aa r X i v : . [ m a t h . R A ] M a y ON THE J ´ONSSON DISTRIBUTIVITY SPECTRUM
PAOLO LIPPARINI
Abstract.
Suppose throughout that V is a congruence distribu-tive variety. If m ≥
1, let J V ( m ) be the smallest natural number k such that the congruence identity α ( β ◦ γ ◦ β . . . ) ⊆ αβ ◦ αγ ◦ αβ ◦ . . . holds in V , with m occurrences of ◦ on the left and k occurrencesof ◦ on the right. We show that if J V ( m ) = k , then J V ( mℓ ) ≤ kℓ ,for every natural number ℓ . The key to the proof is an identitywhich, through a variety, is equivalent to the above congruenceidentity, but involves also reflexive and admissible relations. If J V (1) = 2, that is, V is 3-distributive, then J V ( m ) ≤ m , for every m ≥ V is m -modular, that is, con-gruence modularity of V is witnessed by m + 1 Day terms, then J V (2) ≤ J V (1) + 2 m − m −
1. Various problems are stated atvarious places. The J´onsson distributivity spectrum
Obviously, an algebra A is congruence distributive if and only if, forevery natural number m ≥
2, the congruence identity α ( β ◦ m γ ) ⊆ αβ + αγ holds in Con ( A ) (more precisely, in the algebra of reflexiveand admissible relations on A ). Here α , β , . . . are intended to varyamong congruences of A , juxtaposition denotes intersection, + is joinin the congruence lattice and β ◦ m γ denotes β ◦ γ ◦ β . . . with m factors,that is, with m − ◦ .Let us say that a congruence identity holds in some variety V if itholds in every algebra in V . By a celebrated theorem by J´onsson [10], amilestone both in the theory of Maltsev conditions and in the theory ofcongruence distributive varieties, a variety V is congruence distributiveif and only if there is some n such that the congruence identity(1) α ( β ◦ γ ) ⊆ αβ ◦ n αγ Mathematics Subject Classification.
Key words and phrases.
Congruence distributive variety; (directed) J´onssonterms; J´onsson distributivity spectrum; congruence identity; identities for reflexiveand admissible relations.Work performed under the auspices of G.N.S.A.G.A. Work partially supportedby PRIN 2012 “Logica, Modelli e Insiemi”. holds in V . In other words, for varieties, taking m = 2 in the aboveparagraph is already enough. J´onsson actual statement in [10] is abouta set of terms naturally arising from identity (1), rather than aboutthe identity itself. J´onsson terms shall be recalled later. Stating resultswith regard to congruence identities rather than terms is simpler andeasier to understand, while proofs usually require the correspondingterms. Compare the perspicuous discussion in Tschantz [26].J´onsson proof in [10] goes on by showing that if some variety V hasterms witnessing (1), then, for every m , the inclusion α ( β ◦ m γ ) ⊆ αβ + αγ holds in V . (By the way, let us mention that J´onsson paper[10] contains a big deal of other fundamental results about distribu-tive varieties with significant and unexpected applications to lattices,among other.) It follows easily from J´onsson proof that, for every m ,there is some k (which depends only on m and on the n given by (1),but otherwise not on the variety) such that( m, k )-dist α ( β ◦ m γ ) ⊆ αβ ◦ k αγ A variety is ∆ k in the sense of [10] if and only if it satisfies (2 , k )-dist.Such varieties are sometimes called k -distributive , or are said to have k + 1 J´onsson terms . If k is minimal with the above property (with m = 2), V is said to be of J´onsson level k in Freese and Valeriote [5].If we slightly modify the proof of J´onsson theorem as presented inBurris and Sankappanavar [2, Theorem 12.6] or in McKenzie, McNulty,and Taylor [20, Theorem 4.144], we see that if a variety V satisfies(2 , k + 1)-dist, then V satisfies ( ℓ + 1 , kℓ + 1)-dist, for every ℓ ≥
1. Thisresult is also a special case of Corollary 2.2 below. To formulate thisand other results in a more concise way, it is natural to introduce thefollowing
J´onsson distributivity function J V of a congruence distributivevariety V . For every positive natural number m , we set J V ( m ) to be theleast k such that V satisfies the identity ( m + 1 , k + 1)-dist. The “shiftby 1” in the above notation will greatly simplify subsequent statements.For example, the above remark is more neatly stated by asserting thatif J V (1) = k , then J V ( ℓ ) ≤ kℓ , for every positive ℓ . We can now askthe following problem. The J´onsson distributivity spectrum problem.
