Lattices do not distribute over powerset

We show that there is no distributive law of the free lattice monad over the powerset monad. The proof presented here also works for other classes of lattices such as (bounded) distributive/modular lattices and also for some variants of the powerset monad such as the (nonempty) finite powerset monad.


Introduction
Distributive laws were introduce by Jon Beck [1], they provide a canonical way to compose monads and also allow to lift functors/monads to the category of algebras for a functor/monad. The classical example is the distributive law of product over addition which defines a distributive law of the free monoid monad over the free abelian group monad. This distributive law allows to compose those two monads in order to obtain the free ring monad.
Distributive laws of a functor/monad over the powerset (monad) are of particular interest in computer science and category theory. In computer science, they are studied in automata theory [9], the so-called powerset construction or determinization, and in (guarded) structural operational semantics [10], SOS and GSOS. In category theory, they are of particular interest in the subject of relation lifting [5,7].
There are well-known canonical laws over the powerset, such as the distributive law of the free monoid monad over the powerset monad given by the law A · B → { a · b | a ∈ A, b ∈ B }. Such a canonical law over the powerset can be defined for any functor [5,Section 4] but in general is not a natural transformation. If the functor preserves weak pullbacks this canonical law is a natural transformation and it is distributive law of the functor over the powerset monad [5]. Additionally, if the functor has a monad structure in which the unit and the multiplication are weakly cartesian, meaning that each naturality square is a weak pullback, then this law is a distributive law between monads.
From the perspective of universal algebra, the previous canonical law, that lifts a basic operation to its subsets by choosing elements in each subset and then applies the basic operation, provides the construction of power algebras (also known as complex algebras or global algebras) [3,4]. It is well-know that such construction only preserves regular linear identities [3,Theorem 5]. In particular, the non-linear axiom x ∨ (x ∧ y) = x of absorption in the theory of lattices is not preserved. As a consequence, the previous canonical law is not a distributive law of the lattice monad over the powerset monad (in fact, it is not even a natural transformation!). Nevertheless, this does not rule out the existence of a distributive law of the free lattice monad over the powerset monad. In this paper, we show that there is no distributive law of the free lattice monad over the powerset monad and between some variants of them.

Preliminaries
Throughout this paper we work only in the category Set of sets and functions. A monad (T, η, μ) (see [8, Chapter VI] for more details) is an endofunctor T together with natural transformations η : id =⇒ T (the unit) and μ : T T =⇒ T (the multiplication) such that the following diagrams commute: The powerset monad (P, η, μ) is defined so that PX is the set of all subsets of X, and: A lattice is an algebra on the signature { ∧, ∨ }, where ∧ and ∨ are both binary operation symbols, that satisfies the following equations: We denote by (L, η, μ) the free lattice monad. That is, LX is the underlying set of the free lattice on X, which is the set T X of terms over X on the signature { ∧, ∨ } modulo the equational theory induced by the lattice axioms. The unit η X maps an element x ∈ X to its equivalence class η X (x) and the multiplication "flattens", in a canonical way, each equivalence class whose parameters are also equivalence classes into a single equivalence class. Explicitly, if we use brackets to denote equivalence classes on T X, then η X (x) = [x] and , for x ∈ X and terms t, s 1 , . . . , s n on the signature { ∧, ∨ }. In the sequel, we omit brackets for equivalence classes and only work with (canonical) representatives. It is worth mentioning that the functor L does not preserve weak pullbacks. Indeed, if we consider the pullback square Furthermore, η is not weakly cartesian since, for the same f as above, . This implies that the canonical law λ X : LPX → PLX defined in [5, Section 4] is not a natural transformation. Let S and T be endofunctors on the category Set. A natural transformation λ : T S =⇒ ST is called a distributive law of T over S. Additionally, if η S : id Set =⇒ S and η T : id Set =⇒ T are natural transformations, we can ask whether the following unit laws hold or not: Similarly, if μ S : SS =⇒ S and μ T : T T =⇒ T are natural transformations, we can also ask whether the following multiplication laws hold or not: Depending on the laws that λ : T S =⇒ ST satisfies, we include the respective natural transformations when mentioning a distributive law. For instance, a distributive law of (T, η T ) over (S, μ S ) is a natural transformation λ : T S =⇒ ST that satisfies the unit law (U 2 ) for η T and the multiplication law (M 1 ) for

Lattices do not distribute over powerset
The main purpose of this section is to show the following.
A key property that we use in our proof is the the following. Varieties for which we can apply the previous lemma include: lattices, distributive lattices, modular lattices and their bounded versions, among others. The proof of Theorem 3.1 we present depends only on the lattice axioms and the conclusion of the previous lemma. Hence, Theorem 3.1 also holds if we consider, instead of the free lattice monad, any monad associated to a lattice variety containing 2. For the sake of simplicity we do the proof for the monad (L, η, μ) and state in Theorem 3.7 its generalization.

