Commuting Polynomial Operations of Distributive Lattices

We describe which pairs of distributive lattice polynomial operations commute.


Introduction
Let f : X m → X and g : X n → X be operations on X and let [x ij ] ∈ X m×n be a matrix of elements of X. By f g([x ij ]) we mean the value obtained by first applying g to the rows of [x ij ] and then applying f to the column of results.
One holds for all x ij ∈ X (1 ≤ i ≤ m, 1 ≤ j ≤ n). A self-commuting operation is one that commutes with itself.
Operations that are self-commuting are also called entropic or medial. If C is a clone on X, then the set of operations on X that commute with each member of C is another clone on X, called the centralizer of C. Centralizer clones are also called bicentrally closed clones. On a finite set X, bicentrally closed clones coincide with primitive positive clones. There is a vast literature about entropic algebras, centralizer clones, and clones consisting of pairwise commuting operations. For entropic algebras, see [13] and the references therein. For commutative clones or centralizer clones, see, for example, [8,9,11,14]. For primitive positive clones, see [2,7,10,15,16].
Aggregation functions return a single representative value from a list of values (such as the maximum or average of a list of real numbers). To aggregate the values in a table, one might use a row aggregation function and a (possibly different) column aggregation function. The commutativity of the row and column aggregation functions asserts that the final value is independent of the order of aggregation. A self-commuting aggregation function is called bisymmetric, and certain sequences of pairwise commuting aggregation functions are called strongly bisymmetric. See [1,6] for more details.
It is easy to determine which pairs of module polynomial operations commute. Suppose that M is an R-module, and that f (x 1 , . . . , x m ) = m i=1 a i x i + c and g(x 1 , . . . , x n ) = n j=1 b j x j + d are module polynomial operations. If f and g commute on the zero matrix, then it must be that (i) ( m i=1 a i )d + c = ( n j=1 b j )c + d holds. If f g([x ij ]) = g f ([x ij ] T ) on just those matrices [x ij ] whose only nonzero entry is in position ij, then (ii) (a i b j − b j a i )x = 0 holds for all x ∈ M. Conversely, if (i) and (ii) hold, then f and g commute on all matrices in M m×n .
The main features of the argument for module polynomials are: a normal form for polynomial operations is used and final results are expressed in terms of this normal form; a commutativity condition on coefficients of the normal form must hold and a condition on the constant term must hold; the commutativity of polynomials on general matrices is equivalent to commutativity on matrices with at most one nonzero entry. All of these features have analogues in our argument for commuting distributive lattice polynomial operations. Our result will be expressed in terms of a distinguished disjunctive normal form for polynomial operations which we call maximal disjunctive normal form, and define in Section 2 (see conditions (2.3) and (2.4)). Using the notation [k] for the set {1, 2, . . . , k} for all natural numbers k, we can state the result as follows: if f (x 1 , . . . , x m ) = S⊆[m] a S i∈S x i and g(x 1 , . . . , x n ) = T⊆[n] b T j∈T x j are distributive lattice polynomial operations written in maximal disjunctive normal form, then f commutes with g if and only if (i) some condition on constant terms and leading coefficients is met and (ii) some type of "commutativity condition" is satisfied by all coefficients. Condition (i) is a ∅ + b ∅ ≤ a [m] b [n] , which asserts that the join of the constant terms is dominated by the meet of the leading coefficients. It turns out that this is equivalent to the condition that the ranges of f and g have nonempty intersection. This is obviously a necessary condition for f to commute with g, and is equivalent to the commutativity of f and g on the zero matrix. The commutativity condition for the other coefficients in the case when . This condition can be shown to hold provided f and g commute on all 0, 1-matrices where the 1's occur precisely in the union of two rectangular subregions . Conversely, we show that any pair of polynomial operations that commute on these "2-rectangle" matrices consisting solely of 0's and 1's must commute on all matrices. Still under the assumption that a ∅ = b ∅ = 0, we show that (1.2) is equivalent to the simpler condition together with the condition obtained from this by interchanging the roles of the a's and the b's. Corollaries of the main theorem include: a characterization of the self-commuting, distributive lattice polynomial operations (generalizing the results of [3]), and a characterization of the pairs of commuting distributive lattice term operations.
