A generalization of Goodstein's theorem: interpolation by polynomial functions of distributive lattices
aa r X i v : . [ m a t h . R A ] O c t A GENERALIZATION OF GOODSTEIN’S THEOREM:INTERPOLATION BY POLYNOMIAL FUNCTIONS OFDISTRIBUTIVE LATTICES
MIGUEL COUCEIRO AND TAM ´AS WALDHAUSER
Abstract.
We consider the problem of interpolating functions partiallydefined over a distributive lattice, by means of lattice polynomial func-tions. Goodstein’s theorem solves a particular instance of this interpola-tion problem on a distributive lattice L with least and greatest elements0 and 1, resp.: Given a function f : { , } n → L , there exists a latticepolynomial function p : L n → L such that p | { , } n = f if an only if f ismonotone; in this case, the interpolating polynomial p is unique.We extend Goodstein’s theorem to a wider class of partial functions f : D → L over a distributive lattice L , not necessarily bounded, andwhere D ⊆ L n is allowed to range over cuboids D = { a , b } × · · · ×{ a n , b n } with a i , b i ∈ L and a i < b i , and determine the class of suchpartial functions which can be interpolated by lattice polynomial func-tions. In this wider setting, interpolating polynomials are not necessarilyunique; we provide explicit descriptions of all possible lattice polynomialfunctions which interpolate these partial functions, when such an inter-polation is available. Introduction
Let L be a distributive lattice and let f : D → L ( D ⊆ L n ) be an n -ary partial function on L . In this paper we are interested in the problem ofextending such partial functions to the whole domain L n by means of latticepolynomial functions, i.e., functions that can be represented as compositionsof the lattice operations ∧ and ∨ and constants. More precisely, we aim atdetermining necessary and sufficient conditions on the partial function f that guarantee the existence of a lattice polynomial function p : L n → L which interpolates f , that is, p | D = f .An instance of this problem was considered by Goodstein [8] in the casewhen L is a bounded distributive lattice, and the functions to be interpolatedwere of the form f : { , } n → L . Goodstein showed that such a function f can be interpolated by lattice polynomial functions if and only if it ismonotone. Furthermore, if such an interpolating polynomial function exists,then it is unique.The general solution to the above mentioned interpolation problem eludesus. However, we are able to generalize Goodstein’s result by allowing L tobe an arbitrary (possibly unbounded) distributive lattice and consideringfunctions f : D → L , where D = { a , b } × · · · × { a n , b n } with a i , b i ∈ L and a i < b i . More precisely, we furnish necessary and sufficient conditionsfor the existence of an interpolating polynomial function. As it will becomeclear, in this more general setting, uniqueness is not guaranteed, and thuswe determine all possible interpolating polynomial functions.The structure of the paper is as follows. In Section 2 we recall basicbackground on polynomial functions over distributive lattices (for generalbackground see [7, 9]) and formalize the interpolation problem that we areinterested in. In Section 3 we state and prove the characterization of thosefunctions that can be interpolated by polynomial functions and we describethe set of all solutions of the interpolation problem. We discuss variations ofthe interpolation problem in Section 4 and relate our work to earlier resultsobtained for finite chains in [11]. Finally, in Section 5 we consider potentialapplications of our results in mathematical modeling of decision making.2. Preliminaries
