x ) =
μa i ( x i ) =
i∈Jˆa
0, if x ≤ a; 1, otherwise.
(12)
Lemma 3.5. Let a ∈ Ln , a = (0, . . . , 0 ), and let f: Ln → L be an aggregation function and g: L2 → L extends a binary capacity c: P({1, 2}) → L with c ({1} ) = f (a ). Then for all x ∈ Ln
haf (x ) = Hg1 (x ) ∧ Ha (x ), where Hg1 is given by (7) and Ha by (11), respectively. If g: L2 → L extends a capacity c with c ({2} ) = f (a ), then
haf (x ) = Hg2 (x ) ∧ Ha (x ), where Hg2 is given by (8). Proof. Assume that g: L2 → L fulfills g(1, 0 ) = c ({1} ) = f (a ). Then, with respect to (7) and (11), we obtain Hg1 (x ) ∧ Ha (x ) = 1 for x = (1, . . . , 1 ), Hg1 (x ) ∧ Ha (x ) = f (a ) for x ≥ a, x = (1, . . . , 1 ), and Hg1 (x ) ∧ Ha (x ) = 0 for xa. This shows that Hg1 (x ) ∧ f
Ha (x ) satisfies (2), i.e., Hg1 (x ) ∧ Ha (x ) = ha (x ).
f
f
Remark 3.6. (i) For a = (0, . . . , 0 ) we obtain either ha (x ) = Hg1 (x ) or ha (x ) = Hg2 (x ), depending on whether f (a ) = g(1, 0 ) = 0 or g(0, 1 ) = 0. (ii) Dually, using the functions G1g , G2g and G defined by (9),(10) and (12), respectively, for any a ∈ Ln , a = (1, . . . , 1 ) we obtain ga (x ) = G1g (x ) ∨ Ga (x ) for g: L2 → L extending a capacity c with c ({1} ) = f (a ) and ga (x ) = G2g (x ) ∨ Ga (x ) for g extending c with c ({2} ) = f (a ). f f (iii) Similarly as in the case (i), for a = (1, . . . , 1 ) we obtain ga (x ) = G1g (x ) or ga (x ) = G2g (x ) depending on whether f (a ) = g(1, 0 ) = 1 or g(0, 1 ) = 1. f
f
As we have shown, the crucial role play binary functions and their values at boolean inputs. In order to generate all aggregation functions, one needs a system of functions attaining a sufficient amount of values at boolean elements. This leads to the following definition of a full system on L. Definition 3.7. Let L be a bounded lattice. We say that a system G = {gi : L2 → L : i ∈ I} of binary aggregation functions is full on L, if the set of elements M = {gi (1, 0 ) : i ∈ I} ∪ {gi (0, 1 ) : i ∈ I} generates L as a lattice, i.e., the smallest sublattice containing M equals to L. Lemma 3.8. Let G = {gi : i ∈ I} be a full system of binary aggregation functions on a bounded lattice L. Then for any a ∈ L there is a binary aggregation function g: L2 → L such that g belongs to the clone generated by the set G ∪ {∨, ∧} and g(1, 0 ) = a. Proof. Denote M = {g(1, 0 ) : g ∈ G } ∪ {g(0, 1 ) : g ∈ G } and let C be the clone generated by G ∪ {∨, ∧}. Since G is full, the set M generates the lattice L. Consequently, each element of a ∈ L can be expressed in the form a = q(a1 , . . . , an ) for some {a1 , . . . , an } ⊆ M and an n-ary lattice polynomial q. For each index j ∈ {1, . . . , n} let gi j ∈ G denotes the function with gi j (1, 0 ) = a j or gi j (0, 1 ) = a j . Further, we put
f j (x1 , x2 ) = gi j (x1 , x2 ) if gi j (1, 0 ) = a j and f j (x1 , x2 ) = gi j ( p22 (x1 , x2 ), p21 (x1 , x2 )) = gi j (x2 , x1 ) provided gi j (0, 1 ) = a j . Note
that f j (1, 0 ) = a j holds for all j ∈ {1, . . . , n}. As the functions fj , j = 1, . . . , n, as well as the function q belong to the clone C, it follows that the composition g = q( f1 , . . . , fn ) given by (1) is a member of C, too. Then for the function g ∈ C the following holds
g(1, 0 ) = q f1 , . . . , fn (1, 0 ) = q f1 (1, 0 ), . . . , fn (1, 0 ) = q(a1 , . . . , an ) = a. Let us mention that using dual arguments, one can show that for any a ∈ L there is a function g ∈ C (C clone generated by the set G ∪ {∨, ∧}) with g (0, 1 ) = a.