The main result of this paper was obtained after the BLAST 2010 conference held at the University of Colorado at Boulder. At this meeting, the report on [3] generated the question that is answered in this paper.

Preliminaries
Throughout the paper [n] denotes {1, 2, . . . , n} if n is a natural number, P(X) denotes the power set of a set X, ⊆ denotes inclusion, and ⊂ denotes proper inclusion for sets.
The join and meet operations of a lattice will be denoted by + and · (or juxtaposition), respectively. If a lattice has a least element, then it will be denoted by 0, and if it has a largest element, it will be denoted by 1. If L is a lattice, then L 01 denotes the smallest bounded lattice that contains L as a sublattice; that is, where 0 is the least element of L if L has a least element, while 0 is a new least element otherwise, and similarly for 1. It is straightforward to check that for a distributive lattice L the lattice L 01 is also distributive.
Recall that a clone of operations on a set X is a set of operations on X that contains the projection operations and is closed under composition. The clone of polynomial operations of an algebra on X is the least clone on X that contains the fundamental operations of the algebra and all constant operations on X. If L is a distributive lattice, then these conditions are satisfied by the collection of all operations on L which can be written as a join of meets where S is a nonempty set of subsets of [m] for some m ≥ 1, a S ∈ L for all S ∈ S, and S = ∅ if M S = i∈S x i . Allowing all elements of L 01 to be coefficients a S , we can write every meet M S = i∈S x i above as M S = a S i∈S x i with a S = 1, and we can expand the join S∈S M S by additional joinands M S = a S i∈S x i with a S = 0 whenever S ⊆ [m] but S / ∈ S. Thus we get the following. such that a ∅ = 1 if 1 / ∈ L, and at least one coefficient a For a distributive lattice L, we will denote the clone of polynomial operations of L by PClo(L), and for each f ∈ PClo(L), we will refer to a representation of f described in Lemma 2.1 as a disjunctive normal form, or briefly, a DNF of f . The joinand a S i∈S x i of a DNF will be called the S-term, and a S the S-coefficient of the DNF. If S = {i} is a singleton, then we will write a i instead of a {i} .
An operation in PClo(L) can have many different DNFs. We will call a DNF of f maximal if the following conditions hold for the coefficients a S : and The next proposition shows that every polynomial operation f of a distributive lattice has a unique maximal DNF. Moreover, it shows that for every S ⊆ [m], the S-coefficient of the maximal DNF of f dominates the S-coefficients of all DNFs of f , which justifies the name "maximal DNF". For bounded distributive lattices, the construction of the maximal DNF of f described in part (3) of the proposition can be found in [5]. Proof We start the proof of (1) by verifying the equality (2.2). For all elements where the first equality follows from the definition of a S , while the second one follows from the absorption laws and the fact that a Q i∈S x i ≤ a Q i∈Q x i whenever Q ⊆ S. This proves (2.2). Condition (2.3) follows immediately from the definition of a S . Finally, by Lemma 2.1, the coefficients a S have to satisfy the conditions that a ∅ = 1 if 1 / ∈ L, and at least one a S = 0 if 0 / ∈ L. Since a ∅ = a ∅ and a [m] = Q⊆[m] a Q , condition (2.4) is just a restatement of these restrictions on a S . Thus, the proof of (1) is complete.
For (2), let f (x 1 , . . . , x m ) = S⊆[m] a S i∈S x i be another maximal DNF of f , and assume that a U = a U for some U ⊆ [m]. By symmetry, we may assume that a U a U . Thus a U = 1 and a U = 0. Now we may choose elements c, d ∈ L such that c ≥ a U , d ≤ a U , and d c. (2.5) Indeed, if a U and a U are in L, then we may let c = a U and d = a U , but c, d ∈ L satisfying (2.5) exist even if a U = 0 / ∈ L or a U = 1 / ∈ L. This is so because a U = 0 / ∈ L and a U = 0 imply that the principal ideal (a] of L contains an infinite descending chain for each a ∈ L, a ≤ a U . Thus there exist c, Then i∈S d i is equal to d if ∅ = S ⊆ U, and to cd or c if S ⊆ U. Thus, from the first maximal DNF of f we get that where the second and third inequalities ≤ follow from the monotonicity (2.3) of the coefficients a S , and the equality follows from c ≥ a U in (2.5). From the second maximal DNF of f we obtain that where the last equality follows from d ≤ a U in (2.5). The last two displayed inequalitites yield that d ≤ c. However, c and d were chosen in (2.5) so that d c. Thus we reached the desired contradiction, which completes the proof of (2).