Which functions(with domain the set of positive natural numbers) can be realized as J V ,for some congruence distributive variety V ? Obviously, J V is a monotone function. By the above comments,if J V (1) = 1, then J V ( m ) ≤ m , for every positive m . Moreover if, forsome k , J V ( k ) < k , then V is k -permutable: just take α = 1 in equation( m, k )-dist. If V is k -permutable, then J V ( m ) < k , for every m . ´ONSSON DISTRIBUTIVITY SPECTRUM 3 As a consequence of the above observations, if J V (1) = 1, then J V is either the identity function, or is the identity up to some point andthen it is a constant function. An example of the first eventuality isthe variety of lattices; on the other hand, in the variety of n -Booleanalgebras from Schmidt [24] we have J V ( m ) = min { m, n } . Indeed, n -Boolean algebras have a lattice operation, are n + 1-permutable but,in general, not n -permutable; see also Hagemann and Mitschke [9] and[11, Example 2.8]. For convenience, we shall use J´onsson paper [11]as a reference for this and other examples. The author believes thatthis is the appropriate place to mention that J´onsson [11] has had aprofound influence in his mathematical formation.Mitschke [21] shows that the variety of implication algebras is 3-permutable, not permutable, ∆ and not ∆ . See also [9, Example 1]and [11, Example 2.6]. Hence in the variety of implication algebraswe have J V ( m ) = 2, for every m . Freese and Valeriote [5], using thereduct of an algebra formerly constructed by Kearnes [13], show that,for every n , there is an n -permutable variety which is ∆ n and not ∆ n − .See [5, p. 70–71]. Thus J V is constantly n − J V , for some variety, is closed under pointwise maximum.This is immediate from the result that the non-indexed product of twovarieties V and V ′ satisfies exactly the same strong Maltsev conditionssatisfied both by V and V ′ . See Neumann [22], Taylor [25] or [11, p.368–369]. We also need the easy fact that, for every m and k , thecondition J V ( m ) ≤ k is equivalent to a strong Maltsev condition; forexample, this is a consequence of the equivalence of (A) and (B) inTheorem 2.1 below. Proposition 1.1. If V and V ′ are congruence distributive varieties,then their non-indexed product V ′′ is such that J V ′′ ( m ) = max { J V ( m ) ,J V ′ ( m ) } , for every positive natural number m . We do not know whether, for every pair V , V ′ , we always have some V ′′ such that J V ′′ ( m ) = min { J V ( m ) , J V ′ ( m ) } , for every m .If in Proposition 1.1 we consider the variety of lattices and the men-tioned variety from [5, p. 70–71], we get J V ′′ ( m ) = max { n − , m } .By taking the non-indexed product of the variety of n ′ -Boolean alge-bras and again the variety from [5, p. 70–71], with n ≤ n ′ , we have J V ′′ ( m ) = n −
1, for m ≤ n − J V ′′ ( m ) = min { m, n ′ } , for m > n − J V ( m ) has little influence on thevalues of J V ( m ′ ), for m ′ < m . On the other hand, we are going to showthat J V ( m ) puts some quite restrictive bounds on J V ( m ′ ), for m ′ > m ,as we already mentioned for the easier case m = 1. PAOLO LIPPARINI Bounds on higher levels of the spectrum
Let R , S , . . . be variables intended to be interpreted as reflexive andadmissible (binary) relations on some algebra. If R is such a relation,let R ` denote the converse of R , that is, b R ` a if and only if a R b . Inthe next theorem we show that, for a variety, the congruence identity( m + 1 , k + 1)-dist is equivalent to the relation identity α ( R ◦ m R ` ) ⊆ αR ◦ k αR ` , with the further provision that if R can be expressed asa composition, then αR and αR ` factor out. See condition (3) in thenext theorem for a formal statement. The latter provision is necessary,since, without it, the case m = 1 ≤ k would be trivially true in everyvariety and, for every m > k suchthat the relation identity α ( R ◦ m R ` ) ⊆ αR ◦ k αR ` holds. Hence thisidentity alone is too weak for our purposes. See [17].Recall that a tolerance is a symmetric and reflexive admissible re-lations. We shall prove Part (C) in the next theorem in the generalcase when α is a tolerance, rather than a congruence. In particular,we get that, through a variety, the identity ( m, k )-dist is equivalent tothe same identity in which α is only assumed to be a tolerance. How-ever this stronger version shall not be used in what follows, hence thereader might always assume to be in the simpler case in which α is acongruence. Theorem 2.1.
For every variety V and integers m, k ≥ , the followingconditions are equivalent.(A) J V ( m ) ≤ k , that is, V satisfies the congruence identity ( m + 1 , k +1) -dist α ( β ◦ m +1 γ ) ⊆ αβ ◦ k +1 αγ (equivalently, we can ask that the free algebra in V generated by m + 2 elements satisfies the above identity.)(B) V has m +2 -ary terms t , . . . , t k +1 such that the following identitieshold in V : x = t ( x, x , x , x , . . . , x m , x m +1 ) , (B1) x = t i ( x, x , x , x , . . . , x m , x ) , for ≤ i ≤ k + 1 , (B2) t i ( x , x , x , x , x , x , . . . ) = t i +1 ( x , x , x , x , x , x , . . . ) , for even i , ≤ i ≤ k,t i ( x , x , x , x , x , . . . ) = t i +1 ( x , x , x , x , x , . . . ) , for odd i , ≤ i ≤ k, (B3) ´ONSSON DISTRIBUTIVITY SPECTRUM 5 t k +1 ( x , x , x , x , . . . , x m , z ) = z (B4) (C) For every algebra A ∈ V , every positive integer ℓ (equivalently,for ℓ = 1 ), every tolerance α of A and all reflexive and admissiblerelations R , S , . . . , S ℓ on A , if R = S ◦ S ◦ · · · ◦ S ℓ and Θ = αS ◦ αS ◦ · · · ◦ αS ℓ , then (C1) α ( R ◦ m R ` ) ⊆ Θ ◦ k Θ ` Proof.