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43
Recall that an element a ∈ L is join-irreducible if a = 0 and a = b ∨ c yields a = b or a = c. The notion of meet-irreducible element is defined dually. For a finite lattice L, J(L) and M(L) denotes the set of all join-irreducible and meet-irreducible elements, respectively. Obviously, for any 0 = a ∈ L, a = X holds for some X⊆J(L). Similarly, b = 1 implies that b = X for some X⊆M(L). Now applying all particular results, we obtain the following statement which describes a relatively wide family of generating sets of the aggregation clone. Theorem 3.9. Let L be a finite bounded lattice and let G be a full system of binary aggregation functions on L. Then both the sets Gχ = G ∪ {χa : a ∈ J(L )} ∪ {∧, ∨} and Gμ = G ∪ {μb : b ∈ M(L )} ∪ {∧, ∨} generate the aggregation clone Agg (L ). Proof. Let C be the clone generated by the set Gχ . We prove that C = Agg (L ). First, assume that b = c ∨ d in L. Then χb (x ) = χc (x ) ∧ χd (x ) for all x ∈ L. Indeed χc (x ) ∧ χd (x ) = 1 if and only χc (x ) = 1 and χd (x ) = 1 which is equivalent to x ≥ c and x ≥ d, i.e., x ≥ c∨d. As every non-zero element is a supremum of join irreducible elements, this shows that χ b ∈ C for all b ∈ L, b = 0. For b = 0 it can be easily seen that χ0 = a∈At(L ) χa , where the supremum is taken over the set of all atoms of L. Hence χ 0 ∈ C as well. f According to Corollary 3.2, any n-ary f ∈ Agg (L ) can be expressed as the finite supremum f = a∈Ln ha , hence it suffices f
to show that ha ∈ C for all a ∈ Ln . Due to Lemma 3.8, for each a ∈ Ln there is a binary aggregation function g ∈ C such that f f g(1, 0 ) = f (a ). Further, from Lemma 3.5 and Remark 3.6 we have ha = Hg1 ∧ Ha for a = (0, . . . , 0 ) and ha (x ) = Hg1 (x ) for 1 n a = (0, . . . , 0 ). However, expressions (7) and (11) yield Hg ∈ C and also Ha ∈ C for all a ∈ L , a = (0, . . . , 0 ). Hence we obtain C = Agg (L ). The fact that Gμ generates the aggregation clone Agg (L ) can be proved by using the dual arguments. Corollary 3.10. Let L be an n-element lattice and let T be a generating set of L with the minimal cardinality. Then Agg (L ) can be generated by the set S consisting of certain unary and binary aggregation functions, where
3n |T | +2≤ + 1. |S| ≤ min |J(L )|, |M(L )| +
2
(13)
2
2 and Gμ = |G | + |M(L )| + 2. Hence it suffices to show that given a generating set T of L, there exists a full system G of T cardinality |2| , where · denotes the ceiling function.
Proof. According to the previous theorem, the sets Gχ and Gμ generate the aggregation clone. Obviously Gχ = |G | + |J(L )| +
For this, let a, b ∈ L be arbitrary elements. We put
ga,b (x1 , x2 ) = (x1 ∧ a ) ∨ (x2 ∧ b) ∨ (x1 ∧ x2 ). Using direct computations, it can be easily seen that ga,b (1, 0 ) = a and ga,b (0, 1 ) = b. Consequently, we can take any two subsets A = {a1 , . . . , am } ⊆ T , B = {b1 , . . . , bm } ⊆ T with m =
|T | 2
, A ∪ B = T and put
G = {gai ,bi : 1 ≤ i ≤ m}. Evidently, G represents a full system, which proves the first inequality of (13). Obviously, |J(L )| ≤ n − 1, |M(L )| ≤ n − 1 and |T| ≤ n for any n-element lattice L. Thus we obtain
n 3n |T | min |J(L )|, |M(L )| + +2≤n−1+ +2= + 1.
