AN ELEMENTARY PROOF THAT n−PERMUTABLE VARIETIES SATISFY LATTICE IDENTITIES

Paolo Lipparini

Very Preliminary version, June 9, 2002 (errors and omissions are possible, but, seemingly, the proofs of the results mentioned in the abstract are complete!)

2000 Mathematical Subject Classification: Primary 08B99, 06B20; Secondary 06B15, 08A30, 08B05.

Keywords and phrases: Congruence n-permutable varieties, congruence (lattice) identities.

Abstract. We show that if A is an algebra in an n + 2-permutable variety, and α, β, γ are congruences on A then all congruences between α(β(γ + α) + γ(β + α)) and α(β + γ) n-permute.

This easily implies that for every n there is a nontrivial lattice identity satisfied by congruence lattices of algebras in n-permutable varieties. This identity does not follow from previously obtained identities.

1. Introduction. The notion of an n-permutable variety is a direct generalization of the better known concept of a variety all whose algebras have permuting congruences [Ma]. Scattered results about n-permutable varieties appeared in the literature (see e.g. [HM, CD, Ta, ???]). The notion of n-permutability is relevant in the study of varieties of topological algebras (see [Ta1, Gu1, Co1, Co2, Be, KS]). The nice characterization in [HMK, Theorem 9.14] of the locally finite varieties which are n-permutable for some n probably will lead to a more systematic treatment of the subject.

Some results connecting n-permutability and identities in congruence lattices were known: Polin’s example of a non modular variety still satisfying a non trivial lattice identity is in fact 4-permutable [Po, DF]; later J. B. Nation and A. Day found an identity satisfied in any n-permutable semilattice [Jo, Lemma 3.10]; more generally, [Cz1] showed that, for n-permutable varieties, both meet-semidistributivity and join-semidistributivity are characterized by a lattice identity. More recently [HMK, Theorem 9.19] proved that every locally finite n-permutable variety satisfies some non trivial lattice identity.

Indeed, in [Lp1] we showed that any n-permutable variety satisfies some lattice identity. The results of [Lp1] rely heavily on commutator theory: the aim of the present paper is to provide a shorter and commutator-free proof of the result. In doing this, we also obtain the structural characterization mentioned in the abstract. As a consequence, e.g., if A ∈ V, V is n + 2-permutable and the smallest non modular lattice N_{5}

∘
/ \
∘ α
β ∘ |
∘ γ
\ /
∘

is a sublattice of Con A, then all congruences δ, δ′ with γ ≤ δ, δ′ ≤ α n-permute.

The identities we get in the present paper are quite different from the ones given in [Lp1]. There we showed that certain lattice intervals are abelian, and that congruences in abelian intervals permute: this immediately yields identities. The identities found in the present paper, however, apply also to non-abelian intervals: the interval I(γ, α) in N_{5} above can be non-abelian, as is the case, for example, of Polin’s algebra (see [DF]).

On the contrary, the identities of the present paper say nothing about modular lattices, while [Lp1] implies that certain intervals consist of permuting congruences, in nondistributive modular congruence lattices.

Thus our present results complement those obtained in [Lp1]: it is conceivable that a combination of the two methods will lead to a satisfying structure theory for n-permutable varieties.

2. Preliminaries. Our notation is standard: see, e.g., [BS, MKMNT].

Intersection is sometimes denoted by juxtaposition; and inclusion sometimes by ≤; ∘ is relational product, and m, n are non-negative integers.

A ⊧_{Con} P means that the property P is satisfied by all congruences of the algebra A. V ⊧_{Con} P means that A ⊧_{Con} P, for every A ∈ V .

We shall make a distinction between a congruence identity and a lattice identity: a lattice identity σ is an identity expressed only in terms of + and ∩: thus A ⊧_{Con} σ iff σ holds in the lattice Con A of all congruences of A . A congruence identity is allowed to contain also ∘. Notice that ∘ is not an operation on Con A, since α ∘ β is not necessarily a congruence.