Let L be a bounded distributive lattice with least element 0 and greatestelement 1. It can be shown that a function p : L n → L is a lattice polynomialfunction if and only if there exist c I ∈ L , I ⊆ [ n ] := { , . . . , n } , such that p can be represented in disjunctive normal form (DNF for short) by(1) p ( x ) = _ I ⊆ [ n ] (cid:0) c I ∧ ^ i ∈ I x i (cid:1) , where x = ( x , . . . , x n ) ∈ L n .It is easy to verify that taking c ′ I = W J ⊆ I c J , we also have p ( x ) = _ I ⊆ [ n ] (cid:0) c ′ I ∧ ^ i ∈ I x i (cid:1) , and hence the coefficients c I can be assumed to be monotone in the sensethat I ⊆ J implies c I ≤ c J . This monotonicity assumption allows us torecover the coefficients of the DNF from certain values of the polynomialfunction p . Indeed, denoting by I the characteristic vector of I ⊆ [ n ] (i.e.,the tuple I ∈ L n whose i -th component is 1 if i ∈ I and 0 if i / ∈ I ), wethen have that p ( I ) = c I . In the sequel, we will always consider latticepolynomials in DNF, and we will implicitly assume that the coefficients aremonotone. These observations contain the essence of Goodstein’s theorem. Theorem 1 (Goodstein [8]) . Let L be a bounded distributive lattice, andlet f be a function f : { , } n → L . There exists a polynomial function p over L such that p | { , } n = f if and only if f is monotone. In this case p isuniquely determined, and can be represented by the DNF p ( x ) = _ I ⊆ [ n ] (cid:0) f ( I ) ∧ ^ i ∈ I x i (cid:1) . Informally, Goodstein’s theorem asserts that polynomial functions areuniquely determined by their restrictions to the hypercube { , } n , and a ENERALIZATION OF GOODSTEIN’S THEOREM 3 function on the hypercube extends to a polynomial function if and only if itis monotone.Let us now consider a distributive lattice L without least and greatestelements. (We omit the analogous discussion of the cases where L has oneboundary element.) Polynomial functions over L can still be given in DNFof the form (1) by allowing the coefficients c I to take also the values 0 and1, which are considered as external boundary elements (see, e.g., [1, 3]). Forexample, a polynomial function p ( x, y ) = a ∨ x ∨ ( b ∧ x ∧ y ) can be rewrittenas p ( x, y ) = a ∨ (1 ∧ x ) ∨ (0 ∧ y ) ∨ ( b ∧ x ∧ y ).We can still assume monotonicity of the coefficients, and any such system c I ∈ L ∪ { , } ( I ⊆ [ n ]) of coefficients gives rise to a polynomial function p over L , provided that c ∅ = 1 and c [ n ] = 0. (The latter two cases corre-spond to the constant 1 and constant 0 functions.) Just like in the case ofbounded lattices, there is a one-to-one correspondence between such DNF’sand polynomial functions, since we can recover the coefficients of the DNFof p from certain values of p . To see this, let us choose elements a < b from L to play the role of 0 and 1, and let e I be the “characteristic vector” of I ⊆ [ n ] (i.e., the tuple e I ∈ L n whose i -th component is b if i ∈ I and a if i / ∈ I ). If a is sufficiently small (less than all non-zero coefficients in theDNF of p ) and b is sufficiently large (greater than all non-one coefficients inthe DNF of p ), then a routine computation shows that p ( e I ) = c I if c I ∈ L,a if c I = 0 ,b if c I = 1 . Thus we can learn the coefficient c I from the behavior of the value p ( e I )by letting a decrease and b increase indefinitely, i.e., the polynomial function p is uniquely determined by its values on a sufficiently large cube { a, b } n (fora more detailed discussion, see [1]). As the next example shows, this doesnot imply that there is only one polynomial function that takes prescribedvalues on a fixed cube { a, b } n . Example 2.
Let L be the lattice of open subsets of a topological space X ,and let a, b ∈ L with a ⊂ b . Since L is a bounded distributive lattice, everyunary polynomial function p over L can be represented by a unique DNF ofthe form p ( x ) = c ∪ ( c ∩ x ) with c , c ∈ L, c ⊆ c . It is straightforwardto verify that such a polynomial function satisfies p ( a ) = p ( b ) = b if andonly if b \ a ⊆ c ⊆ b and b ⊆ c ⊆ X. Thus, there may be infinitely many polynomial functions p whose restrictionto the “one-dimensional cube” { a, b } is constant b (for instance, let X bethe real line, and let a and b be open intervals).Let us go one step further, and choose a “zero” and “one”, possibly differ-ent in each coordinate: Let a i , b i ∈ L with a i < b i for each i ∈ [ n ], and let b e I be the “characteristic vector” of I ⊆ [ n ] (i.e., the tuple b e I ∈ L n whose i -th MIGUEL COUCEIRO AND TAM ´AS WALDHAUSER component is b i if i ∈ I and a i if i / ∈ I ). The task of finding a polynomialfunction (or rather all polynomial functions) that takes prescribed values onthe tuples b e I can be regarded as an interpolation problem. Interpolation Problem.
