n-PERMUTABILITY IS NOT JOIN-PRIME FOR n ≥ 5 GERG ˝O GYENIZSE, MIKLÓS MARÓTI, AND LÁSZLÓ ZÁDORI A

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GERG ˝O GYENIZSE, MIKLÓS MARÓTI, AND LÁSZLÓ ZÁDORI

ABSTRACT. LetVPbe the variety generated by an order primal algebra of finite signature associated with a finite bounded posetPthat admits a near-unanimity operation. LetΛbe a finite set of linear identities that does not interpret inVP. LetVΛbe the variety defined byΛ. We prove thatVP∨VΛisn-permutable for somen. This implies that there is annsuch thatn-permutability is not join- prime in the lattice of interpretability types of varieties. In fact, it follows that n-permutability wherenruns through the integers greater than 1 is not prime in the lattice of interpretability types of varieties.

We strengthen this result by makingPandΛmore special. We letPbe the 6-element bounded poset that is not a lattice andVmthe variety defined by the set of majority identities for a ternary operational symbolm. We prove in this case thatVP∨Vmis 5-permutable. This implies thatn-permutability is not join- prime in the lattice of interpretability types of varieties whenevern≥5. We also provide an example demonstrating thatVP∨Vmis not 4-permutable.

1. INTRODUCTION

LetΓbe a set of identities over a certain signature of a variety. We say thatΓ interprets in a variety K if by replacing the operation symbols in Γby term ex- pressions ofK—same symbols by same terms with arities kept—the so obtained set of identities holds inK. AvarietyK1 interprets in a varietyK2if there is a set of identitiesΓthat definesK1 and interprets inK2. Roughly speaking, a va- rietyK1interprets in a varietyK2ifK2has a richer algebraic structure thanK1. Nevertheless, we have to be cautious with this rough approach, since, for example, the variety of sets with no basic operations and the variety of semigroups are equi-interpretable, meaning that they interpret in each other.

As easily seen, interpretability is a quasiorder and equi-interpretability is an equivalence on the class of varieties. The blocks of equi-interpretability are called theinterpretability types. In [3] Garcia and Taylor introduced thelattice of interpretability types of varieties that is obtained by taking the quotient of the class of varieties quasiordered by interpretability and the equi-interpretabiliy relation. The join in this lattice is described as follows. LetK1andK2be two varieties of disjoint signatures. Let K1 andK2 be defined by the sets Σ1 andΣ2 of identities, respectively. Their joinK1∨K2is the variety defined byΣ₁∪Σ₂. The so defined join is compatible with the interpretability relation of varieties, and naturally yields the definition of the join operation in the lattice of interpretability types of varieties.

Letn≥2 be an integer. AnalgebraAis congruencen-permutable, if for any two congruences α andβ ofA, α β· · ·=β α. . . where each side of the equality

The research of authors was supported by grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary, the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00015, and the NKFIH grants K115518 and K128042.

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consists ofnalternating factors ofα andβ. AnalgebraAis congruence distributive (congruence modular), if for any three congruences α, β andγ ofA (with α ≤γ)

(α∨β)∧γ= (α∧γ)∨(β∧γ).

Avariety isn-permutable (distributive, modular)if all of its members are congruencen-permutable (distributive, modular). Ann-ary operation f is idempotentif it satisfies the identity f(x,x, . . . ,x) =x.Avariety is idempotentif all of its members have idempotent basic operations.

In [3] Garcia and Taylor formulated the conjectures that 2-permutability is join- prime and the interpretability types of the modular varieties form a prime filter in the lattice of interpretability types of varieties. In [15] Tschantz announced a proof of the conjecture on 2-permutability. However, his proof has remained unpublished.

For restricted versions of the conjectures some positive results have been at- tained. Following the pioneering work of Hobby and McKenzie on locally finite varieties in [8], Kearnes and Kiss stepped beyond locally finiteness and gave a clas- sification of varieties in [9]. Their results imply that certain idempotent Mal’cev classes identified in their work form a prime filter in the lattice of interpretability types of idempotent varieties. In [16] Valeriote and Willard, by supplementing the characterization of a Mal’cev class in [9], proved that the interpretability types of the idempotentn-permutable varieties forn≥2 form also a prime filter in the idempotent case. In [14] Opršal obtained a similar result for idempotent modular varieties. In [10] Kearnes and Szendrei proved that for any n having an n-cube term is a join-prime property in the idempotent case.

