Self-commuting lattice polynomial functions on chains

SELF-COMMUTING LATTICE POLYNOMIAL FUNCTIONS arXiv:0912.0478v1 [math.RA] 2 Dec 2009 MIGUEL COUCEIRO AND ERKKO LEHTONEN Abstract. We provide sufficient conditions for a lattice polynomial function to be self-commuting. We explicitly describe self-commuting polynomial functions over chains. 1. Introduction Two operations f : An → A and g : Am → A are said to commute, if for all aij ∈ A (1 ≤ i ≤ n, 1 ≤ j ≤ m), the following identity holds
f g(a11 , a12 , . . . , a1m ), g(a21 , a22 , . . . , a2m ), . . . , g(an1 , an2 , . . . , anm )
= g f (a11 , a21 , . . . , an1 ), f (a12 , a22 , . . . , an2 ), . . . , f (a1m , a2m , . . . , anm ) . For n = m = 2, the above condition stipulates that

f g(a11 , a12 ), g(a21 , a22 ) = g f (a11 , a21 ), f (a12 , a22 ) . The Eckmann-Hilton theorem [12] asserts that if both f and g have an identity element and f ⊥ g, then in fact f = g and (A; f ) is a commutative monoid on A. The relevance of the notion of commutation is made apparent in works of several authors. In particular, commutation is the defining property of entropic algebras [22, 23, 27] (an algebra is entropic if its operations commute pairwise; idempotent entropic algebras are called modes) and centralizer clones [18, 19, 25, 28] (the centralizer of a set F of operations is the set of all operations that commute with every operation in F ; the centralizer of F is a clone). We are interested in functions f that commute with themselves. An algebra (A; f ) where f is a binary operation that satisfies the identity

f f (a11 , a12 ), f (a21 , a22 ) = f f (a11 , a21 ), f (a12 , a22 ) is called a medial groupoid [16, 17]. Hence, self-commutation generalizes the notion of mediality (see, e.g., [14]), and it has been investigated by several authors (see, e.g., [1, 2, 20, 26]). In the realm of aggregation theory, self-commutation is also known as bisymmetry; for motivations and general background, see [14]. In this paper, we address the question of characterizing classes of self-commuting operations. In Section 2, we recall basic notions in the universal-algebraic setting and settle the terminology used throughout the paper. Moreover, by showing that self-commutation is preserved under several operations (e.g., permutation of variables, identification of variables and addition of dummy variables), we develop general tools for tackling the question of describing self-commuting operations. This question is partially answered for lattice polynomial functions (in particular, for the so-called discrete Sugeno integrals, i.e., idempotent polynomial functions; see, e.g., [8, 14]) in Section 3. We start by surveying well-known results concerning normal form representations of these lattice functions which we then use to specify those polynomial functions on bounded chains which are self-commuting. This Date: December 2, 2009. 1 2 MIGUEL COUCEIRO AND ERKKO LEHTONEN explicit description is obtained by providing sufficient conditions for a lattice polynomial function to be self-commuting, and by showing that these conditions are also necessary in the particular case of polynomial functions over bounded chains. In Section 4 we point out problems which are left unsettled, and motivate directions of future research. 