Lemma 3.2. Let V be a lattice variety such that the two element linear order 2 is in
Throughout this section, (P, η, μ) denotes the powerset monad and (L, η, μ) denotes the free lattice monad.
If λ : LP =⇒ PL is a distributive law of the free lattice monad over the powerset monad, then for every set Y and elements y 1 , y 2 ∈ Y we have by the axioms (U 2 ) and (U 1 ) involving η and η, respectively. We have the following properties for the elements in Proof. (1) follows by naturality of λ using the function g : X → { v, w } such that g(v) = g(x) = v and g(w) = g(y) = w, and by applying (3.1a).
To prove s(v, w, x, y) ≤ v ∨ w in (2) (3), assume that s(v, w, v, w) = v. Then, by considering the partition of LX given by the filter F (v, x) and the ideal I(w, y) we have that either s(v, w, x, y) ∈ F (v, x) or s(v, w, x, y) ∈ I(w, y). In the latter case, we obtain s(v, w, x, y)≤w∨y and this implies v=s(v, w, v, w) ≤ w, a contradiction. Therefore, s(v, w, x, y)∈F (v, x), which means that s(v, w, x, y)≥v ∧ x.
We have the following dual version of the previous lemma.
Proof. Dual proof of Lemma 3.3.
The previous three lemmas are the main tool that we use in the sequel.

Our next step is to characterize the set λ({ a, b } ∧ { a }) and the set λ({ a, b } ∨ { a }).
Lemma 3.5. Let λ : LP =⇒ PL be a distributive law of (L, η, μ) over (P, η) and let B = { a, b }. Then: Proof. Let X = { v, w, x, y }. Then, by naturality of λ, using the onto function f : s(a, b, a, a) s(a, b, a, a) in s(a, b, a, a) ≤ a ∨ a = a in LB, where LB is the free lattice on two generators. Since LB has only four elements, namely, a, b, a ∨ b  and a ∧ b, and the only ones less than or equal to a are a and a ∧ b then  λ B (B ∧ { a }) ⊆ { a, a ∧ b }. Furthermore, by using the termŝ of Lemma 3.3 together with (3) of that lemma we haveŝ (a, b, a, a)  Therefore, we have the following cases:

Now, by using item (2) from Lemma 3.3, we obtain that each
Note that by using the axiom (M 2 ) involving μ, if we let t(x, y) = x ∨ y, we have: but using the absorption law we can calculate the left hand side as follows: Now, case (1) cannot happen since we would obtain which contradicts (3.2). Similarly, case (2) cannot happen since it would give us B = { a, a ∨ b }, a contradiction. A dual argument to that of case (2) shows that (3) cannot happen either. Therefore, we are left with case (4) and the proof is finished.
Our last lemma is the following.
Lemma 3.6. Let λ : LP =⇒ PL be a distributive law of (L, η, μ) over (P, η, μ) By naturality of λ, using the onto function g : X → C such that g(v) = a, g(w) = b, and g(x) = g(y) = c, we obtain: Note that by item (2)  Now, by Lemma 3.5 and naturality of λ, using the onto function f : s(a, b, a, a) s (a, b, a, a) = a ∧ b. Consider the partition of LC given by F (a, c) and I(b). If s (a, b, c, c) ∈  F (a, c) then a ∧ c ≤ s (a, b, c, c) which implies a ≤ s (a, b, a, a) = a ∧ b, a  contradiction. Therefore, s (a, b, c, c) ∈ I(b), that is, s (a, b, c, c) s (a, b, c, c) ≤ b ∧ c and therefore s (a, b, c, c) = b ∧ c or s (a, b, c, c) Consider the term t(p, q) = p ∧ q. Then, by the axiom (M 2 ) involving μ, forB = { b, c }, we have: but using the absorption law we have and using Lemma 3.5 we have Hence, the last three equations give us (3.5) In order to finish the proof, by using (3.4), we assume by contradiction that s (a, b, c, c) = b ∧ c, that is, we assume that b ∧ c ∈ λ C (B ∧ { c }). By the axiom (M 1 ) involving μ we have: , using the definition t(p, q) = p ∧ q above of the term t, we have: and therefore (3.5) =B, Now, by using Lemma 3.6 again, we have where the first equality follows from (3.1a) and (3.1b) and the third equality by the axiom (M 2 ) involving μ. Now, (3.6) and (3.7) imply = B, a contradiction.
As we commented after Lemma 3.2, the proof of Theorem 3.1 we just made also works for any monad associated to a lattice variety that contains 2. Also, since the proof above did not involve the empty set or infinite sets, we can also replace the powerset monad for its nonempty version and/or its finite version. We can summarize our results as follows.
Theorem 3.7. Let (T, η T , μ T ) be a monad associated to a lattice variety that contains the two element linear order 2 and let (S, η S , μ S ) be any of the following monads: • the (nonempty) powerset monad, • the (nonempty) finite powerset monad.
Then there is no distributive law of (T, η T , μ T ) over (S, η S , μ S ).
Particular cases of the monad (T, η T , μ T ) in the previous theorem include: the free (bounded) lattice monad, the free (bounded) distributive lattice monad and the free (bounded) modular lattice monad, among others.
instance we considered which cannot be handled with any of the results in [6,12] was the group monad (G, η, μ). Indeed, if we consider the preimage given in Section 2, then we have Gf (xy −1 ) = e = Gf (e), where e is the identity element, but there is no t ∈ G{ (x, x), (x, y) } such that Gπ 2 (t) = xy −1 z, which shows that we cannot apply [6,Theorem 2.4.]. On the other hand, the identity yy −1 x ≈ x holds in the theory of groups but the term yy −1 x contains other variables different than x, which means that we cannot apply the theorems in [12] either. In this case, we found that there is no distributive law of (G, η) over (P, η), whose proof, similar to the lattice case, depends on the very specific setting of group theory. We conjecture that such an example of a distributive law of a monad, whose functor part does not preserve weak pullbacks, over the powerset monad does not exist and leave it as an open problem for further research.
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