Finally, if L is bounded and S ⊆ [m], then for e 1 , . . . , e m as in (3), , then by omitting the joinand a U i∈U x i (i.e., replacing a U by 0) we still have a DNF for f , because the two DNFs yield the same maximal DNF. This justifies the following definition. If L is a distributive lattice and f is a polynomial operation of L then the U-term a U i∈U x i of a DNF f (x 1 , . . . , x m ) = S⊆[m] a S i∈S x i of f will be called inessential if a U ≤ Q⊂U a Q , and essential otherwise. In the maximal DNF f (x 1 , . . . , For a distributive lattice L let PClo * L 01 := f ∈ PClo(L 01 ) : f is not a constant operation with value in L 01 \ L .
The existence and uniqueness of maximal DNFs for polynomial operations of L immediately implies the following corollary.

Corollary 2.4
For any distributive lattice L, the mapping that assigns to each polynomial operation f of L the polynomial operation f * of L 01 which has the same maximal DNF as f , is a clone isomorphism PClo(L) → PClo * (L 01 ). Consequently, every polynomial operation f of L has a unique extension to a polynomial operation of L 01 , namely f * .

Commuting Polynomial Operations of Distributive Lattices
Recall from the introduction that two operations f and g on a set X commute if they satisfy the equality (1.1) for all arguments x ij ∈ X. We will write f ⊥ g to indicate that f and g commute. Clearly, f ⊥ g if and only if g ⊥ f .
From now on f , g will be polynomial operations of a distributive lattice. First we will rewrite the condition defining f ⊥ g in terms of the maximal DNFs of f and g.
The following notation will be useful:  To determine the coefficients of the maximal DNF of f g([x ij ]), first we will use Corollary 2.4 to extend all operations involved to L 01 . Since extension to L 01 preserves • and the coefficients of maximal DNFs, no generality is lost if we assume for the rest of the proof that L = L 01 .
Using the maximal DNFs of f and g we see that i.e., if T ⊆ R(i, −), and we have j∈T e ij = 0 otherwise. This, combined with the monotonicity of the coefficients b T , yields that which is the left-hand side of (3.1). The fact that the R-coefficient of the maximal DNF of g f ([x ij ] T ) is the righthand side of (3.1) follows from the result in the preceding paragraph by observing that the operation by switching the roles of f and g and simultaneously switching the subscripts of the variables. (i) f and g are commuting polynomial operations of L; (ii) the unique extensions f * and g * of f and g to L 01 are commuting polynomial operations of L 01 ; (iii) the restrictions f * | C and g * | C of f * and g * to C are commuting polynomial operations of the finite lattice C.
Proof Clearly, the lattice C is finitely generated, hence it is finite. We will use the notation of Proposition 3.1 for the maximal DNFs of f and g. By the definition of f * , f * has the same maximal DNF as f . Furthermore, since C contains the coefficients of the maximal DNF of f * , f * can be restricted to C, and f * | C is a polynomial operation of C with the same maximal DNF as f * (and f ). Similarly, g * ∈ PClo(L 01 ) and g * | C ∈ PClo(C) have the same maximal DNF as g. Therefore, by Proposition 3.1, each one of the commutativity conditions f ⊥ g, f * ⊥ g * , and f * | C ⊥ g * | C is equivalent to the requirement that (3.1) holds for the coefficients of their maximal DNFs for all R ⊆ Corollary 3.2 shows that when studying the relation f ⊥ g for polynomial operations f , g of distributive lattices, no generality is lost in restricting to bounded distributive lattices, or even to finite lattices.
Next we will establish some necessary conditions for two polynomial operations to commute. The equivalence of some of conditions (i)-(vi) below for unary polynomial operations of distributive lattices appears in [4].