2. Preliminaries In this section, we introduce some notions and terminology as well as establish some preliminary results that will be used in the sequel. For an integer n ≥ 1, set [n] := {1, 2, . . . , n}. With no danger of ambiguity, we denote the tuple (x1 , . . . , xn ) of any length by x. 2.1. Operations and algebras. Let A be an arbitrary nonempty set. An operation on A is a map f : An → A for some integer n ≥ 1, called the arity of f . We (n) denote by OA the set of all n-ary operations on A, and we denote by OA the set S (n) of all finitary operations on A, i.e., OA := n≥1 OA . We assume that the reader is familiar with basic notions of universal algebra and lattice theory. In particular, the concepts of term operation and polynomial operation will not be defined in the current paper, and we refer the reader to [3, 4, 9, 10, 11, 15, 24] for general background on universal algebra and lattice theory. (n) (m) 2.2. Simple minors. Let f ∈ OA , g ∈ OA . We say that f is obtained from g by simple variable substitution, or f is a simple minor of g, if there is a mapping σ : [m] → [n] such that f (x1 , . . . , xn ) = g(xσ(1) , xσ(2) , . . . , xσ(m) ). If σ is not injective, then we speak of identification of variables. If σ is not surjective, then we speak of addition of inessential variables. If σ is bijective, then we speak of permutation of variables. For distinct indices i, j ∈ [n], the function fi←j : An → A obtained from f by the simple variable substitution fi←j (x1 , . . . , xn ) := f (x1 , . . . , xi−1 , xj , xi+1 , . . . , xn ) is called a variable identification minor of f , obtained by identifying xi with xj . For studies of classes of operations that are closed under taking simple minors, see, e.g., [5, 21]. 2.3. Self-commutation. Let f : An → A and g : Am → A be operations on A. We say that f commutes with g, denoted f ⊥ g, if for all aij (i ∈ [n], j ∈ [m]), it holds that
f g(a11 , a12 , . . . , a1m ), g(a21 , a22 , . . . , a2m ), . . . , g(an1 , an2 , . . . , anm )
= g f (a11 , a21 , . . . , an1 ), f (a12 , a22 , . . . , an2 ), . . . , f (a1m , a2m , . . . , anm ) . If f ⊥ f , then we say that f is self-commuting. (n) (m) Lemma 2.1. Let f ∈ OA , g ∈ OA , and let σ : [n] → [ν] and τ : [m] → [µ] be (µ) (ν) arbitrary mappings. Let fσ ∈ OA and gτ ∈ OA be the operations defined by fσ (x1 , . . . , xν ) = f (xσ(1) , . . . , xσ(n) ), gτ (x1 , . . . , xµ ) = g(xτ (1) , . . . , xτ (m) ). If f ⊥ g, then fσ ⊥ gτ . SELF-COMMUTING LATTICE POLYNOMIAL FUNCTIONS 3 Proof. By the definition of fσ and gτ ,
fσ gτ (a11 , a12 , . . . , a1m ), gτ (a21 , a22 , . . . , a2m ), . . . , gτ (an1 , an2 , . . . , anm ) = f g(aσ(1)τ (1) , aσ(1)τ (2) , . . . , aσ(1)τ (m) ), g(aσ(2)τ (1) , aσ(2)τ (2) , . . . , aσ(2)τ (m) ), . . . ,
g(aσ(n)τ (1) , aσ(n)τ (2) , . . . , aσ(n)τ (m) ) = g f (aσ(1)τ (1) , aσ(2)τ (1) , . . . , aσ(m)τ (1) ), f (aσ(1)τ (2) , aσ(2)τ (2) , . . . , aσ(m)τ (2) ), . . . ,
f (aσ(1)τ (n) , aσ(2)τ (n) , . . . , aσ(n)τ (m) ) =