An algebra is said to be n-permutable iff it satisfies the congruence identity α∘β … ≤ β ∘ α … (n factors on each side) or, equivalently, α + β = α ∘ β … (n factors). A variety is n-permutable in case all its members are n-permutable. Permutability is the same as 2-permutability.

As usual, we shall sometimes write aαb in place of (a, b) ∈ α to mean that α and β are in the same α-class.

As the reader might expect, our starting point will be the following well-known Mal’cev characterization of n-permutable varieties, due to Hagemann and Mitschke.

2.1. THEOREM. [HM] A variety is n-permutable iff it has terms t_{0}, …, t_{n} satisfying:

(i) x = t_{0}(x, y, z),

(ii) t_{i}(x, x, y) = t_{i+1}(x, y, y), for i = 0, …, n − 1

(iii) t_{n}(x, y, z) = z.

Notice that the equations in Theorem 2.1 imply that all the terms t_{i} are idempotent (that is, t_{i}(x, x, x) = x).

3. Some simpler cases: n ≤ 5. It is well known that permutable and 3-permutable varieties are congruence modular; indeed, every congruence modular variety satisfy the Arguesian identity [FJ]; actually there are more identities satisfied by congruence modular varieties (see [FHH, CHL]). In particular, permutable and 3-permutable varieties satisfy even those stronger identities.

By the way, ??? studied identities satisfied in lattices of permuting equivalence relations; seemingly we can show that every congruence modular variety satisfies all the identities valid in lattices of permuting equivalence relations; hence 3-permutable varieties satisfy all the identities valid in lattices of permuting equivalence relations. We do not know, however, whether there is a lattice identity satisfied by permutable varieties which fails in some 3-permutable variety. More generally, we do not know [Lp2, p. 864] whether, for n even, n-permutable and n + 1-permutable varieties satisfy the same lattice identities.

Having dealt with n = 2 and n = 3, the next case to be considered is n = 4. We give here a very short proof that 4-permutable and 5-permutable varieties satisfy a non trivial lattice identity. The proof is taken from [Lp2, Theorem 3.2].

[...]

4. n-permutable intervals in n + 2-permutable varieties.

4.1. DEFINITION. If t is any ternary term, define the m-th iterated t^{m} of t as follows:

t^{1}(x, y, z) = t(x, y, z)
t^{m+1}(x, y, z) = t(x, t^{m}(x, y, y), t^{m}(x, y, z))

(compare [Gu, p. 64; Lp3, p. 187; Lp2, p. 855])

4.2. LEMMA. Suppose that t_{0}, …, t_{n} are terms as given in Theorem 2.1. Then, for every m > 0, also t_{0}^{m}, …, t_{n}^{m} satisfy the conditions in Theorem 2.1.

Proof. By induction on m. The case m = 1, as well as conditions (i) and (iii) are trivial. For the rest, it is enough to prove (for i = 0, …, n − 1, noticing that all the t_{i}^{m}’s are idempotent):

t_{i}^{m+1}(x, x, y) = t_{i}(x, t_{i}^{m}(x, x, x), t_{i}^{m}(x, x, y)) = t_{i}(x, x, t_{i+1}^{m}(x, y, y))

= t_{i+1}(x, t_{i+1}^{m}(x, y, y), t_{i+1}^{m}(x, y, y)) = t_{i+1}^{m+1}(x, y, y)

⊔

4.3. LEMMA. Suppose that A is an algebra, and t is a term satisfying x = t(x, y, y) for every x, y ∈ A. If α, β, γ ∈ Con A and (a, b) ∈ α(β ∘ γ ∘ …) (m + 2 factors), m > 0, then

aα(γ ∘ β)t^{m}(a, a, b) for m odd, and aα(β ∘ γ)t^{m}(a, a, b) for m even.

Proof. First we show that

(*) if t satisfies x = t(x, y, y) and (a, b) ∈ α(β ∘ γ ∘ β …) (m + 2 factors) for m > 0, then (a, t(a, a, b)) ∈ α(γ ∘ β …) (m + 1 factors).