In the present note we give some negative results related to the conjecture on permutability in the general case (where idempotency is not assumed). We shall prove that the filter of the interpretability types of then-permutable varieties where nruns through the integers greater than 1 is not prime in the lattice of interpretability types of varieties. We shall also prove that for anyn≥5,n-permutability is not join-prime in the lattice of interpretability types of varieties.

2. n-PERMUTABILITY FOR SOMEn

An n-ary operation f, n≥3, is a near-unanimity operation if it satisfies the identities

f(y,x, . . . ,x) =f(x,y, . . . ,x) =· · ·= f(x,x, . . . ,y) =x.

A ternary near-unanimity operation is called a majority operation. Clearly, the near-unanimity operations are idempotent. It is well known that on a finite set any clone that contains an n-ary near-unanimity operation is finitely generated. It is also known that any algebra that has a near-unanimity term operation is congruence distributive.

Let Pdenote a finite poset. Let Pbe an algebra whose underlying set equals that ofPand whose basic operations form a generating set of the clone ofP. We call such an algebra anorder primalalgebra with respect toP. LetVP denote the variety generated byP. IfPadmits a near-unanimity operation, then the clone of monotone operations ofPis finitely generated, and so the algebraPcan be chosen to be of finite signature. We always make this choice throughout the present paper for any finite posetPwhenPadmits a near-unanimity operation. Then by Baker’s

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finite basis theorem in [1] for finite algebras of finite signature in a congruence distributive variety, there exists a finite set Σ of identities that serves as a finite basis forVP.

An identity is called alinear identityif on both sides it has at most one operation symbol. LetPbe a finite bounded poset that admits a near-unanimity operation and Λa finite set of linear identities that does not interpret inVP. We assume thatΛis given in a signature disjoint from that ofVP. LetV =VP∨VΛ, soV is the variety that is defined by the identitiesΣ∪Λ. Aterm reduct of an algebraAis an algebra whose underlying set coincides with that ofAand whose basic operations are term operations ofA. We note that by Jónsson’s theorem for congruence distributive varieties in [2], every algebra inV has a term reduct that is isomorphic to a subdirect power ofPwherePis an order primal algebra forP. Later in the proof of our main result we use this observation.

Our aim is to prove in this section thatV isn-permutable for somen. It follows immediately that there exists annsuch thatn-permutability is not join-prime in the lattice of interpretability types of varieties. For the proof of the main result of this section we require two simple propositions.

Proposition 2.1. Let Q be finite bounded poset, andQan order-primal algebra determined by Q. The compatible quasiorders ofQare the equality, the full relation,

≤and≥.

Proof. It is enough to prove that the quasiorders generated by a single pair are of the above form. Let p and q be two different elements of Q. When p≤q, then (p,q) clearly generates≤. Similarly, when q≤ p, then(p,q)generates ≥.

Finally, observe that any pair(p,q) where pandqare incomparable inQcan be mapped into any pair by a monotone map, sinceQis bounded. Thus the quasiorder generated by such a pair(p,q)is the full relation.

Proposition 2.2. Let P be a finite bounded poset that admits a near-unanimity operation. LetD≤Pⁿandδ a compatible quasiorder ofD. Thenδ is aproduct quasiorder, i.e., there are compatible quasiordersδ₁, . . . ,δn ofPsuch that for all a= (a₁, . . . ,an)and a⁰= (a⁰₁, . . . ,a⁰_n)in D

(a,a⁰)∈δ ⇔ ∀i:(ai,a⁰_i)∈δi.

Proof. For any 1≤i≤n, letηi denote the kernel of the projection fromDto the i-th coordinate. By Theorem 2.6 in [5], the quasiorder lattice ofDis a distributive lattice, so

δ=δ∨0B=δ∨(^{^}ηi) =^{^}(δ∨ηi).