Given D := { b e I : I ⊆ [ n ] } and f : D → L , findall polynomial functions p : L n → L such that p | D = f . Note that here the function f is given on the vertices of a rectangular box(cuboid) instead of a cube as in Theorem 1. We will solve this problem inSection 3, thereby generalizing Goodstein’s theorem. Let us note that theproblem can be interesting also in the case of bounded lattices, for instance,if we do not have access to the values of the polynomial function on { , } n ,but only on some “internal” points. We will discuss such applications inSection 5. 3. Main results
In the sequel we assume that D := { b e I : I ⊆ [ n ] } and f : D → L are given,and our goal is to find (the DNF of) all n -ary polynomial functions p over L that satisfy p | D = f . Clearly, monotonicity of f is a necessary conditionfor the existence of a solution of the Interpolation Problem, but, in contrastwith Goodstein’s theorem, monotonicity is not always sufficient in this moregeneral setting. We will prove that the extra condition that we need is thefollowing:( ⋆ ) f (cid:0)b e I ∪{ k } (cid:1) ∧ a k ≤ f ( b e I ) ≤ f (cid:0)b e I \{ k } (cid:1) ∨ b k for all I ⊆ [ n ] , k ∈ [ n ] . Observe that the first inequality is trivial if k ∈ I , and the second inequalityis trivial if k / ∈ I .Our first lemma shows how to obtain inequalities between f ( b e S ) and f ( b e T ) for S ⊆ T by repeated applications of ( ⋆ ). Lemma 3.
If the function f satisfies ( ⋆ ), then for all S ⊆ T ⊆ [ n ] we have f ( b e T ) ∧ ^ k ∈ T \ S a k ≤ f ( b e S ) and f ( b e T ) ≤ f ( b e S ) ∨ _ k ∈ T \ S b k . Proof.
We only prove the first inequality; the second one follows similarly.Let T \ S = { k , . . . , k r } , and let us apply (the first inequality of) condition( ⋆ ) with I = S ∪ { k , . . . , k m − } and k = k m for m = 1 , . . . , r : f (cid:0)b e S ∪{ k } (cid:1) ∧ a k ≤ f ( b e S ) ,f (cid:0)b e S ∪{ k ,k } (cid:1) ∧ a k ≤ f (cid:0)b e S ∪{ k } (cid:1) , ... f (cid:0)b e S ∪{ k ,...,k r } (cid:1) ∧ a k r ≤ f (cid:0)b e S ∪{ k ,...,k r − } (cid:1) . Combining these r inequalities, we get f (cid:0)b e S ∪{ k ,...,k r } (cid:1) ∧ a k ∧ · · · ∧ a k r ≤ f ( b e S ) . (cid:3) ENERALIZATION OF GOODSTEIN’S THEOREM 5
Let us now show that ( ⋆ ) is a necessary condition for the existence of asolution of the Interpolation Problem. Lemma 4.
If there is a polynomial function p over L such that p | D = f ,then f is monotone and satisfies ( ⋆ ). Proof.
Assume that p is a polynomial function that extends f . Since p ismonotone, f is also monotone. To show that ( ⋆ ) holds, let us fix I ⊆ [ n ]and k ∈ [ n ], and let us assume that k / ∈ I (the case k ∈ I can be dealt withsimilarly). Let ( b e I ) xk ∈ L n denote the n -tuple obtained from b e I by replacingits k -th component by the variable x . We can define a unary polynomialfunction u over L by u ( x ) := p (( b e I ) xk ). Using this notation, ( ⋆ ) takes theform u ( b k ) ∧ a k ≤ u ( a k ). The DNF of u is of the form u ( x ) = c ∨ ( c ∧ x ),where c , c ∈ L ∪ { , } . Using distributivity and the fact that a k < b k , wecan now easily prove the desired inequality: u ( b k ) ∧ a k = ( c ∨ ( c ∧ b k )) ∧ a k = ( c ∧ a k ) ∨ ( c ∧ b k ∧ a k )= ( c ∧ a k ) ∨ ( c ∧ a k ) ≤ c ∨ ( c ∧ a k ) = u ( a k ) . (cid:3) To find all polynomial functions p satisfying p | D = f , we will make useof the Birkhoff-Priestley representation theorem to embed L into a Booleanalgebra B . For the sake of canonicity, we assume that L generates B ; underthis assumption B is uniquely determined up to isomorphism. The boundaryelements of B will be denoted by 0 and 1. This notation will not leadto ambiguity since if L has a least (resp. greatest) element, then it mustcoincide with 0 (resp. 1). The complement of an element a ∈ B is denotedby a ′ . Given a function f : D → L , we define the following two elements in B for each I ⊆ [ n ]: c − I := f ( b e I ) ∧ ^ i/ ∈ I a ′ i , c + I := f ( b e I ) ∨ _ i ∈ I b ′ i . Observe that c − I ≤ c + I , and if f is monotone, then I ⊆ J implies c − I ≤ c − J and c + I ≤ c + J . Let p − and p + be the polynomial functions over B which aregiven by these two systems of coefficients. We will see that p − and p + arethe least and greatest polynomial functions over B whose restriction to D coincides with f (whenever there exists such a polynomial function). Lemma 5. If f is monotone and satisfies ( ⋆ ), then p + ( b e J ) ≤ f ( b e J ) for all J ⊆ [ n ]. Proof.