Lemma 3.3 Let f , g be polynomial operations of a distributive lattice L with maximal
DNFs as in Proposition 3.1. If f and g commute, then they must satisfy the following equivalent conditions: (iii) the coefficients of the maximal DNFs of f and g satisfy (3.1) for R = ∅; Proof If f and g commute on all matrices, then they commute on all constant matrices, which is easily seen to be equivalent to condition (i). It remains to show that all conditions are equivalent. First we make some remarks about distributive lattice polynomials.
Since a ∅ is a joinand of the maximal DNF for f , it follows that a ∅ ≤ c holds for all c ∈ im( f ). Since f and a [m] Using Corollary 2.3 we can easily see that this holds if and only if the constant terms If R = ∅ in (3.1), then, due to the monotonicity of the coefficients, (3.1) reduces to a ∅ + a [m] then by the observations of the second paragraph of this proof we have that c ∈ I fg ⊆ im( f ) and c ∈ I g f ⊆ im(g). It follows that c ∈ im( f ) ∩ im(g). If, on the other hand, c = 0 / ∈ L, then both I fg and I g f are nonempty downward closed sets. Hence if d ∈ I fg and e ∈ I g f , then de ∈ I fg ∩ I g f ⊆ im( f ) ∩ im(g). This shows that (iii) implies (iv).
where the first equality follows from the monotonicity of coefficients in a maximal DNF, the second follows from distributivity, and the third follows from (v):

2)
Consequently, (3.2) holds for f and g whenever f and g commute.
Since (3.2) is obtained from the special case of (3.1) when R is a union of two rectangles U i × V i (i = 1, 2), we will refer to (3.2) as the 2-rectangle condition.
Proof Throughout the proof, . First we will simplify the left-hand side of (3.1) for this R. We want to show that We will use the monotonicity of the coefficients of the maximal DNFs of f and g, namely that (2.3) holds for the a's, and analogously, for the b's. Also, notice that the shape of R implies that The fact that the left-hand side of (3.3) is dominated by the right-hand side will follow if we verify that every joinand on the left-hand side is bounded above by a joinand on the right-hand side. Let S ⊆ [m]. If S = ∅, then a S i∈S b R(i,−) = a ∅ , which is a joinand on the right-hand side. If S = ∅ but S ⊆ U 1 ∩ U 2 , then by the description of R(i, −) above we have that R(i, −) = V 1 ∪ V 2 for each i ∈ S. Since S = ∅, the monotonicity of the a's and b's implies that . Since S \ (U 1 ∩ U 2 ) = ∅, the monotonicity of the a's and b's implies again that This proves ≤ in (3.3).
To prove the reverse inequality ≥ in (3.3) it suffices to establish that every joinand on the right-hand side is bounded above by a joinand a S i∈S b R(i,−) on the left-hand side. The first joinand a ∅ appears as a S i∈S b R(i,−) for S = ∅. The last joinand satisfies All other joinands on the right-hand side of (3.3) are of the form This proves the equality (3.3), which simplifies the left-hand side of (3.1) for the . By switching the roles of f and g we get an analogous equality for the right-hand side of (3.1): Thus we obtain from (3.3) and (3.4) that for R = (U 1 × V 1 ) ∪ (U 2 × V 2 ), condition (3.1) is equivalent to (3.2), as claimed.
After these preparations we can state the main theorem of this paper, which characterizes commuting pairs of polynomial operations of distributive lattices. We will show that two polynomial operations commute if and only if they satisfy the 2-rectangle condition (3.2). We will also present a more transparent condition characterizing commutativity. As the proof progresses we will find other necessary and sufficient conditions for commutativity, which we will summarize in Corollary 3.10. The following conditions on f and g are equivalent: (i) f ⊥ g; (ii) the 2-rectangle condition (3.2) holds for all U 1 , U 2 ⊆ [m] and V 1 , The proof of Theorem 3.5 will occupy most of this section. Since the implication (i) ⇒ (ii) has been established already in Lemma 3.4, we will first focus on the implication (ii) ⇒ (iii). Lemma 3.6 Let f, g ∈ PClo(L) be as in Theorem 3.5. If f and g satisfy condition (ii) in Theorem 3.5, then they also satisfy the following condition: , we take the meet of the left-hand side of (3.2) with a U1 a U2 , and apply the monotonicity of the coefficients and the distributive and absorption laws to get that a U1 a U2 a ∅ + a [m] Taking the meet of the right-hand side of (3.2) with a U1 a U2 and applying the distributive and absorption laws again we obtain that Since a ∅ ≤ b [n] , and hence b [n] a ∅ = a ∅ , the equality (3.7) follows. The equality (3.8) can be proved in a similar way. Lemma 3.7 Let f, g ∈ PClo(L) be as in Theorem 3.5. If f and g satisfy condition ( †) 2 in Lemma 3.6, then they also satisfy condition (iii) in Theorem 3.5.