SincePhas no proper subalgebras and is simple,D/ηi∼=P. Hence the quotient of δ∨ηibyηicorresponds to a compatible quasiorderδi ofPby the correspondence theorem of quasiorders. This gives that for everyi

(a,a⁰)∈δ∨ηi⇔(ai,a⁰_i)∈δi.

Ann-arycompatible relationof an algebraAis a subuniverse ofAⁿ. LetDbe an n-ary compatible relation of P. A representation of D is a pair (R,S) where R is a finite quasiordered set, S is an n-element subset of R and there exists an enumerations₁, . . . ,snof the elements ofSsuch that

D={(f(s1), . . . ,f(sn))|f:R→Pis a monotone map}.

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Notice that if(R,S)and(R⁰,S⁰)are two representations ofD(even with possibly different base sets) andPis not an antichain, then the restrictions ofRtoSandR⁰ toS⁰={s⁰₁, . . . ,s⁰_n}are isomorphic via the maps_i7→s⁰_i. Indeed, for anys_i,s_j∈S, si ≤sj if and only if for all monotone f:R→P, f(si) ≤ f(sj), and the latter condition only depends on the projections of Donto itsiand jcoordinates. We note that every finitary compatible relation ofPhas a representation obtained in an obvious way from a primitive positive formula defining the relation in the language {≤_P},see [4].

LetCandDbe twon-ary compatible relations ofPsuch thatC⊆D. Then every representation(R,S)ofDextends to a representation(T,S)ofCby adding suitable vertices and edges to(R,S). Indeed, if(R,S)is a representation ofDand(R⁰,S)is an arbitrary representation ofCsuch that the only elements shared byRandR⁰are the elements ofS, thenT is obtained fromR∪R⁰by taking the transitive closure of the relation ofR∪R⁰.

LetQandRbe two quasiordered sets. A monotone mapg:Q→Ris called a retractionif there is a monotone maph:R→Qsuch thatgh=idR. Then the maph is called acoretraction. If there is retraction fromQtoR, thenRis called aretract ofQ. Now we present the main result of the section.

Theorem 2.3. Let P be a finite bounded poset that admits a near-unanimity operation andPan associated order primal algebra of finite signature. LetVP denote the variety generated byP. LetΣbe a finite basis forVPandΛa finite set of linear identities in a disjoint signature. IfΛdoes not interpret in the varietyVP, then the varietyV defined byΣ∪Λisn-permutable for somen.

Proof. In order to prove the statement we invoke a theorem of Hagemann in [6], an implicit proof is given by Lakser in [12]: a variety is n-permutable for some n if and only if every compatible quasiorder of any algebra of the variety is a congruence. So it suffices to prove that every compatible quasiorder of any algebra inV is a congruence. Let us suppose to the contrary that there is an algebraBin V that has a compatible quasiorderδ that is not a congruence. As we mentioned earlier, by Jónsson’s theorem, without loss of generality we may assume thatBis a subuniverse ofP^Ifor some setI.

We pick an edge(y₀,z₀)in the quasiordered set(B,δ)such that(z₀,y₀)6∈δ. We define two subalgebras ofP^I whose underlying sets are contained inB. LetC₀be the subalgebra generated by{y₀,z0}inP^I, and letD0be the subalgebra generated by

C₀∪ {f(u₁, . . . ,uk)|f is an operation symbol of aritykinΛ,u₁, . . . ,uk∈C₀} also in P^I. Clearly,y₀ andz₀ are inC0, andC₀⊆D₀⊆B.Since C0 andD0 are finitely generated algebras in a locally finite varietyVP, bothC₀andD₀are finite subalgebras ofP^I. Hence there is a finite subset ofI such that the projection ofD₀ to this subset is a bijective map. Lety,z,CandDbe the images ofy₀,z₀,C0andD0

under this projection, respectively. The restrictions ofδ toCandDproject down to the quasiorders δCandδD, respectively. SoCandDare subalgebras ofP^mfor some finitem, andδ_Candδ_Dare compatible quasiorders ofCandD, respectively.

Moreover,(y,z)∈δC⊆δDand(z,y)∈/δD.