2
2
2
Observe that for an n-element chain L we have the equality |J(L )| = |M(L )| = n − 1 and |T | = n, since L can be generated by itself only. Hence in this case, the second inequality of (13) is in fact the equality. Let us remark that the inequality (13) improves the result from [12], where the generating set having 2n + 2 elements was described. Further, for any finite lattice L, the set J(L) eventually together with the bottom element (in the case that L possesses the only atom, otherwise 0 can be omitted) generates L. Similarly, M(L) with the top element generates L. Hence from (13) we obtain that Agg (L ) can be generated by the set S, consisting of functions of arity at most 2, and satisfying
|S| ≤ min
3|J(L )| + 1 , 2
3|M(L )| + 1 2
+2
At the end of this section we provide two examples illustrating the obtained results for particular lattices. Example 3.11. Consider a finite distributive lattice L. Observe that the functions ga, b used in the proof of Corollary 3.10 given by
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R. Halaš et al. / Information Sciences 476 (2019) 38–47
ga,b (x1 , x2 ) =
m (I ) ∧
I⊆{1,2}
xi
i∈I
coincide with binary Sugeno integrals determined by the capacity m: P({1, 2}) → L with m({1} ) = a, m({2} ) = b. Moreover, we have
g0,0 ( x1 , x2 ) = ( x1 ∧ 0 ) ∨ ( x2 ∧ 0 ) ∨ ( x1 ∧ x2 ) = x1 ∧ x2 and
g1,1 ( x1 , x2 ) = ( x1 ∧ 1 ) ∨ ( x2 ∧ 1 ) ∨ ( x1 ∧ x2 ) = x1 ∨ x2 . Hence, as a consequence of Theorem 3.9 we obtain the following corollary. Corollary 3.12. The aggregation clone Agg (L ) on a finite distributive lattice L is generated by the set consisting of the functions χ a , a ∈ J(L) (or functions μb , b ∈ M(L)), and any full system of binary Sugeno integrals (which must contain the two Sugeno integrals g0,0 and g1,1 extending the smallest and the greatest capacity on L, respectively). Observe that the expression (3) of Corollary 3.2 is valid for any aggregation function on an arbitrary lattice. Hence, it can be used to express a given aggregation function by means of a full system also in the case of an infinite lattice, although the infinitary operation of supremum must be involved. To illustrate this, we present an example concerning aggregation functions defined on the unit interval of reals. Example 3.13. Let L = [0, 1] be the unit interval of reals. It is obvious that the only generating set (by means of finitary lattice operations) of L is L itself. One of the full systems on L is given by binary Sugeno integrals, similarly as in Example 3.11. We present another binary full system of this lattice. For each a ∈ [0, 12 ] denote by ga : [0, 1]2 → [0, 1] the weighted arithmetic mean