Proof Assume that condition ( †) 2 in Lemma 3.6 holds for f and g. We have to verify the equalities (3.5) and (3.6). Since (3.5) and (3.6) can be obtained from one another by interchanging the roles of the a S 's and the b T 's (i.e., the roles of f and g), it is enough to prove (3.5). Let U 1 , U 2 ⊆ [m] be fixed, and let V ⊆ [n]. We will prove the equality (3.5) by induction on |V|. For V = ∅, (3.5) is the equality which is clearly true, as a U1∩U2 ≤ a U1 a U2 .
Next let |V| ≥ 1, say V = W ∪ {z} with z / ∈ W. We will prove (3.5) for V, assuming that (3.5) is true for W in place of V. Applying the assumption (3.7) to V 1 = W and V 2 = {z} to get the second equality below, the absorption and distributive laws to get the third, the induction hypothesis to get the fourth, and again the absorption and distributive laws in the fifth, we deduce that completing the proof.
To prepare the proof of the implication (iii) ⇒ (i) in Theorem 3.5, we will show now that the equalities (3.5) and (3.6) extend to any finite number of U i 's and V j 's.

Lemma 3.8 Let f, g ∈ PClo(L) be as in Theorem 3.5. If f and g satisfy condition (iii)
in Theorem 3.5, then they also satisfy the following condition: Proof Assume that condition (iii) in Theorem 3.5 holds for f and g. Again, by symmetry, it suffices to prove equality (3.9). We will proceed by induction on k. Let For k = 2, (3.9) coincides with the equality (3.5), which holds by assumption. From now on let k ≥ 3, and suppose that (3.9) is true for k − 1 in place of k, that is, Taking the meet of both sides with a Uk and using the distributive law together with a ∅ ≤ a Uk we see that Applying the equality (3.5) to the sets k−1 i=1 U i , U k , and V on the right-hand side we obtain that where the last equality follows by observing that the monotonicity of the coefficients in a maximal DNF implies that a k−1 i=1 Ui ≤ k−1 i=1 a Ui , and hence that the joinand a k−1 i=1 Ui a Uk b ∅ + v∈V b v can be omitted. This completes the proof of Lemma 3.8. , 3, so (3.11) holds in this case. From now on we will assume that T = ∅, and set S = j∈T R(−, j). Thus, applying (3.9) to the right-hand side of (3.11), simplifying the last joinand by taking into account that b ∅ + t∈T b t = t∈T b t if T = ∅, and then using the distributive law, we get that Here Thus a S b T is bounded above by one of the joinands on the left-hand side of (3.11). We will argue similarly that every joinand j∈T a R(−, j) b t in (3.12), where t ∈ T, is dominated by a joinand on the left-hand side of (3.11). First, we get that where the right-hand side is the joinand for S = R(−, t) on the left-hand side of (3.11).
This proves (3.11), and therefore completes the proof of Lemma 3.9.