gτ fσ (a11 , a21 , . . . , am1 ), fσ (a12 , a22 , . . . , am2 ), . . . , fσ (a1n , a2n , . . . , anm ) , where the second equality holds by the assumption that f ⊥ g.  Corollary 2.2. If f ∈ OA is self-commuting, then every simple minor of f is self-commuting. In the particular case when A is finite, Corollary 2.2 translates into saying that the class of self-commuting operations on A is definable by functional equations (see [6]). The set of self-commuting operations is also closed under special type of substitutions of constants for variables, as described by the following lemma. Let f : An → A and c ∈ A. For i ∈ [n], we define fci : An−1 → A to be the operation fci (a1 , . . . , an−1 ) = f (a1 , . . . , ai−1 , c, ai , . . . , an−1 ). Lemma 2.3. Assume that f : An → A preserves c ∈ A, i.e., f (c, . . . , c) = c. If f is self-commuting, then for every i ∈ [n], fci is self-commuting. Proof. We will show that the claim holds for i = 1. It the follows from Lemma 2.1, by considering suitable permutations of variables, that the claim holds for all i ∈ [n]. By the definition of fc1 and by the assumption that f (c, . . . , c) = c, we have
fc1 fc1 (a11 , . . . , a1,n−1 ), . . . , fc1 (an−1,1 , . . . , an−1,n−1 ) =
f f (c, c, . . . , c), f (c, a11 , . . . , a1,n−1 ), . . . , f (c, an−1,1 , . . . , an−1,n−1 ) =
f f (c, c, . . . , c), f (c, a11 , . . . , an−1,1 ), . . . , f (c, a1,n−1 , . . . , an−1,n−1 ) =
fc1 fc1 (a11 , . . . , an−1,1 ), . . . , fc1 (a1,n−1 , . . . , an−1,n−1 ) , where the second equality holds by the assumption that f is self-commuting.  3. Self-commuting lattice polynomial functions Let (L; ∧, ∨) be a lattice. With no danger of ambiguity, we denote lattices by their universes. In this section we study the self-commutation property on lattice polynomial functions, i.e., mappings f : Ln → L which can be obtained as compositions of the lattice operations and applied to variables (projections) and constants. As shown by Goodstein [13], lattice polynomial functions have neat normal form representations in the case when L is a bounded distributive lattice. Thus, in what follows we assume that L is a bounded distributive lattice with least and greatest elements 0 and 1, respectively. We recall the necessary representation results concerning the representation of lattice polynomials as well as introduce some related concepts and terminology in Subsection 3.1. Then, we consider the property of self-commutation on these functions. We start by providing sufficient conditions for a lattice polynomial function to be self-commuting, which we then use to obtain explicit descriptions of those polynomial functions on chains which satisfy this self-commutation property. 4 MIGUEL COUCEIRO AND ERKKO LEHTONEN 3.1. Preliminary results: representations of lattice polynomials. An n-ary (lattice) polynomial function from Ln to L is defined recursively as follows: (i) For each i ∈ [n] and each c ∈ L, the projection x 7→ xi and the constant function x 7→ c are polynomial functions from Ln to L. (ii) If f and g are polynomial functions from Ln to L, then f ∨ g and f ∧ g are polynomial functions from Ln to L. (iii) Any polynomial function from Ln to L is obtained by finitely many applications of the rules (i) and (ii). If rule (i) is only applied for projections, then the resulting polynomial functions are called (lattice) term functions [4, 15, 10]. Idempotent polynomial functions are also referred to as (discrete) Sugeno integrals [8, 14]. In the case of bounded distributive lattices, Goodstein [13] showed