The equivalence of (A) and (B) is an instance of the Pixley-Wille algorithm [23, 27] (actually, it can be seen as a good exercise tocheck a student’s understanding of the algorithm). We shall need hereonly (A) ⇒ (B), which can be proved as follows. Consider the freealgebra F V ( m + 2) in V over m + 2 generators y , . . . , y m +1 and let α bethe congruence generated by ( y , y m +1 ), β be the congruence generatedby { ( y , y ) , ( y , y ) , ( y , y ) , . . . } and γ be the congruence generated by { ( y , y ) , ( y , y ) , . . . } . Thus ( y , y m +1 ) ∈ α ( β ◦ m +1 γ ) hence, by (A),( y , y m +1 ) ∈ αβ ◦ k +1 αγ . This latter relation is witnessed by k + 2elements of F V ( m + 2) which give rise to terms witnessing (B).(B) ⇒ (C) Let ( a, c ) ∈ α ( R ◦ m R ` ) in some algebra A ∈ V . This iswitnessed by elements a = b , b , b , . . . , b m = c such that b R b R ` b R b R ` b . . . Furthermore a α c .First suppose that m is even. For 0 ≤ i ≤ k , we shall consider theelements e i = t i ( b , b , b , b , . . . , b m − , b m − , b m , b m ) = t i +1 ( b , b , b , b , . . . , b m − , b m − , b m , b m ) for i even, e i = t i ( b , b , b , b , . . . , b m − , b m − , b m − , b m ) = t i +1 ( b , b , b , b , . . . , b m − , b m − , b m − , b m ) for i odd,where the identities follow from (B3). In writing the above formula weare supposing that m is large enough; otherwise, say, for m = 2 and i odd, e i should be set equal to t i ( b , b , b , b ).If we had to show only α ( R ◦ m R ` ) ⊆ αR ◦ k αR ` , for α a congruence,it would be enough to consider the above elements, since, e = b = a by(B1), e k = b m = c by (B4) and, say, for i even, e i = t i +1 ( b , b , b , b , . . . ,b m − , b m − , b m , b m ) R t i +1 ( b , b , b , b , . . . , b m − , b m − , b m − , b m ) = e i +1 ,since R is admissible. Similarly e i R ` e i +1 , for i odd. Notice that b h R b h +1 for h even and b h R ` b h +1 for h odd, hence b h +1 R ` b h for h even and b h +1 R b h for h odd. Moreover, for i even, if α isa congruence, then e i = t i ( a, a, b , . . . , c ) α t i ( a, a, b , . . . , a ) = a = t i +1 ( a, b , b , . . . , a ) α t i +1 ( a, b , b , . . . , c ) = e i +1 , by (B2), hence e i αe i +1 . Similarly, e i α e i +1 , for i odd. Hence the elements e i , for 0 ≤ i ≤ k , witness ( a, c ) ∈ αR ◦ k αR ` . However, as we mentioned, the PAOLO LIPPARINI identity α ( R ◦ m R ` ) ⊆ αR ◦ k αR ` is too weak to chain back to theother conditions.Hence we need to use the assumption R = S ◦ S ◦ · · · ◦ S ℓ to provethe stronger conclusion α ( R ◦ m R ` ) ⊆ Θ ◦ k Θ ` . Moreover, we shall alsoextend some ideas from Cz´edli and Horv´ath [3] in order to prove thecase of (C) in which α is only assumed to be a tolerance.By the assumption R = S ◦ S ◦ · · · ◦ S ℓ and given the elements b , b , b , . . . , b m introduced at the beginning, we have that, for every h with 0 ≤ h < m , there are elements b h, , b h, , . . . , b h,ℓ +1 such that b h = b h, S b h, S . . . S ℓ b h,ℓ +1 = b h +1 , for h even, and b h +1 = b h, S b h, S . . . S ℓ b h,ℓ +1 = b h , for h odd, since in this latter case b h R ` b h +1 ,that is, b h +1 R b h . Hence, with the e i ’s defined as in the above-displayedformula, we have, for i even, e i = t i +1 ( b , b , b , . . . , b m − , b m , b m ) = t i +1 ( b , b , , b , , . . . , b m − , , b m − , , b m ) S t i +1 ( b , b , , b , , . . . , b m − , , b m − , , b m ) S t i +1 ( b , b , , b , , . . . , b m − , , b m − , , b m ) S . . .S ℓ t i +1 ( b , b ,ℓ +1 , b ,ℓ +1 , . . . , b m − ,ℓ +1 , b m − ,ℓ +1 , b m ) = t i +1 ( b , b , b , . . . , b m − , b m − , b m ) = e i +1 Moreover, for 0 ≤ q ≤ ℓ and 0 ≤ i ≤ k , we have, by (B2): t i +1 ( a, b ,q , b ,q , . . . , c ) = t i +1 ( t i +1 ( a, b ,q , b ,q , . . . , c ) , b ,q +1 , b ,q +1 , . . . , t i +1 ( a , b ,q , b ,q , . . . , c )) αt i +1 ( t i +1 ( a, b ,q , b ,q , . . . , a ) , b ,q +1 , b ,q +1 , . . . , t i +1 ( c , b ,q , b ,q , . . . , c )) = t i +1 ( a, b ,q +1 , b ,q +1 , . . . , c )(elements in bold are those moved by α ). Hence, for i even, the elements t i +1 ( a, b ,q , b ,q , b ,q , . . . , c ), for 0 ≤ q ≤ ℓ , witness e i Θ e i +1 , recallingthat Θ = αS ◦ αS ◦ · · · ◦ αS ℓ and that a = b , b m = c . Similarly e i Θ ` e i +1 , for i odd. After the above considerations, we see that theelements e i , for 0 ≤ i ≤ k , witness ( a, c ) ∈ α Θ ◦ k α Θ ` , what we had toshow.The case m odd is similar. This time, the e i ’s are defined as follows,again for 0 ≤ i ≤ k . e i = t i ( b , b , b , b , . . . , b m − , b m − , b m − , b m ) for i even, e i = t i ( b , b , b , b , . . . , b m − , b m − , b m , b m ) for i odd . The rest is similar. ´ONSSON DISTRIBUTIVITY SPECTRUM 7 (C) ⇒ (A) Take ℓ = 1, S = β and S = γ in (C). Apparently,computing R ◦ m R ` = ( β ◦ γ ) ◦ m ( γ ◦ β ) gives 2 m factors, but we have m − R ◦ m R ` = β ◦ m +1 γ .Similarly, Θ ◦ k Θ ` = αβ ◦ k +1 αγ and we get (A). (cid:3) Corollary 2.2. If J V ( m ) = k and ℓ > , then J V ( mℓ ) ≤ kℓ .Proof. We assume J V ( m ) = k and we have to show that α ( β ◦ mℓ +1 γ ) ⊆ αβ ◦ kℓ +1 αγ . We apply Theorem 2.1(A) ⇒ (C), taking S = S = · · · = β and S = S = · · · = γ . We have R ◦ m R ` = β ◦ mℓ +1 γ ,since, as in the proof of (B) ⇒ (C) above, we apparently have m ( ℓ + 1)factors, but m − m ( ℓ + 1) − ( m −
1) = mℓ + 1 factors. The definition of Θ in (C) becomesΘ = αβ ◦ ℓ +1 αγ , hence, arguing as above, Θ ◦ k Θ ` has kℓ + 1 actualfactors, that is, Θ ◦ Θ ` = αβ ◦ kℓ +1 αγ . The inclusion (C1) thus givesthe corollary. (cid:3) Stronger results can be proved in the case of 3-distributivity. Actu-ally, the arguments work in a more general setting. In the followingtheorem we do not need the assumption that V is congruence distribu-tive. Theorem 2.3.