Now we are ready to complete the proof of Theorem 3.5.  In addition to conditions (ii)-(iii) in Theorem 3.5 the conditions listed below are also equivalent to f ⊥ g: (iv) condition ( ‡) in Lemma 3.8, which strengthens (iii); (v) condition ( †) 2 in Lemma 3.6; (vi) the following conditions, which strengthen (v): hold for all k, ≥ 1, Proof Theorem 3.5 was proved via the implications are all equivalent to f ⊥ g. Clearly, (v) is the special case k = 2 = of (vi), therefore (vi) ⇒ (v). Finally, we show that (iv) ⇒ (vi). To this end it will be enough to prove that for arbitrary sets U 1 , . . . , U k ⊆ [m] and V 1 , . . . , V ⊆ [n] (k, ≥ 1), the equality (3.9) with V = j=1 V j implies (3.13). Then similarly, the equality (3.10) with U = k i=1 U i implies (3.14), completing the proof. So, let U 1 , . . . , U k ⊆ [m] and V 1 , . . . , V ⊆ [n] (k, ≥ 1), and define V = j=1 V j . Then (3.13) and (3.9) have the same left-hand sides. Therefore, the equality (3.13) will follow from (3.9) if we prove that the right-hand side of (3.13) is trapped between the right-hand side of (3.9) and the common left-hand sides of (3.9) and (3.13). The right-hand side of (3.13) is greater than or equal to the right-hand side of (3.9), because j=1 b Vj ≥ b ∅ + v∈V b v . The right-hand side of (3.13) is less than or equal to the common left-hand sides of (3.9) and (3.13), because a k i=1 Ui ≤ k i=1 a Ui and

Self-Commuting Lattice Polynomial Operations
Let L be a distributive lattice, and let f , g be polynomial operations of L. Applying the characterizations of f ⊥ g in Theorem 3.5 and Corollary 3.10 to the case when f = g, we can obtain analogous characterizations of self-commuting polynomial operations of distributive lattices. The conditions obtained in this way can be simplified by observing that the requirement im( f ) ∩ im(g) = ∅ holds automatically for f = g. Moreover, in the remaining requirements the joinands a ∅ = b ∅ on both sides of the equalities can be omitted, since they are dominated by the remaining joinands on both sides. In the corollary below we will state only the characterizations obtained from Theorem 3.5. The following conditions on f are equivalent: Next we will apply Corollary 4.1 to deduce the main result of [3], which is an explicit description of all self-commuting polynomial operations of a bounded chain. We will state the result for a wider class of polynomial operations, but in view of Corollary 3.2 this is equivalent to the original formulation.
such that a i < a S1 < · · · < a Sr .

Remark 4.3 Condition (ii)
is stated here in a slightly different form than in [3], but the two formulations are equivalent.
Proof of Corollary 4.2 To prove the implication (i) ⇒ (ii) assume that f ⊥ f , and let be the maximal DNF of f . By the definition of maximal DNF, the coefficients a S ∈ L 01 satisfy a S ≤ a T whenever S ⊆ T ⊆ [m]. We will use this property without further reference. In addition, since f ⊥ f , the coefficients also satisfy (4.2) for all . Notice also that by Proposition 2.2 the coefficients a U are obtained from the coefficients a S of the given DNF by taking joins. Therefore, the hypothesis that { a S : S ⊆ [m]} is a chain in L 01 implies that {a U : U ⊆ [m]} is also a chain in L 01 . Let E denote the set of all S ⊆ [m] such that |S| ≥ 2 and the S-term of (4.4) is essential, i.e., a S > U⊂S a U . First we will prove that Since the implication ⇐ is clear, suppose for a contradiction that ⇒ is false, that is, for some S, T ∈ E we have a S ≤ a T but S T. Then S ∩ T ⊂ S, and hence the fact that a S is essential implies that a S∩T < a S . Now, applying (4.2) to U 1 = S, U 2 = T, and V = S we get that a S a T a S = a S∩T a S + a S a T a ∅ + s∈S a s .
We have a ∅ + s∈S a s ≤ a S , and we saw earlier that a S∩T < a S ≤ a T , therefore the displayed equality simplifies to a S = a S∩T + a ∅ + s∈S a s . Since |S| ≥ 2 and S ∩ T ⊂ S, this equality shows that, contrary to the choice of S, the S-term of (4.4) is inessential. This proves (4.5).
Therefore, it remains to show the equality a ∅ + i∈S1 a i = a ∅ + i∈[m] a i from (2). Suppose that the equality fails, that is, a ∅ + i∈S1 a i < a ∅ + i∈[m] a i . The fact that the set {a ∅ } ∪ {a i : i ∈ [m]} of coefficients is a chain implies then that a ∅ + i∈[m] a i = a p for some p ∈ [m], and p / ∈ S 1 . Thus, applying (4.2) to U 1 = S 1 , U 2 = {p}, and V = S 1 we get that a S1 a { p} a S1 = a S1∩{ p} a S1 + a S1 a { p} a ∅ + s∈S1 a s .