that polynomial functions are exactly those which allow representations in disjunctive normal form (see Proposition 3.1 below, first appearing in [13, Lemma 2.2]; see also Rudeanu [24, Chapter 3, §3] for a later reference). Proposition 3.1. Let L be a bounded distributive lattice. A function f : Ln → L is a polynomial function if and only if there exist aI ∈ L, I ⊆ [n], such that, for every x ∈ Ln , ^ _ (aI ∧ xi ). f (x) = I⊆[n] i∈I The expression given in Proposition 3.1 is usually referred to as the disjunctive normal form (DNF) representation of the polynomial function f . In order to simplify notation, if I is a singleton or a two-element set, then we write ai and aij for a{i} and a{i,j} , respectively. The following corollaries belong to the folklore of lattice theory and are immediate consequences of Theorems D and E in [13]. Corollary 3.2. Every polynomial function is completely determined by its restriction to {0, 1}n . Corollary 3.3. A function g : {0, 1}n → L can be extended to a polynomial function f : Ln → L if and only if it is nondecreasing. In this case, the extension is unique. It is easy to see that the DNF representations of a polynomial function f : Ln → L are not necessarily unique. For instance, in Proposition 3.1, if for some I ⊆ [n] W we have aI = J(I aJ , then for every x ∈ Ln , ^ _ (aJ ∧ xi ). f (x) = I6=J⊆[n] i∈J V We refer to the term aI i∈I x W we say that |I| is its size. Vi as the I-th term of f , and We say that the I-th term aI i∈I xi is essential if aI > J(I aJ ; otherwise, we say that it is inessential. (For a discussion on the uniqueness of DNF representations of lattice polynomial functions see [8].) However, using Corollaries 3.2 and 3.3, one can easily set canonical ways of constructing these normal form representations of polynomial functions. Let 2[n] denote the set of all subsets of [n]. For I ⊆ [n], let eI be the characteristic vector of I, i.e., the n-tuple in Ln whose i-th component is 1 if i ∈ I, and 0 otherwise. Note that the mapping α : 2[n] → {0, 1}n given by α(I) = eI , for every I ∈ 2[n] , is an order-isomorphism. Proposition 3.4 (Goodstein [13]). Let L be a bounded distributive lattice. A function f : Ln → L is a polynomial function if and only if for every x ∈ Ln , ^
_ f (eI ) ∧ xi . f (x) = I⊆[n] i∈I SELF-COMMUTING LATTICE POLYNOMIAL FUNCTIONS 5 It is noteworthy that Proposition 3.4 leads to the following characterization of the essential arguments of polynomial functions in terms of necessary and sufficient conditions [7]. Proposition 3.5. Let L be a bounded distributive lattice and let f : Ln → L be a polynomial function. Then for each j ∈ [n], xj is essential in f if and only if there exists a set J ⊆ [n] \ {j} such that f (eJ ) < f (eJ∪{j} ). Remark 1. The assumption that the lattice L is bounded is not very crucial. Let L′ be the lattice obtained from L by adjoining new top and bottom elements ⊤ and ⊥, if necessary. Then, if f is a polynomial function over L induced by a polynomial p, then p induces a polynomial function f ′ on L′ , and it holds that the restriction of f ′ to L coincides with f . Similarly, if L′ is a distributive lattice and f ′ is a polynomial function on L′ represented by the DNF ^ _ (aI ∧ xi ), I⊆[n] i∈I V then V by omitting each term