Let us fix J ⊆ [ n ] and consider the value of p + at b e J : p + ( b e J ) = _ I ⊆ [ n ] (cid:0) c + I ∧ ^ j ∈ I ( b e J ) j (cid:1) = _ I ⊆ [ n ] (cid:0) c + I ∧ ^ j ∈ I \ J a j ∧ ^ j ∈ I ∩ J b j (cid:1) . It is sufficient to verify that each joinand is at most f ( b e J ). Taking intoaccount the definition of c + I , this amounts to showing that(2) (cid:0) f ( b e I ) ∨ _ i ∈ I b ′ i (cid:1) ∧ ^ j ∈ I \ J a j ∧ ^ j ∈ I ∩ J b j ≤ f ( b e J ) MIGUEL COUCEIRO AND TAM ´AS WALDHAUSER holds for all I ⊆ [ n ]. Distributing meets over joins, the left hand side of (2)becomes(3) (cid:0) f ( b e I ) ∧ ^ j ∈ I \ J a j ∧ ^ j ∈ I ∩ J b j (cid:1) ∨ _ i ∈ I (cid:0) b ′ i ∧ ^ j ∈ I \ J a j ∧ ^ j ∈ I ∩ J b j (cid:1) . Let us examine each joinand of this expression. For each i ∈ I , the joinandinvolving b ′ i equals 0, since b ′ i ∧ ^ j ∈ I \ J a j ∧ ^ j ∈ I ∩ J b j ≤ b ′ i ∧ ^ j ∈ I \ J b j ∧ ^ j ∈ I ∩ J b j = b ′ i ∧ ^ j ∈ I b j ≤ b ′ i ∧ b i = 0 . The joinand of (3) that involves f ( b e I ) can be estimated using ( ⋆ ) andLemma 3 (with T = I and S = I ∩ J ): f ( b e I ) ∧ ^ j ∈ I \ J a j ∧ ^ j ∈ I ∩ J b j ≤ f ( b e I ) ∧ ^ j ∈ I \ ( I ∩ J ) a j ≤ f ( b e I ∩ J ) . Since f is monotone, we have f ( b e I ∩ J ) ≤ f ( b e J ), and this proves (2). (cid:3) The following lemma is the dual of Lemma 5, and it can be proved byusing the conjunctive normal form of p − . Lemma 6. If f is monotone and satisfies ( ⋆ ), then p − ( b e J ) ≥ f ( b e J ) for all J ⊆ [ n ].The estimates obtained in the previous two lemmas allow us to find allsolutions of our interpolation problem over B , whenever a solution exists. Theorem 7.