ga ( x1 , x2 ) = a · x1 + ( 1 − a ) · x2 . Obviously, the system {ga : a ∈ [0, 12 ]} is full on L. Using the presented full system, we provide the formula (3) for the function of binary multiplication. According to Lemma 3.5, for (a1 , a2 ) ∈ [0, 1]2 the function ha1 ·a2 (we omit the lower case (a1 , a2 )) has the expression
⎧
1 ⎪ g ai , if 0 ≤ a1 · a2 ≤ , χ (x ) ∨ χ0 (x2 ), χ1 (x1 ) ∧ χ1 (x2 ) ∧ ⎪ ⎨ a1 ·a2 0 1 2 i∈J(a ,a ) 1 2 ha1 ·a2 (x1 , x2 ) =
1 ⎪ ⎪ ai , if < a1 · a2 ≤ 1. ⎩ga1 ·a2 χ1 (x1 ) ∧ χ1 (x2 ), χ0 (x1 ) ∨ χ0 (x2 ) ∧ i∈J(a
Hence, we obtain
x1 · x2 =
ha1 ·a2 (x1 , x2 ) =
(a1 ,a2 )∈[0,1]2
=
a1 · a2 ·
(a1 ,a2 )∈[0,1]2 a1 ·a2 ≤ 12
∨
ha1 ·a2 (x1 , x2 ) ∨
2
ha1 ·a2 (x1 , x2 )
(a1 ,a2 )∈[0,1]2 1 2
ai χ0 (x1 ) ∨ χ0 (x2 ) + (1 − a1 · a2 ) χ1 (x1 ) ∧ χ1 (x2 ) ∧
(a1 ,a2 )∈[0,1]2 a1 ·a2 ≤ 12
1 ,a2 )
i∈J(a
a1 · a2 ·
1 ,a2 )
ai . χ1 (x1 ) ∧ χ1 (x2 ) + (1 − a1 · a2 ) χ0 (x1 ) ∨ χ0 (x2 ) ∧
(a1 ,a2 )∈[0,1]2 1 2
i∈J(a
1 ,a2 )
4. Generation with higher arities In what follows we extend the results from the previous section. We focus on lowering the number of generating functions of the aggregation clone by using of functions with higher than binary arity. Given a bounded lattice L, for a subset I ⊆ {1, . . . , n} = [n], n ≥ 1, we denote the (L-valued) characteristic function of I by 1I , i.e., 1I ∈ Ln and for all i ∈ [n], 1I (i ) = 1 if i ∈ I and 1I (i ) = 0 for i ∈ I. In accordance with Definition 3.7 we say that a system of aggregation functions G = {g j : Ln j → L : j ∈ J} is full on a bounded lattice L, if the set M = j∈J {g j (1I ) : I ⊆ [n j ]} generates the lattice L. Similarly to the case of generation of the aggregation clone by a binary full system, we show that the same result can be obtained by using of arbitrary full system of aggregation functions. Theorem 4.1. Let L be a finite lattice and G be a full system of aggregation functions on L. Then both the sets Gχ = G ∪ {χa : a ∈ J(L )} ∪ {∧, ∨} and Gμ = G ∪ {μb : b ∈ M(L )} ∪ {∧, ∨} generate the aggregation clone Agg (L ).
R. Halaš et al. / Information Sciences 476 (2019) 38–47
45
Proof. We show that for a clone C generated by the set Gχ , the equality C = Agg (L ) holds. Let n ≥ 1 be fixed and b ∈ L be such that b = g(1I ) for some g: Lk → L, g ∈ G and I⊆[k]. For any x = (x1 , . . . , xn ) ∈ Ln we put
HgI (x1 , . . . , xn ) = g y1 (x ), . . . , yk (x ) , where the functions yi : Ln → L, i ∈ [k] are given by stipulation
yi ( x1 , . . . , xn ) =
⎧ n ⎪ ⎪ ⎨ χ0 (xl ), if i ∈ I; l=1
n
⎪ ⎪ ⎩ χ1 (xl ), if i ∈/ I. l=1
It can be easily seen that for any x ∈ Ln , (0, . . . , 0 ) = x = (1, . . . , 1 ), yi (x ) = 1 if i ∈ I and yi (x ) = 0 otherwise. Hence in this case y1 (x ), . . . , yk (x ) = 1I and we obtain
HgI
0, if x = (0, . . . , 0 ); (x1 , . . . , xn ) = g y1 (x ), . . . , yk (x ) = 1, if x = (1, . . . , 1 );
b, otherwise.