aIV∧ i∈I xi where aI = ⊥ and replacing each term aI ∧ i∈I xi where aI = ⊤ by i∈I xi , we obtain an equivalent polynomial representation for f ′ . Unless f ′ is the constant function that takes value ⊤ or ⊥ and this element is not in L, the function f on L induced by this new polynomial coincides with the restriction of f ′ to L. 3.2. Self-commuting polynomial functions on chains. In this subsection we provide explicit descriptions of self-commuting polynomial functions on chains. A lattice polynomial function f : Ln → L is said to be a weighted disjunction if it is of the form _ ai xi (1) f (x1 , x2 , . . . , xn ) = a∅ ∨ i∈[n] for some elements a∅ , ai (i ∈ [n]) of L. We say that f has chain form if _ ^ _ ai xi ∨ aS ℓ xi , (2) f (x1 , x2 , . . . , xn ) = a∅ ∨ 1≤ℓ≤r i∈[n] i∈Sℓ for a chain of subsets S1 ⊆ S2 ⊆ · · · ⊆ Sr ⊆ [n], r ≥ 1, |S1 | ≥ 2, and some elements a∅ , ai (i ∈ [n]), aSℓ (1 ≤ ℓ ≤ r) of L such that aI ≤ aJ whenever I ⊆ J, and for all i∈ / S1 , there is a j ∈ S1 such that ai ≤ aj . Theorem 3.6. Let L be a bounded chain. A polynomial function f : Ln → L is self-commuting if and only if it is a weighted disjunction or it has chain form. Theorem 3.6 will be a consequence of the following two results. We start with a lemma that provides sufficient conditions for a polynomial to be self-commuting in the general case of bounded distributive lattices. Lemma 3.7. Let L be a distributive lattice. Assume that a function f : Ln → L is a weighted disjunction or has chain form. Then f is self-commuting. Proof. Assume first that f is a weighted disjunction. We have that
f f (x11 , x12 , . . . , x1n ), . . . , f (xn1 , xn2 , . . . , xnn ) _ _ _ _ ai aj xij aj xij ) = a∅ ∨ ai (a∅ ∨ = a∅ ∨ = a∅ ∨ i∈[n] j∈[n] j∈[n] i∈[n] _ _ j∈[n] i∈[n] aj ai xij = a∅ ∨ _ j∈[n] aj (a∅ ∨ _ ai xij ) i∈[n]
= f f (x11 , x21 , . . . , xn1 ), . . . , f (x1n , x2n , . . . , xnn ) . 6 MIGUEL COUCEIRO AND ERKKO LEHTONEN Thus, f is self-commuting. Assume then that f has chain form. The assumption that for every i ∈ / S1 there is a j ∈ S1 such that ai ≤ aj implies that ai ≤ aSℓ (and hence ai aSℓ = ai ) for all i ∈ [n] and for all ℓ ∈ [r]. Using this observation and the distributive laws we get
f f (x11 , x12 , . . . , x1n ), f (x21 , x22 , . . . , x2n ), . . . , f (xn1 , xn2 , . . . , xnn ) i _ ^ _ _ h aj xij ∨ aS ℓ xij ai a∅ ∨ = a∅ ∨ _ ∨ aS t _ _ i∈St j∈[n] ai aj xij ∨ _ ∨ {z (I) _ 1≤ℓ≤r _ ai i∈[n] 1≤ℓ≤r i∈[n] j∈[n] | j∈Sℓ i ^h _ ^ _ aj xij ∨ aS ℓ xij a∅ ∨ 1≤t≤r = a∅ ∨ 1≤ℓ≤r j∈[n] i∈[n] } ^ xij j∈Sℓ {z | j∈Sℓ (II) } i _ ^ ^h _ aj xij ∨ xij . aS t aS ℓ a∅ ∨ 1≤t≤r i∈St 1≤ℓ≤r j∈[n] j∈Sℓ {z | } (III) Every term in (II) is absorbed by a term in (I): for every i ∈ [n],V there is a k ∈ S1 such that ai ≤ ak , and hence for any ℓ ∈ [r], the term ai j∈Sℓ xij = V ai ak xik j∈Sℓ \{k} xij in (II) is absorbed by the term ai ak xik in (I). V V In (III), for a fixed t, if ℓ > t, then the term aSt aSℓ j∈Sℓ xij = aSt j∈Sℓ xij is V V absorbed by aSt j∈St xij = aSt aSt j∈St xij , and hence (III) simplifies to i _ ^ _ _ ^h aj xij ∨ xij . aS ℓ a∅ ∨ (3) 1≤t≤r i∈St 1≤ℓ≤t j∈[n] | j∈Sℓ {z } (IV) V For a fixed t, (IV) expands to the disjunction of all possible conjunctionsV i∈St φi of |St | terms, where each φi is one of a∅ , aj xij for some j ∈ [n], or aSℓ j∈Sℓ xij for some 1 ≤ ℓ ≤ t. If φi = a∅ for some i ∈ St , then the conjunction is absorbed by a∅ . If φi = ai xii for some i ∈ St , then the conjunction is absorbed by the term ai ai xii = ai xii in (I). V Consider then such a conjunction i∈St φi where for all i ∈ St , φi is not equal to a∅ nor to ai xii , but for some i ∈ St , φi = aj xij for some j 6= i. By our assumption, there is a k ∈ S1 such that aj ≤ ak Vand hence aj = aj ak . We have that φk equals either aℓ xkℓ for some ℓ 6= k or aSℓ m∈Sℓ xkm for some 1 ≤ ℓ ≤ t. In the former V case, φi φk = aj ak xij aℓ xkℓ , and hence the conjunction i∈SV φi is absorbed by the t term ak aℓ xkℓ in (I). In the latter case, φi φk = aj ak xij aSℓ m∈Sℓ xkm , and hence V the conjunction i∈St φi is absorbed by the term ak ak xkk = ak xkk in (I). The remaining conjunctions that arise from the expansion of (IV) are of the form ^ ^ aS ℓ i xij i∈St j∈Sℓi ′ ′ where 1 ≤ ℓi ≤ tV(i ∈ SV t ). Let ℓ = mini∈St ℓi . If ℓ < t, then this conjunction is absorbed by aSℓ′ i∈Sℓ′ j∈Sℓ′ xij , which arises from the expansion of i ^ h _ ^ _ aj xij ∨ aS ℓ xij a∅ ∨ i∈Sℓ′ j∈[n] 1≤ℓ≤ℓ′ j∈Sℓ in (3).VThus,Vthe only remaining conjunction that arises from the expansion of (IV) is aSt i∈St j∈St xij . SELF-COMMUTING LATTICE POLYNOMIAL FUNCTIONS 7 Thus, we have that
(4) f f (x11 , x12 , . . . , x1n ), f (x21 , x22 , . . . , x2n ), . . . , f (xn1 , xn2 , . . . , xnn ) = _ ^ ^ _ _ ai aj xij ∨ aS ℓ a∅ ∨ xij . i∈[n] j∈[n] 1≤ℓ≤r i∈Sℓ j∈Sℓ In a similar way, we can deduce that
(5) f f (x11 , x21 , . . . , xn1 ), f (x12 , x22 , . . . , xn2 ), . . . , f (x1n , x2n , . . . , xnn ) = _ ^ ^ _ _ ai aj xij ∨ aS ℓ a∅ ∨ xij . j∈[n] i∈[n] 1≤ℓ≤r j∈Sℓ i∈Sℓ The right hand sides of (4) and (5) are clearly equal, and we conclude that f is self-commuting.  The necessity of the conditions in Theorem 3.6 follows from our next lemma. Lemma 3.8. Let L be a bounded chain. If a polynomial function f : Ln → L is self-commuting, then it is a weighted disjunction or it has chain form. Proof. The statement clearly holds for n = 1 and n = 2, since every unary or binary polynomial function is a weighted disjunction or has chain form. Suppose n = 3. Then (6) f = a∅ ∨ a1 x1 ∨ a2 x2 ∨ a3 x3 ∨ a12 x1 x2 ∨ a13 x1 x3 ∨ x23 x2 x3 ∨ a123 x1 x2 x3 , where aI ≤ aJ whenever I ⊆ J. If for all i, j ∈ {1, 2, 3}, ai ∨ aj = aij , then each term aij xi xj in (6) equals (ai ∨ aj )xi xj = ai xi xj ∨ aj xi xj and gets absorbed by ai xi and aj xj , and hence f has the desired form (1) or (2). Otherwise, there exist i, j such that ai ∨ aj < aij ; without loss of generality, assume that a1 ∨ a2 < a12 . We have that
(7) f f (1, 1, 0), f (0, 1, 1), f (0, 0, 0) = a1 ∨ a2 ∨ a12 a23 ,
f f (1, 0, 0), f (1, 1, 0), f (0, 1, 0) = a1 ∨ a2 , and since f is self-commuting, we have a1 ∨ a2 ∨ a12 a23 = a1 ∨ a2 . This equality translates into a12 a23 ≤ a1 ∨ a2 . In a similar way, after suitably permuting the rows and columns of the 3 × 3 matrix used in (7), we can deduce that (8) aij ajk ≤ ai ∨ aj ≤ aij for {i, j, k} = {1, 2, 3}. Since L is a chain, we have for some choice of {α, β, γ} = {1, 2, 3} that aαβ ≤ aβγ ≤ aαγ . Inequalities (8) then imply aα ∨ aβ = aαβ and aβ ∨ aγ = aαγ , i.e., the terms associated with sets {α, β} and {α, γ} are inessential. Thus, f has at most one essential term of size 2. If