Let D = { b e I : I ⊆ [ n ] } and f : D → L be given, as in theInterpolation Problem. Suppose that f is monotone and satisfies ( ⋆ ) , and let p be an n -ary polynomial function over B given by the DNF corresponding toa system of coefficients c I ∈ B ( I ⊆ [ n ]) . Then the following three conditionsare equivalent: (i) p | D = f ; (ii) for all I ⊆ [ n ] the inequalities c − I ≤ c I ≤ c + I hold; (iii) for all x ∈ L n we have p − ( x ) ≤ p ( x ) ≤ p + ( x ) .Proof. Implication (ii) = ⇒ (iii) is trivial. To prove (i) = ⇒ (ii), assume that p | D = f , i.e., p ( b e J ) = f ( b e J ) for all J ⊆ [ n ]. Then we can replace f ( b e J ) by p ( b e J ) in the definition of c − J , and we can compute its value by substituting b e J into the DNF of p : c − J = f ( b e J ) ∧ ^ j / ∈ J a ′ j = p ( b e J ) ∧ ^ j / ∈ J a ′ j = (cid:16) _ I ⊆ [ n ] (cid:0) c I ∧ ^ i ∈ I ( b e J ) i (cid:1)(cid:17) ∧ ^ j / ∈ J a ′ j = _ I ⊆ [ n ] (cid:0) c I ∧ ^ i ∈ I \ J a i ∧ ^ i ∈ I ∩ J b i ∧ ^ j / ∈ J a ′ j (cid:1) . If there exists i ∈ I \ J , then a i ∧ a ′ i = 0 appears in the joinand correspondingto I , hence we can omit each of these terms from the join, and keep only ENERALIZATION OF GOODSTEIN’S THEOREM 7 those where I \ J = ∅ : c − J = _ I ⊆ J (cid:0) c I ∧ ^ i ∈ I \ J a i ∧ ^ i ∈ I ∩ J b i ∧ ^ j / ∈ J a ′ j (cid:1) ≤ _ I ⊆ J c I = c J . This proves c − J ≤ c J . The inequality c J ≤ c + J can be proved by a dualargument.Finally, to prove (iii) = ⇒ (i), let us assume that p − ≤ p ≤ p + holds inthe pointwise ordering of functions. Applying Lemma 5 and Lemma 6, weget the following chain of inequalities for every I ⊆ [ n ]: f ( b e I ) ≤ p − ( b e I ) ≤ p ( b e I ) ≤ p + ( b e I ) ≤ f ( b e I ) . This implies p ( b e I ) = f ( b e I ) for all I ⊆ [ n ], therefore we have p | D = f . (cid:3) Note that in Lemma 4 we did not make use of the fact that p is a polyno-mial function over L : the proof works also for polynomial functions over B .This fact together with Theorem 7 shows that monotonicity and property( ⋆ ) of f are necessary and sufficient for the existence of a solution of ourinterpolation problem over B . This observation leads to the following result. Theorem 8.
The Interpolation Problem has a solution if and only if f ismonotone and satisfies ( ⋆ ) . In this case a polynomial function p over L verifies p | D = f if and only if c − I ≤ c I ≤ c + I holds for the coefficients c I ofthe DNF of p for all I ⊆ [ n ] . In particular, p can be chosen as the polynomialfunction p given by the coefficients c I = f ( b e I ) : p ( x ) = _ I ⊆ [ n ] (cid:0) f ( b e I ) ∧ ^ i ∈ I x i (cid:1) .Proof. The necessity of the conditions has been established in Lemma 4.To prove the sufficiency, we just need to observe that if f is monotone andsatisfies ( ⋆ ), then the polynomial function p is a solution of the InterpolationProblem by Theorem 7, as c − I ≤ f ( b e I ) ≤ c + I follows immediately from thedefinition of c − I and c + I . Since f ( b e I ) ∈ L for all I ⊆ [ n ], the polynomialfunction p is actually a polynomial function over L . The description of theset of all solutions over L also follows from Theorem 7. (cid:3) Let us note that if L is bounded and a i = 0 , b i = 1 for all i ∈ [ n ], thenTheorem 8 reduces to Goodstein’s theorem. Indeed, in this case ( ⋆ ) holdstrivially, hence a solution exists if and only if f is monotone. Moreover,we have c − I = c + I = f ( b e I ), hence p (which is the same as the polyno-mial function given in Theorem 1) is the only solution of the InterpolationProblem. 4. Variations
We have seen that monotonicity and property ( ⋆ ) are necessary and suffi-cient to guarantee the existence of a solution of the Interpolation Problem.The following example shows that these two conditions are independent,hence neither of them can be dropped. MIGUEL COUCEIRO AND TAM ´AS WALDHAUSER
Example 9.