Suppose that V satisfies the congruence identity (2) α ( β ◦ γ ) ⊆ α ( γ ◦ β ) ◦ αγ (the above identity is equivalent to the existence of Gumm terms p, j , j in the terminology of Section 4 below). Then V satisfies α ( β ◦ γ ) ◦ αβ = αβ ◦ α ( γ ◦ β )(3) α ( β ◦ γ ◦ β ) ◦ αγ = α ( β ◦ γ ) ◦ αβ ◦ αγ (4) α ( β ◦ m +2 γ ) = α ( β ◦ γ ) ◦ ( αβ ◦ m αγ ) , for m ≥ . (5)Notice that the identity (2) is weaker than 3-distributivity, whichcorresponds to the identity α ( γ ◦ β ) ⊆ αγ ◦ αβ ◦ αγ , equivalently,taking converse, α ( β ◦ γ ) ⊆ αγ ◦ αβ ◦ αγ . Proof.
By applying (2) with β and γ exchanged, we get αβ ◦ α ( γ ◦ β ) ⊆ αβ ◦ α ( β ◦ γ ) ◦ αβ = α ( β ◦ γ ) ◦ αβ , since obviously αβ ◦ α ( β ◦ γ ) = α ( β ◦ γ ).The reverse inclusion in (3) follows by symmetry.It is easy and standard to show that if V satisfies the congruenceidentity (2) then V has terms p and j such that the identities x = p ( x, y, y ), p ( x, x, y ) = j ( x, x, y ), j ( x, y, y ) = y and x = j ( x, y, x ) holdin V . Were V in addition 3-distributive, we would also have the identity x = p ( x, y, x ). Cf. the J´onsson terms which shall be recalled at the PAOLO LIPPARINI beginning of the next section. However, here the identity x = p ( x, y, x )shall not be needed.We now prove (4). One inclusion is trivial and, in order to provethe nontrivial inclusion, it is enough to prove α ( β ◦ γ ◦ β ) ⊆ α ( β ◦ γ ) ◦ αβ ◦ αγ , since αγ is a congruence, hence transitive. So let ( a, d ) ∈ α ( β ◦ γ ◦ β ), hence a α d and a β b γ c β d , for some b and c . Let uscompute a = j ( d, a, a ) β j ( d, b, a ) γ j ( d, c, a ) β j ( c, c, b ) = p ( c, c, b ) βp ( d, c, b ) γ p ( d, c, c ) = d . Moreover, a = j ( a, b, a ) α j ( d, b, a ), hence a αβ j ( d, b, a ). Furthermore, j ( d, b, a ) α a = j ( a, c, a ) α j ( d, c, a )hence also j ( d, b, a ) αγ j ( d, c, a ). Finally, j ( d, c, a ) α a α d , hence j ( d, c, a ) α ( β ◦ γ ) d . Hence the elements j ( d, b, a ) and j ( d, c, a ) witness( a, d ) ∈ αβ ◦ αγ ◦ α ( β ◦ γ ). By applying (3) twice, we get αβ ◦ αγ ◦ α ( β ◦ γ ) = αβ ◦ α ( γ ◦ β ) ◦ αγ = α ( β ◦ γ ) ◦ αβ ◦ αγ . In conclusion,( a, d ) ∈ α ( β ◦ γ ) ◦ αβ ◦ αγ and (4) is proved.In order to prove (5) we need a claim. Claim. If A ∈ V , α is a congruence, R , S , T are reflexive and admis-sible relations on A , R ⊆ T and S ⊆ T ` , then (6) α ( T ◦ T ` ◦ R ◦ S ) = α ( T ◦ T ` ) ◦ αR ◦ αS To prove the claim, first notice that an inclusion is trivial, since α is assumed to be a congruence. To prove the nontrivial inclusion,let ( a, e ) ∈ α ( T ◦ T ` ◦ R ◦ S ), with a α e and a T b T ` c R d Se . We have a = p ( a, c, c ) T p ( b, b, d ) = j ( b, b, d ) T ` j ( a, c, e ), since R ⊆ T and S ⊆ T ` . Moreover, a = j ( a, c, a ) α j ( a, c, e ), hence a α ( T ◦ T ` ) j ( a, c, e ). Furthermore, j ( a, c, e ) R j ( a, d, e ) S j ( a, e, e ) = e and j ( a, c, e ) α a α e = j ( e, d, e ) α j ( a, d, e ), hence j ( a, c, e ) αRj ( a, d, e ) and j ( a, d, e ) αS = e , thus the elements j ( a, c, e ) and j ( a, d, e )witness ( a, e ) ∈ α ( T ◦ T ` ) ◦ αR ◦ αS and the claim is proved.Having proved the claim, we go on by proving the case m = 2 ofequation (5). By taking T = β ◦ γ , R = 0 and S = γ in equation (6),we get α ( β ◦ γ ◦ β ◦ γ ) = α ( β ◦ γ ◦ γ ◦ β ◦ γ ) = α ( T ◦ T ` ◦ R ◦ S ) = α ( T ◦ T ` ) ◦ αR ◦ αS = α ( β ◦ γ ◦ γ ◦ β ) ◦ αγ = α ( β ◦ γ ◦ β ) ◦ αγ = α ( β ◦ γ ) ◦ αβ ◦ αγ ,where in the last identity we have used (4).The rest of the proof now follows quite easily by induction. Supposethat m ≥ n with2 ≤ n < m . If m is odd, take T = β ◦ m +12 γ , R = γ and S = β in (6),getting α ( β ◦ m +2 γ ) = α ( T ◦ T ` ◦ R ◦ S ) = α ( T ◦ T ` ) ◦ αR ◦ αS = α ( β ◦ m γ ) ◦ αγ ◦ αβ = α ( β ◦ γ ) ◦ ( αβ ◦ m αγ ), where the last identityfollows from the inductive hypothesis, except in case m = 3, where weneed (4). If m is even, take T = β ◦ m +22 γ , R = 0 and S = γ in (6),getting α ( β ◦ m +2 γ ) = α ( T ◦ T ` ◦ R ◦ S ) = α ( T ◦ T ` ) ◦ αR ◦ αS = ´ONSSON DISTRIBUTIVITY SPECTRUM 9 α ( β ◦ m +1 γ ) ◦ αγ = α ( β ◦ γ ) ◦ ( αβ ◦ m αγ ), where we have used theinductive hypothesis again to obtain the last identity. (cid:3) Corollary 2.4. If J V (1) = 2 , that is, V is -distributive, then J V ( n ) ≤ n , for every n ≥ . Moreover the following congruence identity holdsin V α ( β ◦ γ ◦ β ) ◦ αγ = αβ ◦ αγ ◦ αβ ◦ αγ Proof.
Immediate from Theorem 2.3. Indeed, as we mentioned, if V is3-distributive, then the hypothesis of Theorem 2.3 holds. Then, using α ( β ◦ γ ) ⊆ αβ ◦ αγ ◦ αβ in equation (4), we get α ( β ◦ γ ◦ β ) ◦ αγ = α ( β ◦ γ ) ◦ αβ ◦ αγ ⊆ αβ ◦ αγ ◦ αβ ◦ αβ ◦ αγ = αβ ◦ αγ ◦ αβ ◦ αγ . Theconverse inclusion is trivial. All the other inclusions are similar; asabove, in any case, a pair of αβ ’s is absorbed into a single occurrence(the case n = 3 here corresponds to m = 2 in (5) and so on). (cid:3) Variants of the spectrum
One can consider an alternative function J ` V which is defined in sucha way that J ` V ( m ) is the smallest k such that the identity( m + 1 , k + 1)-dist ` α ( β ◦ m +1 γ ) ⊆ αγ ◦ k +1 αβ holds in V . Here γ and β are exchanged on the right-hand side, incomparison with ( m + 1 , k + 1)-dist. It is easy to see that J ` V and J V are different functions; indeed, J ` V ( m ) = m , for some m , implies m + 1-permutabilty, while J V ( m ) = m holds in lattices, for every m .Obviously, however, J ` V ( m ) and J V ( m ) differ at most by 1. There aresome further simple relations connecting J ` V and J V . For example, if m and J V ( m ) have the same parity, then J ` V ( m ) ≥ J V ( m ). Comparethe parallel discussion (corresponding to the case m = 1 here) in [5, p.63].A theorem analogous to Theorem 2.1 holds for J ` V : just exchangeeven and odd in (B3) and replace identity (C1) by α ( R ◦ m R ` ) ⊆ Θ ` ◦ k Θ. Arguing as in Corollary 2.2, we then get that if J ` V ( m ) = k and ℓ is odd, then J ` V ( mℓ ) ≤ kℓ . If ℓ is even, then J ` V ( m ) = k implies J V ( mℓ ) ≤ kℓ .A probably more significant variant is suggested by the use of di-rected J´onsson terms. See Z´adori [28] and Kazda, Kozik, McKenzieand Moore [12]. Let us recall the definitions. J´onsson terms [10] are terms j , . . . , j k satisfying x = j ( x, y, z ) , (J1) x = j i ( x, y, x ) , for 0 ≤ i ≤ k, (J2) j i ( x, x, z ) = j i +1 ( x, x, z ) , for even i , 0 ≤ i < k,j i ( x, z, z ) = j i +1 ( x, z, z ) , for odd i , 0 ≤ i < k, (J3) j k ( x, y, z ) = z (J4)Notice that this is exactly condition (B) in Theorem 2.1 in the par-ticular case m = 1 and with k in place of k + 1. We get directedJ´onsson terms , or Z´adori terms [28, 12] if in the above set of identitieswe replace condition (J3) by(D) j i ( x, z, z ) = j i +1 ( x, x, z ) for 0 ≤ i < k Seemingly, directed J´onsson terms first appeared (unnamed) in [28,Theorem 4.1], whose proof relies on McKenzie [19]. In [28] it is shown,among other, that a finite bounded poset admits J´onsson operationsfor some k if and only if it admits directed J´onsson operations for some k ′ . Kazda, Kozik, McKenzie and Moore [12] proved the equivalencefor terms in an arbitrary variety, thus, by [10], a variety is congruencedistributive if and only if it has directed J´onsson terms.For a binary relation R , let R k = R ◦ R ◦ R . . . with k factors. Inother words, R k = R ◦ k R . Proposition 3.1.