Let(T,S)and(R,S)be representations of them-ary compatible relationsCand D, respectively. Without loss of generality—by the note preceding the present theorem—we assume thatR⊆T where containment means that(T,S)is obtained

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from (R,S) by adding suitable vertices and edges. The quasiorder δ_D is not a congruence ofD, and by Proposition 2.2,δDis a product quasiorder. So there exist quasiordersδi,i∈S, as in Proposition 2.2. Letr∈Ssuch thatδr=≤_Porδr=≥_P and, in both cases, the r-th coordinates ofyandzare different. There exists such anr, since(y,z)∈δDand(z,y)6∈δD. Forδ_Cthere also exist quasiordersδ_i⁰,i∈S, witnessing the claim in Proposition 2.2. Notice that δ_i⁰ ⊆δi, i∈S, and hence δ_r⁰=δr. Without loss of generality we may assumeδr=≤_P.

Letg be ther-th projection fromDtoP. We prove thatg is a retraction from the quasiordered set(D,δD)onto the posetP. Certainly,gis monotone. We define a maphfromPtoC: for any p∈Pleth(p) = (h₁(p), . . . ,h_m(p)),where for any i∈T

h_i(p):=







p, ifi≤r≤iinT, 1, ifi6≤r≤iinT, 0, otherwise.

Observe that the sequences(hi(p))i∈T, p∈P, are monotone maps fromT toP.

Therefore, for all p∈P, h(p)∈C. Moreover, the sequences (hi(p))i∈T, p∈P, differ only on the r-block ofT where they are defined to equal the constant p, respectively. So by applying Proposition 2.2 and taking into account thatδr=≤_P, we get that h(p)−→^δ h(q) for all p≤q inP. Hence h is monotone. Obviously, gh=idPby the definitions ofgandh. Thus,gis indeed a retraction. Moreover,h is a corresponding coretraction with the property that its image is contained inC.

By the facts proved in the preceding two paragraphs, since(D₀,δ|_D₀)is isomorphic to (D,δ_D)under an isomorphism that maps(C₀,δ|_C₀) to(C,δ_C), there exist a retractiong₀from(D0,δ|D₀)ontoPand a corresponding coretractionh₀fromP into(D₀,δ|_D₀)such thath₀(P)⊆C₀.

It is well known that linear identities are preserved under taking retract. Basi- cally, the proof of this fact will work in our case as well to get thatΛinterprets in VP. Here is the proof adapted to our situation. For any operation symbol f inΛwe define a term operationt_f onPby

tf(x₁, . . . ,xk):=g₀(f(h₀(x₁), . . . ,h₀(xk)))

provided f is of arityk. The operationt_f is well-defined, ash₀(P)⊆C₀and by the definition ofD₀, f mapsk-tuples overC₀intoD₀. Moreover,tf is monotone onP, sinceg₀,h₀and f are monotone. Now thet_f satisfy the identities ofΛonP, since the fsatisfy them on the tuples overC₀. ThusΛinterprets inVP, which contradicts

our assumption onΛ.

We conjecture that the statement of the preceding theorem extends to every finite bounded posetPwherePgenerates a join semi-distributive variety.

For the rest of the paper we letPdenote the poset with underlying set{0,a,b,c,d,1}

and covering relation

0≺a,b≺c,d≺1,

see the first poset in Figure 1. It is well known that Padmits a 5-ary but no 4- ary near-unanimity operation, see [17]. LetΛbe a finite basis for the variety VQ

whereQis a two-element chain. Clearly,Λdoes not interpretVP, sinceQadmits a majority operation and Pdoes not. Then by the preceding theoremVP∨VQ is n-permutable for somen. On the other hand, by Hagemann’s above mentioned

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1

0

FIGURE1. PosetsP,Xand the 4-crown

result in [6], both ofVP andVQare notn-permutable for anyn. Thus we have the following corollaries.

Corollary 2.4. The filter of the interpretability types of then-permutable varieties where n runs through the integers greater than 1 is not prime in the lattice of interpretability types of varieties

Corollary 2.5. There is ann such that congruence n-permutability is not join- prime in the lattice of interpretability types of varieties.