Let L be a distributive lattice, let a, b, c ∈ L such that a < b Let L = { , , a, b } with 0 < a, b < a, b incomparable.Let n = 1 and D = { , b } , and define f : D → L by f (0) = b , f ( b ) = a and g : D → L by g (0) = a , g ( b ) = 1 Then f trivially satisfies (4), but f is notmonotone, hence it is not the restriction of any polynomial function. Onthe other hand, g does not satisfy (4), although it is the restriction of thepolynomial function p ( x ) = x ∨ a to D .Observe that if L is a chain, then (4) implies that f is monotone , but thisis not true for arbitrary distributive lattices (see the example above). Thuswe may want to require that f is a monotone function satisfying (4). Wewill prove below that if D is of “rectangular” shape, then monotonicity of f and condition (4) are sufficient to ensure that f extends to a polynomialfunction (but (4) is not necessary, as we have seen in Example 12). Proposition 13. Let L be a distributive lattice and D = { b e I : I ⊆ [ n ] } asin the Interpolation Problem. If f : D → L is monotone and satisfies (4) ,then there exists a polynomial function p over L such that p | D = f .Proof. Let f : D → L be a monotone function satisfying (4). By Theorem 8,we only have to prove that f also satisfies ( ⋆ ). Let us assume that k / ∈ I ; theother case is similar. Then only f (cid:0)b e I ∪{ k } (cid:1) ∧ a k ≤ f ( b e I ) needs to be verified,as the second inequality of ( ⋆ ) is trivial in this case. Since f is monotone,we have f ( b e I ) ≤ f (cid:0)b e I ∪{ k } (cid:1) , and if equality holds here, then we are done.On the other hand, if f ( b e I ) < f (cid:0)b e I ∪{ k } (cid:1) , then (4) implies that there is an i ∈ [ n ] such that(5) ( b e I ) i ≤ f ( b e I ) < f (cid:0)b e I ∪{ k } (cid:1) ≤ (cid:0)b e I ∪{ k } (cid:1) i . This is clearly impossible for i = k , since then the i -th component of b e I and b e I ∪{ k } is the same (namely, a i ). Thus we must have i = k , and then (5)reads as a k ≤ f ( b e I ) < f (cid:0)b e I ∪{ k } (cid:1) ≤ b k . From this we immediately obtain the desired inequality: f (cid:0)b e I ∪{ k } (cid:1) ∧ a k ≤ a k ≤ f ( b e I ) . (cid:3) Finally, we give an example that shows that monotonicity and condition(4) together do not guarantee the existence of a solution of the InterpolationProblem if L is an arbitrary distributive lattice and D is an arbitrary subsetof L n . Thus it remains as a topic of further research to find an appropriatecriterion for the existence of an interpolating lattice polynomial function inthis general setting. Of course, this follows from Theorem 11, but it is also easy to verify directly. Example 14. Let L be the same lattice as in Example 12, and let D = { a, b } . Then the function f : D → L defined by f ( a ) = b , f ( b ) = a ismonotone and satisfies (4), but it is not the restriction of a polynomialfunction. 5. Application in decision making The original motivation for considering the Interpolation Problem lies inthe following mathematical model of multicriteria decision making. Let usassume that we have a set of alternatives from which we would like to choosethe best one (e.g., a house to buy). Several properties of these alternativescould be important in making the decision (e.g., the size, price, etc., ofa house), and this very fact can make the decision difficult (for instance,maybe it is not clear whether a cheap and small house is better than a bigand expensive one). To overcome this difficulty, the values correspondingto the various properties of each alternative should be combined to a singlevalue, which can then be easily compared.To formalize this situation, let us assume that there are n criteria alongwhich the alternatives are evaluated, and these take their values in linearlyordered sets L , . . . , L n . These linearly ordered sets could be quantitativescales (e.g., L could be the real interval [40 , L could be the finitechain { very small < small < big < very big } ). Thus, to each alternative cor-responds a profile x ∈ L × · · · × L n . Since this product is usually not alinearly ordered set, some alternatives may be incomparable. Therefore, wechoose a common scale L , and monotone functions ϕ i : L i → L ( i ∈ [ n ]) totranslate the values corresponding to the different criteria (which may havedifferent units of measure, e.g., square meters, euros, etc.) to this commonscale, and which are then combined into a single value (for each alternative)by a so-called aggregation function p : L n → L . In this way we obtain afunction U : L × · · · × L n → L defined by(6) U ( x ) = p ( ϕ ( x ) , . . . , ϕ n ( x n )) , and we can choose the alternative that maximizes U . The function U iscalled a global utility function, whereas the maps ϕ i are called local utilityfunctions. The relevance of such functions is attested by their many applica-tions in decision making, in particular, in representing preference relations[2].It