If some variety V has k + 1 directed J´onsson terms d , . . . , d k , with k ≥ , then, for every ℓ ≥ , V satisfies the identity α ( S ◦ S ◦ . . . ◦ S ℓ ) ⊆ ( αS ◦ αS ◦ . . . ◦ αS ℓ ) k − where α varies among tolerances and S , S , . . . vary among reflexiveand admissible relations on some algebra in V . In particular, for ℓ even, V satisfies (7) α ( S ◦ ℓ T ) ⊆ αS ◦ ℓ ( k − αT and, for ℓ odd, α ( β ◦ ℓ T ) ⊆ αβ ◦ k ′ αT, where β varies among congruences and k ′ = ℓ ( k − − k + 2 .Proof. Let us work in some fixed algebra belonging to V . Let ( a, c ) ∈ α ( S ◦ S ◦ · · · ◦ S ℓ ). Thus a α c and there are elements b , b , b , . . . , b ℓ such that a = b S b S b . . . b ℓ − S ℓ b ℓ = c . First suppose that α isa congruence. For every i and h , we have d i ( a, b h , c ) α d i ( a, b h , a ) = a ,hence all such elements are α -related. Moreover, for every i < k , d i ( a, a, c ) S d i ( a, b , c ) S d i ( a, b , c ) S . . . S ℓ d i ( a, c, c ) = d i +1 ( a, a, c ). ´ONSSON DISTRIBUTIVITY SPECTRUM 11 This shows that, for every i < k , ( d i ( a, a, c ) , d i +1 ( a, a, c )) ∈ αS ◦ αS ◦· · · ◦ αS ℓ .Since a = d ( a, c, c ) = d ( a, a, c ) and d k ( a, a, c ) = c , then the ele-ments d i ( a, a, c ), for 1 ≤ i ≤ k , witness ( a, c ) ∈ ( αS ◦ αS ◦· · ·◦ αS ℓ ) k − .The last statement follows immediately. We just mention that inthe last equation k − β istransitive.The case when α is just a tolerance is treated as in Cz´edli andHorv´ath [3]. Indeed, for every i, j, h, h ′ , we have d i ( a, b h , c ) = d i ( d j ( a,b h ′ , a ) , b h , d j ( c, b h ′ , c )) α d i ( d j ( a, b h ′ , c ) , b h , d j ( a, b h ′ , c )) = d j ( a, b h ′ , c ).The rest is the same. (cid:3) Since congruences and tolerances are, in particular, reflexive andadmissible, we obtain corresponding congruence/tolerance identitiesfrom the above identities about relations. In particular, by [10, 12]and Proposition 3.1, a variety is congruence distributive if and only ifequation (7) holds for some ℓ ≥ k , equivalently, for ℓ = 2 andsome k . If ∗ denotes transitive closure, we then get that a variety V iscongruence distributive if and only if ( α ( S ◦ T )) ∗ = ( αS ◦ αT ) ∗ holdsin V , if and only if α ∗ ( S ◦ T ) ∗ = ( αS ◦ αT ) ∗ holds in V . Notice that, onthe other hand, neither the identity ( α ( S ◦ S )) ∗ = ( αS ) ∗ nor the iden-tity α ∗ S ∗ = ( αS ) ∗ imply congruence distributivity, since the identitieshold, e. g., in permutable varieties (in fact, we have a proof that bothidentities are equivalent to congruence modularity [17]). See however[18] for a variation actually equivalent to congruence distributivity. Inall the above statements α can be taken equivalently as a tolerance oras a congruence, while S and T vary among reflexive and admissiblerelations.We do not know whether, for every congruence distributive variety V ,there is k such that V satisfies the relation identity R ( S ◦ T ) ⊆ RS ◦ k RT .The existence of some k as above is equivalent to R ∗ ( S ◦ T ) ∗ = ( RS ◦ RT ) ∗ . The arguments from Gyenizse and Mar´oti [8] can be adapted toshow that if V is congruence distributive and the free algebra F V (3) in V over 3 elements is finite, then R ∗ ( S ◦ T ) ∗ = ( R ∗ S ◦ R ∗ T ) ∗ . It is easyto see that 2-distributivity does imply R ( S ◦ T ) ⊆ RS ◦ RT . Cf. [15,Remark 17]. We have a proof that if V has 4 directed J´onsson terms d , d , d , d and F V (2) is finite, then R ( S ◦ T ) ⊆ RS ◦ k RT , for some k which depends on the variety.Notice also that, since the composition of reflexive and admissiblerelations is still reflexive and admissible, we can get new identities bysubstitution, without recurring to terms. E. g., from α ( S ◦ T ) ⊆ αS ◦ αT ◦ αS , replacing T by T ◦ S , we get α ( S ◦ T ◦ S ) ⊆ αS ◦ α ( T ◦ S ) ◦ αS ⊆ αS ◦ αT ◦ αS ◦ αT ◦ αS . We leave similar computations to the interestedreader, observing that arguments using terms, though more involved,generally produce stronger results.By the above considerations, one would be tempted to believe that itis always more convenient to use directed J´onsson terms, rather thanJ´onsson terms. However, the exact relation between the (minimal)number of directed J´onsson terms and of J´onsson terms in a variety isnot clear; see [12, Observation 1.2 and Section 7]. Even in the case whenwe have the same number of J´onsson and of directed J´onsson terms,there are situations in which it is more convenient to use the formerterms. For example, with 5 J´onsson terms we have α ( β ◦ γ ) ⊆ αβ ◦ αγ ,while from 5 J´onsson directed terms we obtain only α ( β ◦ γ ) ⊆ αβ ◦ αγ from (7). Notice that here we count terms including the initial and finalprojections (that is, 5 terms correspond to the case k = 4).On the other hand, an example in which it is more convenient to usedirected terms is Baker example from [1], the variety V generated bythe polynomial reducts of lattices in which only the ternary operation f ( a, b, c ) = a ∧ ( b ∨ c ) is considered. See also [11, Example 2.12]. Bakershowed that V is ∆ but not ∆ , that is, in the present terminology, J V (1) = 3, from which we get from Corollary 2.2 that J V ( ℓ ) ≤ ℓ .However, taking d ( x, y, z ) = x ∧ ( y ∨ z ), d ( x, y, z ) = z ∧ ( x ∨ y ) and d and d the first and last projections, we get a sequence of directedJ´onsson terms, hence we can apply Proposition 3.1 with k = 3 inorder to get the better evaluation J V ( ℓ − ≤ ℓ −