The first joinand on the right-hand side can be omitted, since S 1 ∩ {p} = ∅. Furthermore, a ∅ + s∈S1 a s < a p by the choice of a p , and a ∅ + s∈S1 a s < a S1 , since S 1 ∈ E. This implies that a ∅ + s∈S1 a s < a S1 a p , because a S1 and a p are comparable. Thus the displayed equality simplifies to a S1 a p = a ∅ + s∈S1 a s , contradicting the conclusion of the last sentence. This completes the proof of (i) ⇒ (ii).
For the reverse implication (ii) ⇒ (i) let us assume that (ii) holds. Condition (ii) remains valid if we replace each coefficient a i (i ∈ [m]) with a ∅ + a i , therefore we may assume without loss of generality that a ∅ ≤ a i holds for all i ∈ [m]. Under this additional assumption one can easily see, using Proposition 2.2, that for each one of the sets S = ∅, S = {i} with i ∈ [m], and S = S with 1 ≤ ≤ r, the S-coefficient of the maximal DNF of f is a S . Therefore, we can describe all coefficients of the maximal DNF of f as follows: In view of Corollary 4.1, the proof of (i) will be complete if we verify that (4.2) holds for all U 1 , Since the inequality ≥ is clearly true in (4.2), we will prove equality by showing that the left-hand side a U1 a U2 a V of (4.2) is dominated by one of the joinands on the right-hand side of (4.2). If S 1 V, then a V = a ∅ + v∈V a v , so a U1 a U2 a V is clearly one of the joinands on the right-hand side of (4.2). Therefore, we will assume from now on that S 1 ⊆ V. Let be such that S ⊆ V, but S +1 V. Thus a V = a S . If one of U 1 , U 2 fails to contain S 1 , say, S 1 U 1 , then where the second = follows from condition (2) in (ii), and the succeeding ≤ follows from S 1 ⊆ V. Hence a U1 a U2 a V = a U1 a U2 = a U1 a U2 a U1 ≤ a U1 a U2 a ∅ + v∈V a v , completing the proof in this case. Finally, let S 1 ⊆ U 1 , U 2 , say S i ⊆ U 1 , S i+1 U 1 , and S j ⊆ U 2 , S j+1 U 2 . We may assume without loss of generality that i ≤ j. Then S i ⊆ U 1 ∩ U 2 and S i+1 U 1 ∩ U 2 , so a U1∩U2 = a Si = a U1 and a U1 = a Si ≤ a S j = a U2 . Hence a U1 a U2 a V = a Si a V = a U1∩U2 a V , which completes the proof of Corollary 4.2.

Corollary 4.4 A polynomial operation f of a distributive lattice is symmetric and selfcommuting if and only if it admits a representation of the form
i∈ [m] x i for some a ∅ , a 1 , a [m] ∈ L 01 with a ∅ ≤ a 1 ≤ a [m] .
Proof If f has such a representation, then f is symmetric. Furthermore, if a 1 < a [m] , then the given expression for f satisfies condition (ii) in Corollary 4.2 with r = 1, while if a 1 = a [m] , then the [m]-term of the given expression for f is inessential and can be omitted, so the resulting expression satisfies condition (ii) in Corollary 4.2 with r = 0. In either case, f is self-commuting.
Conversely, assume that f is symmetric and self-commuting. As we observed above, symmetry implies that in the maximal DNF (4.6) of f we have a I = a J whenever |I| = |J| (I, J ⊆ [m]). Now, if an I-term in (4.6) is essential, then so are all J-terms with |J| = |I|. However, as we have seen in the proof of (ii) ⇒ (i) in Corollary 4.2, if f is self-commuting, then the sets S ⊆ [m] with |S| ≥ 2 for which the S-terms of the maximal DNF of f are essential form a chain. This forces that the only set S with |S| ≥ 2 for which the S-term in (4.6) may be essential is S = [m]. Thus an S-term in (4.6) is essential only if |S| ∈ {0, 1, m}, so f has the prescribed form.

Commutativity of Special Lattice Polynomial Operations
We will now use our theorem on commuting pairs of distributive lattice polynomial operations to determine all commuting pairs of distributive lattice term operations. At the end of this subsection we will outline a second proof of the same result that does not use our theorem on commuting polynomials.