is common to choose the real interval [0 , 1] for L , and consider ϕ i ( x i )as a kind of “score” with respect to the i -th criterion. In this case, simpleaggregation functions p are for instance the weighted arithmetic means, butthere are of course other, more elaborate ways of aggregating the scoressuch as the so-called Choquet integrals. However, in the qualitative ap-proach, where only the ordering between scores is taken into account (forinstance, when L = { bad < OK < good < excellent } ), such operators are oflittle use since they rely heavily on the arithmetic structure of the real unit ENERALIZATION OF GOODSTEIN’S THEOREM 11 interval. In the latter setting, one of the most prominent class of aggregationfunctions is that of discrete Sugeno integrals, which coincides with the classof idempotent lattice polynomial functions (see [10]).In [5] and [6] a more general situation was considered: L is an arbitraryfinite distributive lattice, the lattice polynomial functions are not assumedto be idempotent, and the local utility functions are not assumed to bemonotone (instead they have to satisfy the boundary conditions ϕ i (0 i ) ≤ ϕ i ( x i ) ≤ ϕ i (1 i ) for all x i ∈ L i , where 0 i and 1 i denote the least and greatestelement of L i ). The corresponding compositions (6) were called pseudo-polynomial functions, and several axiomatizations were given for this classof functions. Besides axiomatization, another noteworthy problem is thefactorization of such functions: given a function U : L × · · · × L n → L , findall factorizations of U in the form (6). Such a factorization can be useful inreal-life applications, when only the function U is available (from empiricalobservations), and an analysis of the behavior of the local utility functions ϕ i and of the aggregation function p could give valuable information aboutthe decision maker’s attitude.Suppose that we have already found the local utility functions ϕ i (see[5] and [6] for a method to find them), and let a i = ϕ i (0 i ) , b i = ϕ i (1 i ).If x ∈ L × · · · × L n is such that x i = 1 i if i ∈ I and x i = 0 i if i / ∈ I ,then U ( x ) = p ( b e I ). Thus, knowing the global utility function U , we haveinformation about p | D , and we can use Theorem 8 to find all possible latticepolynomial functions p that can appear in a factorization (6) of U . (Ofcourse, one has to take into account the other values of U as well, but thiscan be done by using the boundary conditions.) Acknowledgments. The first named author is supported by the internal re-search project F1R-MTH-PUL-09MRDO of the University of Luxembourg.The second named author acknowledges that the present project is sup-ported by the T ´AMOP-4.2.1/B-09/1/KONV-2010-0005 program of the Na-tional Development Agency of Hungary, by the Hungarian National Foun-dation for Scientific Research under grants no. K77409 and K83219, by theNational Research Fund of Luxembourg, and cofunded under the MarieCurie Actions of the European Commission (FP7-COFUND). References [1] Behrisch, M., Couceiro, M., Kearnes, K., Lehtonen, E., Szendrei, ´A.: Commutingpolynomial operations of distributive lattices. To appear in Order[2] Bouyssou, D., Dubois, D., Prade, H., Pirlot, M. (eds): Decision-Making Process –Concepts and Methods. ISTE/John Wiley (2009)[3] Couceiro, M., Lehtonen, E.: Self-commuting lattice polynomial functions on chains.Aequationes Math. 81(3), 263–278 (2011)[4] Couceiro, M., Marichal, J.-L.: Characterizations of discrete Sugeno integrals as poly-nomial functions over distributive lattices. Fuzzy Sets and Systems 161(5), 694–707(2010) [5] Couceiro, M., Waldhauser, T.: Axiomatizations and factorizations of Sugeno utilityfunctions. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 19(4), 635–658(2011)[6] Couceiro, M., Waldhauser, T.: Pseudo-polynomial functions over finite distributivelattices. In: Liu, W. (ed.) ECSQARU 2011. LNCS (LNAI), vol. 6717, pp. 545–556.Springer (2011)[7] Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Cambridge Uni-versity Press, New York (2002)[8] Goodstein, R. L.: The Solution of Equations in a Lattice. Proc. Roy. Soc. EdinburghSection A, 67, 231–242 (1965/1967)[9] Gr¨atzer, G.: General Lattice Theory. Birkh¨auser Verlag, Berlin (2003)[10] Marichal, J.-L.: Weighted lattice polynomials. Discrete Mathematics 309(4), 814–820(2009)[11] Rico, A., Grabisch, M., Labreuche, Ch., Chateauneuf, A.: Preference modeling ontotally ordered sets by the Sugeno integral. Discrete Applied Math. 147(1), 113–124(2005)(Miguel Couceiro) Mathematics Research Unit, FSTC, University of Luxem-bourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg E-mail address : miguel.couceiro[at]uni.lu (Tam´as Waldhauser) Mathematics Research Unit, FSTC, University of Lux-embourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg, andBolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, H-6720 Szeged,Hungary E-mail address ::