2, for ℓ odd and J V ( ℓ − ≤ ℓ −
1, for ℓ even.We can introduce a version of J V which takes into account reflexiveand admissible relations in place of congruences. For m ≥
1, we let J r V ( m ) be the least k such that α ( S ◦ m +1 T ) ⊆ αS ◦ k +1 αT ; similarly, J r ` V is defined by considering αT ◦ k +1 αS on the right-hand side. Thedefinitions are justified in view of [10, 12] and Proposition 3.1. We canthus formulate the following general problem. The generalized J´onsson distributivity spectrum problem.
Which quadruplets of functions can be realized as ( J V , J ` V , J r V , J r ` V ) , forsome congruence distributive variety V ? In the above problem we could take into account additional condi-tions. For example, consider the formula α ( β ◦ γ ◦ δ ◦ β . . . ) ⊆ αβ ◦ αγ ◦ αδ ◦ αβ . . . , with m factors on the left-hand side. Which is the smallestnumber of factors on the right-hand side such that the formula holds ina specific congruence distributive variety? Similarly, which is the bestbound in ( α ◦ m β )( α ◦ m β ) ⊆ α α ◦ α β ◦ β α ◦ β β ◦ α α . . . ? ´ONSSON DISTRIBUTIVITY SPECTRUM 13 Or, more generally, for ( α ◦ m β )( α ◦ m β ) . . . ( α h ◦ m β h )? What about( α ◦ m β )( α ◦ m γ )( β ◦ m γ ) ⊆ αβ ◦ βγ ◦ αγ ◦ αβ . . . ?Of course, the analogue of Proposition 1.1 holds for J ` V , J r V and J r ` V ,too, using similar arguments.4. An unexpected connection with the modularityspectra
One can also define functions analogous to J V in the case of congru-ence modular varieties. In the present section we give up the conventionthat every variety at hand is congruence distributive. By a fundamen-tal theorem by A. Day [4], a variety V is congruence modular if andonly there is some k such that the congruence identity(D k ) α ( β ◦ αγ ◦ β ) ⊆ αβ ◦ k αγ holds in V . Again, Day result is stated in a form involving a certainnumber of terms, but we shall not need the explicit Day terms here.A variety V is k -modular if V satisfies equation (D k ), and V has Daylevel k if such a k is minimal. For a congruence modular variety V ,we can define the Day modularity function D V as follows. For m ≥ D V ( m ) is the least k such that α ( β ◦ m αγ ) ⊆ αβ ◦ k αγ holds in V . Thearguments from [4] show that D V ( m ) is defined for every m and everycongruence modular variety V , but the methods from [4] do not furnishthe best value. See [16].The case of congruence modularity is substantially more involvedthan the distributivity case treated here. Gumm [6, 7] provided anothercharacterization of congruence modular varieties by considering termswhich “compose permutability with distributivity”. In detail, Gummterms are terms p, j , . . . , j k satisfying the above conditions (J2)-(J4)(the distributivity part, involving only the j ’s), together with the fol-lowing permutability part:(P) x = p ( x, z, z ) , and p ( x, x, z ) = j ( x, x, z )Notice that the definition given here is slightly different from Gummoriginal conditions in [6, Theorem 1.4], where odd and even are ex-changed and where p is considered after the j i ’s (the q i ’s in the notationfrom [6]). This is not simply a matter of symmetry: in the formulationby Gumm, when n is odd, one gets an unnecessary term which can bediscarded, hence Gumm actual condition is interesting only for n even.Of course, the above remark is interesting only when one is concernedwith the smallest k (or n ) for which there are Gumm terms. If one isconcerned just with the existence of some k for which such terms exist, then the remark is irrelevant. As far as we know, the above formulationfirst appeared in Lakser, Taylor and Tschantz [14] and Tschantz [26].Using Gumm terms, Tschantz [26] showed that a variety V is con-gruence modular if and only if the congruence identity α ( β + γ ) ⊆ α ( γ ◦ β ) ◦ ( αγ + αβ ) holds in V . Hence it is also natural to introducethe Tschantz modularity function T V for a congruence modular variety V in such a way that, for m ≥ T V ( m ) is the least k such that thefollowing congruence identity holds in V α ( β ◦ m γ ) ⊆ α ( γ ◦ β ) ◦ ( αγ ◦ k αβ )Notice that T V (2) = 0 is equivalent to congruence permutability, justtake α = 1 (by convention we let β ◦ γ = 0). More generally, T V (2) = k if and only if V has k + 2 Gumm terms p, j , . . . , j k +1 , but not k + 1Gumm terms. In the above terminology, equation (5) in Theorem 2.3states that if T V (2) = 1, then T V ( m + 2) ≤ m , for m ≥ D V and T V appear rather involved. Adetailed study of their connection goes beyond the scope of the presentnote; we refer to [16] for further details. We just mention that we candefine the ` -variants of the above notions. Moreover, Kazda, Kozik,McKenzie and Moore [12] introduced the directed variants of Gummterms, too. Using their result we can see that a variety is congruencemodular if and only if, for every m ≥
1, there is some h such that α ( S ◦ m +1 T ) ⊆ α ( T ◦ S ) ◦ ( αT ◦ h αS ), where, as above, S and T varyamong reflexive and admissible relations. See [17].We finally note an intriguing connection among the J´onsson and themodularity spectra. We first need a preliminary lemma. Lemma 4.1.