Obviously HgI ∈ C as the function HgI is a composition of functions belonging to C. Note that concerning the functions χ 0 and χ 1 respectively, we use the results from the proof of Theorem 3.9. Further, let f: Ln → L be an arbitrary aggregation function. As G is a full system, for any a ∈ Ln there exist a system of m functions g j : Ln j → L, g j ∈ G, j = 1, . . . , m, an m-ary lattice polynomial q with q(a1 , . . . , am ) = f (a ), where for all j = 1, . . . , m, a j = g j (1I j ) for some Ij ⊆[nj ]. Consequently, for x ∈ Ln we obtain
haf (x ) = Ha (x ) ∧ q HgI11 (x ), . . . , HgImm (x ) , I
where the function Ha is given by (11). Since all the functions Ha , q and Hgjj , j = 1, . . . , m, belong to the clone C, it follows f
that ha ∈ C as well. According to Corollary 3.2, this yields f ∈ C, i.e., C = Agg (L ). The fact that the set Gμ generates the clone Agg (L ) can be proved analogously.
Allowing higher than binary arities for functions in a full system, it is natural to ask whether singleton full systems can exist and what are the arities of aggregation functions belonging to such systems. In the following theorem we answer this problem positively, and we also give an upper bound for the corresponding arities. Theorem 4.2. Let L be a finite lattice and T be a generating set of L with a minimal cardinality. Then there exists a singleton full system G = {g} on L, where g is a k-ary aggregation function with
k = min n : |T | ≤
nn .
(14)
2
Proof. At first we show that any L-valued capacity can be extended to an aggregation function. Let m: P([n]) → L be an L-valued capacity defined on the power set of [n] = {1, . . . , n}, n ≥ 1. Consider the following function gm : Ln → L given by
gm ( x1 , . . . , xn ) =
m (I ) ∧
xi
(15)
i∈I
I⊆[n]
for all x1 , . . . , xn ∈ L. We show that the function gm extends the capacity m, i.e., gm (1I ) = m(I ) for all subsets I⊆[n]. Let I⊆[n] be a fixed subset. Then
gm ( 1I ) =
J ⊆[n],J ⊆I
m (J ) ∧
j∈J
1I ( j )
∨
J ⊆[n],J I
m (J ) ∧
j∈J
1I ( j ) =
J ⊆[n],J ⊆I
m (J ) ∨
0 = m ( I ),
J ⊆[n],J I
since 1I ( j ) = 1 for all j ∈ J⊆I and 1I ( j ) = 0 for some j ∈ JI. In what follows we find a capacity m, range of which contains the set T. In the most general case, elements of T can form an antichain in the lattice L. According to Sperner’s theorem, cf. eg [1]., for any family F of subsets of an n-element set such that XY for all distinct X, Y ∈ F (F forms an antichain of subsets of an n-element set), the following inequality holds:
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R. Halaš et al. / Information Sciences 476 (2019) 38–47
n . |F | ≤ n 2
Note that the family of all n2 -element subsets on an n-set forms an antichain and that for this family the equality is attained. Now let k be a positive integer given by (14). Then it is possible to define a capacity m: P([k]) → L in such a way that for all X⊆[k]
⎧ ⎨= 0, if |X | < 2k , m(X ) ∈ T , if |X | = 2k , ⎩ = 1, if |X | > 2k ,
while T ⊆ {m(X ) : |X | > 2k }. With respect to the first part of the proof, the k-ary function gm given by (15), which extends m, forms a full system on the lattice L. Corollary 4.3. Let L be a finite lattice n-element lattice. Then the aggregation clone Agg (L ) can be generated by the set consisting of the lattice operations, min {|J(L)|, |M(L)|} unary aggregation functions and one k-ary aggregation function, where k satisfies (14). Let us mention that the presented upper bound k for arities applies for arbitrary finite bounded lattices. However, if the order structure of an underlying lattice is known, this upper bound can be lowered. In the following example we illustrate this fact in the case of a finite chain. Note that if L is a finite chain, L is the only generating set of L. Example 4.4. Let L be an n-element chain L = {a1 , . . . , an }, n ≥ 1, where a1 < < an , and k be a positive integer such that 2k ≥ n. Consider the cardinality-lexicographic order of P([k]), i.e., for X, Y⊆[k] we put X < Y provided |X| < |Y| or |X | = |Y | and min (XY) < min {YX}. Note that with respect to this ordering, P([k]) becomes a linearly ordered set. Let X1 , . . . , X2k be the enumeration of P([k]) such that X1 < · · · < X2k . Define a capacity