3. 5-PERMUTABILITY

Recall thatPdenotes the six-element poset in Figure 1. Letmbe a ternary operation symbol, and letVmbe the variety defined by the setΛof majority identities m(y,x,x) =m(x,y,x) =m(x,x,y) =xform. We saw thatΛ does not interpret in VP. We letV =VP∨Vm. In this section we shall prove thatV is 5-permutable, and hence for anyn≥5 congruencen-permutability is not join-prime in the lattice of interpretability types of varieties.

In [7] Hagemann and Mitschke proved that for a given n,n-permutability of a variety is a strong Mal’cev condition. Moreover, they gave the following characterization ofn-permutability of a variety.

Theorem 3.1 (Hagemann, Mitschke (1973)). LetK be a variety and n≥2 an integer. Then the following are equivalent.

(1) K isn-permutable.

(2) Any edge of a reflexive compatible binary relationρof any algebraA∈K is in a directedn-cycle of the digraph(A,ρ).

LetBbe an algebra in varietyV, andρa binary reflexive compatible relation of B. In this section we are going to prove that every edge fromρis in a directed cycle of length 5 in the digraph(B,ρ). If this is done, the above result of Hagemann and Mitschke yields thatV is 5-permutable. We already saw that neither varietyVPnor the variety generated by an order primal algebra associated with the two element chain are congruencen-permutable for anyn. On the other hand the join of these two varieties is 5-permutable. Thusn-permutability is not join-prime in the lattice of interpretability types of varieties for anyn≥5.

Acolored digraph(R,f)is a digraphRwith a partial map f fromRtoP. The map fis called thecoloring of(R,f), and the elements in the domain offare called the colored elements of (R,f). If f extends to a fully defined edge preserving map from Rto P, then (R,f) and f are called extendible. Most of the time we

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deal with colored quasiordered sets. A finite colored quasiordered set is called an obstruction, if it is non-extendible, but the restriction of its coloring to any quasiordered set properly contained in it is extendible. Containmenthere means subdigraph containment opposed to spanned subdigraph containment.

In the next lemma we characterize the obstructions among the finite colored quasiordered sets. The importance of this lemma is given by the fact that a finite colored quasiordered set is extendible if and only if it does not contain any obstruction. We introduce some types of obstructions before stating the lemma and launching into its proof. The simplest obstructions are thetwisted edges, that is, 2-element chains with a fully defined non-monotone coloring. Let X be the 5- element poset that looks like anX, see the second item in Figure 1. If we color the minimal elements ofX byaandband its maximal elements bycandd, we obtain an obstruction called thetight X.

Lemma 3.2. In the class of finite colored quasiordered sets the obstructions are the twisted edges and the tight X .

Proof. Let(Q,f)be a finite colored quasiordered set. We assume that(Q,f)contains none of the twisted edges and a tight X. We then prove that f is extendible.

Letαbe the largest equivalence contained in the quasiorder ofQ, andQ⁰the quotient posetQ/α. On each block ofα, f must take on at most one value since there are no twisted edges contained in(Q,f). Hence finduces a partial map f⁰from the posetQ⁰toP. Observe that, by the assumption, the colored poset(Q⁰,f⁰)contains none of the twisted edges and a tightX. The statement of the present lemma was verified for the case whenQis a poset in [17], cf. the remark on fences on page 89 and Theorem 3.3. Hence f⁰extends to a fully defined monotone map fromQ⁰toP.

By composing a full extension of f⁰ with the natural map fromQtoQ⁰ we obtain

a required extension of f.

Let D be a subalgebra of Pⁿ and ρ a reflexive compatible binary relation on D. Then, we conceive ρ as a compatible 2n-ary relation ofP, and hence it has a representation of the form(R,S∪S⁰), whereRis a quasiordered set,

S={s₁, . . . ,sn}, S⁰={s⁰₁, . . . ,s⁰_n} are disjointn-element subsets ofRand

ρ={((f(s₁), . . . ,f(sn)),(f(s⁰₁), . . . ,f(s⁰_n)))|f:R→Pis monotone}.

Now, by reflexivity,(R,S)and(R,S⁰)are two representations ofD, and the restrictions ofRto the setsSandS⁰are isomorphic quasiordered sets via the maps_i7→s⁰_i. LetCbe a subalgebra ofD. Then any representation(R,S∪S⁰)ofρ extends to a representation(R,S∪S⁰)ofρ|Cby adding suitable vertices and edges to(R,S∪S⁰).