If a variety V has k + 2 Gumm terms p, j , . . . , j k +1 , then V satisfies the following congruence identity (as above, α can be takenas a tolerance). α ( β ◦ γ ◦ β ) ⊆ α ( γ ◦ β ) ◦ ( αγ ◦ k αβ ) Proof.
If we do not care about the exact number of factors on the right-hand side, the proposition follows from [26]. The arguments from [26]seem to produce a much larger number of factors, anyway.Suppose that ( a, d ) ∈ α ( β ◦ γ ◦ β ), thus a α d and a β b γ c β d , forsome b and c . Using equations (J2)-(J4), no essentially new argumentis needed to show that ( j ( a, a, d ) , d ) ∈ ( αβ ◦ αγ ◦ αβ ) k .Indeed, by (J2), j i ( a, e, d ) = j i ( j i ′ ( a, e ′ , a ) , e, j i ′ ( d , e ′ , d )) α j i ( j i ′ ( a, e ′ , d ) , e, j i ′ ( a , e ′ , d )) = j i ′ ( a, e ′ , d ), for all indices i, i ′ and elements e, e ′ ,hence all the elements of the above form are α -related. For i odd,we have j i ( a, a, d ) β j i ( a, b, d ) γ j i ( a, c, d ) β j i ( a, d, d ), thus ( j i ( a, a, d ) ,j i ( a, d, d )) ∈ αβ ◦ αγ ◦ αβ . Similarly, ( j i ( a, d, d ) , j i ( a, a, d )) ∈ αβ ◦ αγ ◦ ´ONSSON DISTRIBUTIVITY SPECTRUM 15 αβ , for i even. Hence j ( a, a, d ), j ( a, d, d ) = j ( a, d, d ), j ( a, a, d ) = j ( a, a, d ), . . . witness ( j ( a, a, d ) , d ) ∈ ( αβ ◦ αγ ◦ αβ ) k = αβ ◦ k +1 αγ .Moreover, a = p ( a, b, b ) γ p ( a, b , c ) β p ( a, a , d ) = j ( a, a, d ), by(P), hence ( a, j ( a, a, d )) ∈ α ( γ ◦ β ), thus ( a, d ) ∈ α ( γ ◦ β ) ◦ ( αβ ◦ k +1 αγ ). Finally, α ( γ ◦ β ) ◦ αβ = α ( γ ◦ β ) (or, better, use j ( a, b, d ) inplace of j ( a, a, d )), hence one more factor is absorbed and we get theconclusion. (cid:3) Theorem 4.2.
Suppose that V is a congruence distributive variety.(1) If m ≥ and V is m -modular, then J V (2) ≤ J ` V (1)+2 m − m − and J ` V (2) ≤ J V (1) + 2 m − m − .(2) If V has k + 2 Gumm terms, then J V (2) ≤ J ` V (1) + 2 k and J ` V (2) ≤ J V (1) + 2 k . The bounds can be improved by if J ` V (1) is odd, respectively, if J V (1) is even.Proof. Part (2) is immediate from Lemma 4.1.In [14, Theorem 2] it is proved that if a variety is m -modular, then V has ≤ m − m + 1 Gumm terms. Hence (1) follows from (2). (cid:3) There are versions of Lemma 4.1 and of Theorem 4.2 obtained usingGumm directed terms [12] in place of Gumm terms. See [17].As far as we know, there is the possibility that Theorem 4.2 is anempty result, namely that, say, for m ≥
3, every congruence distribu-tive variety with Day level m has J´onsson level < m − m − J V (2)). However, this would be a quite unexpected result. What The-orem 4.2 and the above comment do show is that in any case the Daylevel of a congruence distributive variety has an influence on the verylow levels of the J´onsson spectrum. This connection appears rathersurprising, whichever of the above cases occurs. Acknowledgement.
We thank Antonio Pasini for encouragement andfor providing us with a lot of material when we were unexperienced inuniversal algebra. We thank Keith Kearnes for useful correspondence.We thank the students of Tor Vergata University for stimulating dis-cussions.
Though the author has done his best efforts to compile the following list of referencesin the most accurate way, he acknowledges that the list might turn out to be incompleteor partially inaccurate, possibly for reasons not depending on him. It is not intended thateach work in the list has given equally significant contributions to the discipline. Hence-forth the author disagrees with the use of the list (even in aggregate forms in combinationwith similar lists) in order to determine rankings or other indicators of, e. g., journals,individuals or institutions. In particular, the author considers that it is highly inappro-priate, and strongly discourages, the use (even in partial, preliminary or auxiliary forms)6 PAOLO LIPPARINIof indicators extracted from the list in decisions about individuals (especially, job oppor-tunities, career progressions etc.), attributions of funds, and selections or evaluations ofresearch projects.
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