f has no essential term of size 2, then either it is a weighted disjunction or it has chain form with S1 = {1, 2, 3}. Otherwise f has precisely one essential term of size 2, say, associated with S1 = {1, 2}. Then a12 > a1 ∨ a2 and a3 ≤ a13 = a13 a12 ≤ a1 ∨ a2 . Since L is a chain, a3 ≤ a1 or a3 ≤ a2 , and we conclude that f has chain form. We proceed by induction on n. Assume that the claim holds for n < ℓ for some ℓ ≤ 4. We show that V for n = ℓ. W it holds Let f = a∅ ∨ I⊆[ℓ] aI i∈I xi be self-commuting, and assume that aI ≤ aJ whenever I ⊆ J. If f has no essential terms of size at least 2, then f is a weighted disjunction. Thus, we suppose that f has an essential term of size at least 2. First we show that the essential terms of f of size at least 2 are associated with a chain 8 MIGUEL COUCEIRO AND ERKKO LEHTONEN S1 ⊆ S2 ⊆ · · · ⊆ Sq . For a contradiction, suppose that there are I, J ⊆ [k] such that |I| ≥ 2, |J| ≥ 2, I k J and the I-th and the J-th terms of f are essential. Fix such I and J so that |I ∩ J| is the largest possible, |I| ≤ |J| and |J| is the largest among such pairs. We will consider several cases. Case 1: |I ∩ J| ≥ 2. Take distinct i, j ∈ I ∩ J, and V consider fi←j .VThis function is a polynomial function having essential terms bI ′ i∈I ′ xi and bJ ′ i∈J ′ xi where I ′ = I \ {i}, J ′ = J \ {i}. Since I ′ k J ′ , the induction hypothesis implies that fi←j is not self-commuting, which contradicts Corollary 2.2 which asserts that selfcommutation is preserved by taking simple minors. Case 2: |I ∩ J| ≤ 1 and |J| ≥ 3. Take distinct i, j ∈ J \ I, and considert fi←j . As V in Case 1, we derive V a contradiction, because this function has essential terms bI ′ i∈I ′ xi and bJ ′ i∈J ′ xi where I ′ = I, J ′ = J \ {i} and I ′ k J ′ . Case 3: |I ∩ J| = 0 and |J| = 2. Take i ∈ I, j ∈ J, and consider fV i←j . Again, we V derive a contradiction, because this function has essential terms bI ′ i∈I ′ xi and bJ ′ i∈J ′ xi where I ′ = (I \ {i}) ∪ {j}, J ′ = J and I ′ k J ′ . Case 4: |I ∩ J| = 1, |J| = 2 and ℓ ≥ 5. Take distinct i, j ∈ [ℓ] \ (I ∪ J), and consider fV a contradiction, because this function has essential i←j . Again, we derive V terms bI ′ i∈I ′ xi and bJ ′ i∈J ′ xi where I ′ = I, J ′ = J and I ′ k J ′ . Case 5: |I ∩ J| = 1, |J| = 2 and ℓ = 4. We have that
(9) f f (1, 1, 0, 0), f (0, 1, 1, 0), f (0, 0, 0, 0), f(0, 0, 0, 0) = a1 ∨ a2 ∨ a12 a23 ,
f f (1, 0, 0, 0), f (1, 1, 0, 0), f (0, 1, 0, 0), f(0, 0, 0, 0) = a1 ∨ a2 , and since f is self-commuting, we have a1 ∨ a2 ∨ a12 a23 = a1 ∨ a2 . This equality translates into a12 a23 ≤ a1 ∨ a2 . In a similar way, after suitably permuting the rows and columns of the 4 × 4 matrix used in (9), we can deduce that (10) aij ajk ≤ ai ∨ aj ≤ aij for distinct i, j, k ∈ {1, 2, 3, 4}. Assume, without loss of generality, that I = {1, 2}, J = {2, 3}. Since L is a chain, we either have a12 ≤ a23 or a23 < a12 . In the former case, by (10), we have a12 = a12 a23 ≤ a1 ∨ a2 ≤ a12 , which implies that a12 = a1 ∨ a2 , which contradicts the assumption that the I-th term of f is essential. In the latter case, we have a23 = a23 a12 ≤ a2 ∨ a3 ≤ a23 , which implies that a23 = a2 ∨ a3 , which contradicts the assumption that the J-th term of f is essential. Thus, the essential terms of f of size