When determining pairs of commuting term operations, one special case that is key to the general argument is the case where one term is a join of two variables. We shall work out that case first in a bit more generality than necessary, namely we will describe those polynomial operations of a distributive lattice which commute with an arbitrary linear polynomial operation. It seems plausible that this case will find application some day.
By a linear polynomial operation of a distributive lattice we mean a polynomial operation of the form Since φ ij f : L 01 → L 01 are lattice homomorphisms, condition (b) here is easily seen to be equivalent to (4.8) In the special case when U 1 = {i} and i.e., φ Conversely, the equality (4.8) can be obtained by joining to the obvious equality a ∅ + a U1∩U2 b V = a ∅ + a U1∩U2 b V all equalities (4.9) where {i, j} = {u 1 , u 2 }, u 1 ∈ U 1 \ U 2 , u 2 ∈ U 2 \ U 1 , and 1 ≤ i < j ≤ m. This proves Corollary 4.5.
Applying Corollary 4.5 to f and g, both considered as polynomial operations of L ∂ , we get that they commute if and only if In view of (4.11), (4.12) and (4.13), condition (a) ∂ coincides with condition (a) in the statement, and with the notation V = [n] \ W, the equality in (b) ∂ coincides with the equality in (b). Since V = [n] \ W runs over all subsets of [n] as W does, the proof of Corollary 4.6 is complete.
Now we are ready to determine all pairs of commuting term operations of a distributive lattice. We will distinguish two cases. Suppose first that F has a least element, S 0 . Then S 0 = ∅ and the S 0 -term is the only essential term of the maximal DNF of f , so f (x 1 , . . . , x m ) = i∈S0 x i . By assumption, f is not a projection, therefore |S 0 | ≥ 2.
Since f generates the same clone as x 1 x 2 , we have f ⊥ g if and only if x 1 x 2 ⊥ g. Applying Corollary 4.6 with m = 2 and a [2] = 1, a 1 = a 2 = 0, we see that ψ 12 is the identity homomorphism, and hence G is closed under ∩ by Corollary 4.6 (b) . The intersection of all elements of G yields a least element T 0 of G, and g(x 1 , . . . , x n ) = j∈T0 x j . This shows that if f ⊥ g and f is a meet of at least two variables, then g is also a meet of variables.
It remains to consider the case when F has two incomparable minimal elements, say U 1 and U 2 . In this case a U1 = a U2 = 1, but a U1∩U2 = 0, while we still have a ∅ = b ∅ = 0. We will substitute these values into condition (3.5) of Theorem 3.5, which we recall here: The result of the substitution is that b V = v∈V b v holds for all V ⊆ [n], which shows that if a V-term of g is essential, then V is a singleton . Hence g(x 1 , . . . , x n ) = t∈T0 x t for some T 0 ⊆ [n]. By assumption, g is not a projection, so it is a join of at least two variables. By the dual of the last sentence of the preceding paragraph, f must also be a join of variables. Now we outline a second proof of Corollary 4.7. Suppose that f and g are term operations of a nontrivial distributive lattice L. The restriction map to a 2-element sublattice of L is a clone isomorphism, because the variety of distributive lattices is minimal, so f and g commute on L if and only if they commute on some (any) 2-element sublattice. Thus no generality is lost in assuming that L = {0, 1} has only two elements.
Let I denote the clone of all idempotent operations on {0, 1}. Let C be the clone generated by f , let D = C ⊥ ∩ I be the clone of idempotent operations centralizing C, and let E = D ⊥ ∩ I be the clone of idempotent operations centralizing D. We have that {D, E} is an unordered pair of idempotent clones, each the "idempotent centralizer" of the other, and that g ∈ D and f ∈ E. By examining Post's lattice of all clones on the 2-element set [12] and by determining the centralizer clones among them [7,10] it is easy to see that there are four such pairs {D, E}: In case (1), one of f or g must be a projection. In case (2), both f and g are monotone affine operations, hence again must be projections. So if neither f nor g is a projection, then both are nonprojections from the same (minimal) clone generated by one of the semilattice operations. This forces them both to be meets of variables or both to be joins of variables.