Indeed, by hypothesis, aαb, and there are a_{1}, …, a_{m+1} such that aβa_{1}γa_{2}βa_{3} … a_{m+1} β^{∗}b, where β^{∗} = γ when m is even, and β^{∗} = β when m is odd. Then

a = t(a, a_{1}, a_{1})γt(a, a_{1}, a_{2})βt(a, a, a_{3})γt(a, a, a_{4})βt(a, a, a_{5}) … t(a, a, a_{m+1})β^{∗}t(a, a, b)

Moreover, a = t(a, b, b)αt(a, a, b). Thus (a, t(a, a, b)) ∈ α(γ ∘ β …) (m + 1 factors). Having proved (*), we get by induction that

(**)_{j} (a, t^{j}(a, a, b)) ∈ α(γ^{∗} ∘ β^{∗} ∘ γ^{∗} …) (m + 2 − j factors), for m > 0 and 0 < j ≤ m, where γ^{∗} = γ, β^{∗} = β when j is odd, and γ^{∗} = β, β^{∗} = γ when j is even.

Indeed, (*) is the basis j = 1 of the induction. If (**)_{j} is true for some j < m then we get (a, t(a, a, t^{j}(a, a, b))) ∈ α(β^{∗} ∘ γ^{∗} …)

(m + 2 − j − 1 factors) by applying (*) with t^{j}(a, a, b) in place of b. But t^{j+1}(a, a, b) = t(a, t^{j}(a, a, a), t^{j}(a, a, b)) = t(a, a, t^{j}(a, a, b)), thus we have showed (**)_{j+1}.

Now (**)_{m} is precisely the statement of the Lemma. ⊔

4.4. REMARKS. (a) In the conclusion of 4.3 we could actually get aα(β ∘ γ) t^{m+1}(a, a, b) for every m: this is because if aα(γ ∘ β)b, say aγa_{1}βb, then a = t(a, a_{1}, a_{1})β t(a, a_{1}, b)γt(a, a, b).

(b) Indeed, we could do with a much smaller exponent of t in the conclusion of 4.3: just use a_{h} in place of a_{1} in the proof of (*), taking h to be closer to m/2.

For example, [...].

(c) Moreover, it is enough to take α, β, γ to be tolerances rather than congruences.

(d) More generally, the arguments in the proof of (*) show that if α is a tolerance, Θ, Ψ are reflexive admissible relations, and aα(Θ ∘ Ψ)b then aα\overline{Θ^{−} ∪ Ψ}t(a, a, b), where \overline{X} means “the least admissible relation containing X”, and Θ^{−} denotes the converse of Θ. [...].

Our main result is now a rather direct consequence of Lemmata 4.2 and 4.3. Suppose that we are in an n + 2-permutable variety with terms t_{0}, …, t_{n+2} given by Theorem 2.1.

Since α(β ∘ γ) ≤ α(β(γ + α) + γ(β + α)) then, for large enough m, the terms t_{2}^{m}, …, t_{n}^{m} give the condition for n-permutability in the interval between α(β(γ + α) + γ(β + α)) and α(β ∘ γ).

In order to make this idea into an accurate proof, we need a definition.

4.5. DEFINITION. If A is any algebra, and B is a subset of A, then the restriction of A to B is the algebra whose base set is B and which has, as operations, {h_{|B}|h : A^{m} → A, m < ω, h is a polynomial operation of A and h(B^{m}) ⊆ B}.

When there is no danger of confusion we shall call this algebra simply B.

What will be important for us is that if B is a block of some congruence on A, and t is an idempotent term, then t is an operation of B.

4.6. THEOREM. Suppose that A ∈ V, V is n + 2-permutable, α, β, γ ∈ Con A, and let α′ = α(β + γ), α′′ = α(β(γ + α) + γ(β + α)). If B is a block of α′ then the restriction of A/α′′ to B/α′′ belongs to an n-permutable variety.