at least 2 are associated with a chain S1 ⊆ S2 ⊆ · · · ⊆ Sq . To complete the proof, we need to show that for every i ∈ / S1 , there is a j ∈ S1 such that ai ≤ aj . For a contradiction, suppose that there is an i∈ / S1 such that ai > aj for every j ∈ S1 . We consider several cases. Case 1: |S1 | ≥ 3. Take distinct k, m ∈ S1 , and consider fk←m . The essential terms of fk←m of size at least 2 are associated with a chain S1′ ⊆ S2′ ⊆ · · · ⊆ Sq′ , where Si′ := Si \ {k} for 1 ≤ i ≤ q, and the m-th term of f is (ak ∨ am )xm . Since ai > aj for every j ∈ S1 , the induction hypothesis implies that fk←m is not selfcommuting. This contradicts Corollery 2.2 which asserts that self-commutation is preserved by taking simple minors. Case 2: |S1 | = 2. Then there is a t ∈ [ℓ]\ (S1 ∪{i}). Consider ft←i . The essential terms of ft←i of size at least 2 are associated with a chain whose least element is S1 , and the i-th term of this function is (ai ∨ at )xi . Since for every j ∈ S1 , ai > aj , we also have ai ∨at > aj , and, as above, we have reached the desired contradiction.  SELF-COMMUTING LATTICE POLYNOMIAL FUNCTIONS 9 Proof of Theorem 3.6. Lemma 3.7, when restricted to chains, shows that the condition is sufficient. Necessity follows from Lemma 3.8.  4. Concluding remarks and future work We have obtained an explicit form of self-commuting polynomial functions on chains (in fact, unique up to addition of inessential terms). As Lemma 3.7 asserts, our condition is sufficient in the general case of polynomial functions over distributive lattices. However, we do not know whether it is also a necessary condition in the general case. This constitutes a topic of ongoing research. Another problem which was not addressed concerns commutation. As mentioned, self-commutation appears within the scope of aggregation function theory under the name S of bisymmetry. In this context, functions are often regarded as mappings f : n≥1 An → A. In this framework, bisymmetry is naturally generalized to what is referred to as strong bisymmetry. Denoting by fn the restriction of f to An , the map f is said to be strongly bisymmetric if for any n, m ≥ 1, we have fn ⊥ fm . This generalization is both natural and useful from the application point of view. To illustrate this, suppose one is given data in tabular form, say an n×m matrix, to be meaningfully fused into a single representative value. One could first aggregate the data by rows and then aggregate the resulting column; or one could first aggregate the columns and then the resulting row. What is expressed by the property of strong bisymmetry is that the final outcome is the same under both procedures. Extending the notion of polynomial functions to such families, we are thus left with the problem of describing those families of polynomial functions which are strongly bisymmetric. 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Couceiro) University of Luxembourg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L–1359 Luxembourg, Luxembourg E-mail address: miguel.couceiro@uni.lu (E. Lehtonen) University of Luxembourg, Computer Science and Communications Research Unit, 6, rue Richard Coudenhove-Kalergi, L–1359 Luxembourg, Luxembourg E-mail address: erkko.lehtonen@uni.lu View publication stats