m: P([k]) → L by m(Xi ) = ai , i = 1, . . . , 2k , where ai = an for all i ≥ n. From the definition of < it can be easily seen that X⊆Y implies X ≤ Y, which yields that m is a monotone set function on P([k]). Since a1 , a2k is the bottom and the top element of L respectively, we obtain that m is a capacity. Then the singleton {gm }, gm given by (15), forms a full system on L. Consequently, the chain L can be generated by the set {∨, ∧} ∪ {χai : 2 ≤ i ≤ n} ∪ {gm }, where gm is log2 n-ary aggregation function. In the proof of Theorem 4.2 we have presented aggregation functions (15) extending an arbitrary capacity m. In the following example we give another type of aggregation functions satisfying this property. Consequently, such functions can be included in an appropriate full system. Example 4.5. Let n ≥ 1 be a positive integer and m: P([n]) → L be an L-valued capacity. For x = (x1 , . . . , xn ) ∈ Ln put Ix = {i ∈ [n] : xi = 1}. Further we can define fm : Ln → L by
fm (x ) = m(Ix ) for all x ∈ Ln . It can be easily seen that x1 ≤ x2 implies Ix1 ⊆ Ix2 , which due to the monotonicity of m yields fm (x1 ) ≤ fm (x2 ). Since I(0,...,0 ) = ∅ and I(1,...,1 ) = [n], the function fm fulfills the boundary conditions as well, i.e., it is an aggregation function. Obviously, fm extends the capacity m, as I1X = X for any subset X⊆[n]. Observe that the ranges of fm and m coincide, thus fm is the smallest aggregation function extending the capacity m. Finally, omitting the finiteness requirement, in the following example we provide a singleton full system for the real unit interval. Example 4.6. Consider the lattice L = [0, 1]. Since an n-ary capacity attains finite number of values only, it is obvious that none of the aggregation functions of finite arity itself forms a full system of L. Hence, any singleton singleton full system of L possesses an infinitary function. We show that it is possible to find an aggregation function involving ℵ0 = |N| arguments, such that it forms a full system of L. To see this, recall that the set N of all positive integers and Q[0,1] of all rationals between 0 and 1 have the same cardinality, i.e., there is a bijection between them. Due to this well-known fact it is sufficient to find an appropriate capacity on the power set of Q[0,1] . Define a set function m : P(Q[0,1] ) → [0, 1] in the following way: m(∅ ) = 0 and m({q} ) = q for any q ∈ Q[0,1] . Further, for ∅ = X ⊆ Q[0,1] we put
m (X ) =
q.
q∈X
It can be easily seen, that the function m fulfills the boundary conditions and it is monotone, i.e., it is a capacity. Moreover, as any real number c ∈ [0, 1] can be expressed as a supremum of some rationals, it follows that for the range of m the
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equality Rng (m ) = [0, 1] holds. Consequently, any (ℵ0 -ary) aggregation function extending m, e.g., the infinitary analogue of the function fm from the previous example, can be seen as a singleton full system of L. 5. Conclusion In this paper, a new general method for searching generating sets of aggregation clones on finite lattices has been presented. As the main tool we developed the so-called full systems of aggregation functions which together with certain unary aggregation functions and lattice operations generate the clone of aggregation functions. We have also discussed the existence and minimal arity of singleton full systems of functions. We believe that our results will be convenient for further analysis of aggregation functions on finite lattices and also for the development of other special classes of aggregation functions. Acknowledgments ˇ The first author was supported by the project of Grant Agency of the Czech Republic (GACR) no. 18-06915S and by the ˇ no. project MSMT Mobility 7AMB17AT054; the second author by the project of Grant Agency of the Czech Republic (GACR) 18-06915S and by the Slovak VEGA Grant 1/0420/15; the third author by the IGA project of the Faculty of Science Palacký University Olomouc PrF2018012 and by the Slovak Research and Development Agency under the contract No. APVV-16-0073. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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