Proof. Let t_{0}, … t_{n+2} be the terms given by Theorem 2.1. Let a, b ∈ B, thus aα′b. By n+2-permutability, β +γ = β ∘γ … (n+2 factors), thus (a, b) ∈ α(β ∘γ …) (n+2 factors). Define u_{i} = t_{i}^{n}, for i ≤ n + 2. By Lemma 4.2, x = u_{1}(x, y, y) holds in A. Then Lemma 4.3 implies that either aα(β ∘ γ)u_{1}(a, a, b), or aα(γ ∘ β)u_{1}(a, a, b). Moreover, Lemma 4.2 implies that u_{1}(a, a, b) = u_{2}(a, b, b).

Since both α(β ∘ γ) and α(γ ∘ β) are contained in α′′ = α(β(γ + α) + γ(β + α)), we have that aα′′u_{2}(a, b, b). Since a, b are arbitrary elements of B, we have that the identity x = u_{2}(x, y, y) holds in B/α′′ (notice that the u_{i}’s are idempotent terms, hence their restrictions are operations on B, hence on B/α′′).

By symmetry, u_{n}(x, x, y) = y holds in B/α′′.

By Lemma 4.2, u_{i}(x, x, y) = u_{i+1}(x, y, y) (i = 2, …, n − 1) hold in A, a fortiori the above relations hold in B/α′′. Thus the terms v_{i} = u_{i+1} (i = 1 … n − 1) witness that B/α′′ belongs to an n-permutable variety, by Theorem 2.1 (defining v_{0} and v_{n} to be the first and the third projection, of course). ⊔

4.7. DEFINITION. If α′′ ≤ α′ are congruences of an algebra A, we let I(α′′, α′) = {γ|γ ∈ Con A and α′′ ≤ γ ≤ α′}.

If σ(α_{1}, α_{2}, …) is a congruence identity, we say that σ holds in I(α′′, α′) if and only if σ(α_{1}, α_{2}, …) holds for every α_{1}, α_{2}, … ∈ I(α′′, α′). If this is the case,we also say that I(α′′, α′) satisfies σ.

Notice that if σ is a lattice (respectively, congruence) identity, then “σ(α_{1}, α_{2}, …) holds in I(α′′, α′)” is equivalent to the lattice (respectively, congruence) identity σ(α′′ + α′α_{1}, α′′ + αα_{2}, …).

4.8. COROLLARY. Suppose that A ∈ V, V is an n + 2-permutable variety, α, β, γ ∈ Con A, α′ = α(β + γ), α′′ = α(β(γ + α) + γ(β + α)). Then I(α′′, α′) satisfies all the congruences identities which hold in every n-permutable variety. In particular, all congruences in I(α′′, α′) n-permute.

Proof. Suppose that the congruence identity ϵ(α_{1}, α_{2}, …) = ϵ′(α_{1}, α_{2}, …) holds in every n-permutable variety. If A is as in the statement of the corollary, α_{1}, α_{2} ∈ Con A, a, b ∈ A, aϵb and α_{1}, α_{2} … ≤ α′, then a and b belong to the same α′ block B. Moreover, ϵ = ϵ′ holds in B/α′′, by Theorem 4.6. But this implies that aϵ′b, if α′′ ≤ α_{1}, α_{2}, . . .. ⊔

4.9. COROLLARY. For every n there is a non trivial identity satisfied by all n-permutable varieties.

Proof. Immediate from Corollary 4.8, the remark in Definition 4.7 and the well known fact that permutable varieties satisfy non trivial identities (see Section 3). ⊔

Other identities shall be found in the next section.

5. Some improvements. [...]

6. Further remarks. [...]

??? were announced in [Lp2]. K. Kearnes [Ke1, Ke2] has provided several characterizations of locally finite varieties

satisfying non trivial lattice identities, and has announced that (joint work with E. Kiss) in most cases the assumption of local finiteness can be dropped. Thus Kearnes and Kiss’ results provide still another proof that n-permutable varieties satisfy identities (however, the identities obtained here do not seem to follow from Kearnes and Kiss ones).

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Paolo Lipparini Dipartimentico di Matematica II Università di Roma (Tor Vergata) Viale della Ricerca scientifica I-00133 ROME ITALY LIPPARIN@AXP.MAT.UNIROMA2.IT