In the proof of our main result we require the following lemma.

Lemma 3.3. Let(R,S∪S⁰)be a representation of a binary reflexive relationρ on DwhereDis a subalgebra ofPⁿwhere

S={s₁, . . . ,sn}and S⁰={s⁰₁, . . . ,s⁰_n}.

(1) If i,j∈S and i≤ j⁰in R, then i≤ j and i⁰≤ j⁰in R.

(2) Let i,j≤k,l in S and i^∗∈ {i,i⁰}, j^∗∈ {j,j⁰}, k^∗∈ {k,k⁰}, l^∗∈ {l,l⁰}. If there is an r^∗∈R such that i^∗,j^∗≤r^∗ ≤k^∗,l^∗, then there exist r,r⁺∈R such that i,j≤r≤k,l and i⁰,j⁰≤r⁺≤k⁰,l⁰.

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k l k' l'

i j i' j'

r r r

S S'

R

= = = =

j i

k l

* +

* *

FIGURE 2. A figure presenting a particular case in the second statement of Lemma 3.3. It may happen that the vertices r and r⁺are outside ofSandS⁰, respectively.

Proof. Notice that ifi6≤ jinR, then there is a monotone map f:R→Psuch that f(i) =1 and f(j) =0. Hence, for the first part of claim (1), it suffices to prove that for any monotone map f:R→P, f(i)≤ f(j). So let f:R→P be an arbitrary monotone map. Then f|S∈D. Sinceρis reflexive, if we color the vertices of both S andS⁰ by f|_SinR, we obtain an extendible colored quasiordered set. Suppose thatg:R→Pis an extension of the coloring of this colored quasiordered set. Then

f(i) =g(i)≤g(j⁰) =g(j) = f(j). Thusi≤ j. We similarly obtain thati⁰≤ j⁰. For the first part of claim (2), we shall prove that there is anrsuch thati,j≤r≤ k,l. If the sub quasiordered set spanned by the elementsi,j,k,lis not a 4-crown—

see the third item in Figure 1—we are clearly done. So we may assume thati,j,k,l spans a 4-crown inR. Suppose that there is nor such thati,j≤r≤k,l. Notice then if we colori,j,k,lbya,b,c,d, respectively, inR, the so obtained coloring f is monotone on its domain and the resulting colored poset does not contain a tightX.

Thus f extends toR. Then f|_S∈D. Sinceρis reflexive, if we color the vertices of bothSandS⁰accordingly to f|_SinR, we obtain an extendible colored quasiordered set. On the other hand i^∗,j^∗,r^∗,k^∗,l^∗ form a tight X in this colored quasiordered set, a contradiction. The existence of r⁺ such that i⁰,j⁰≤r⁺≤k⁰,l⁰ is obtained

similarly.

A subset {i₀, . . . ,il,i⁰₀, . . . ,i⁰_l} of a representation(R,S∪S⁰) is called anl-step tilted ladderif{i₀, . . . ,i_l} ⊆S,{i⁰₀, . . . ,i⁰_l} ⊆S⁰,i₀≤ · · · ≤i_l,i⁰₀≤ · · · ≤i⁰_land either iv≤i⁰_v+1, 0≤v≤l−1 ori_v+1≥i⁰_v, 0≤v≤l−1. See Figure 3. A tilted ladder {i₀, . . . ,i_l,i⁰₀, . . . ,i⁰_l}withi_v≤i⁰_v+1, 0≤v≤l−1 (iv+1≥i⁰_v, 0≤v≤l−1) isnon- crossedifi⁰₀6≤il(i06≤i⁰_l) inR.

Now we have all the tools at our disposal to state and prove our main theorem.

Theorem 3.4. Let P be the 6-element poset in Figure 1,Pan order primal algebra of finite type related to P, andΣa finite basis for the varietyVP generated byP.

LetBbe an algebra that supports a majority term operation m and term operations satisfyingΣ. Letρ be a reflexive compatible binary relation ofB. Then every edge ofρ is in a directed cycle of length 5 in the digraph(B,ρ). Hence every variety that supports a majority term m and terms satisfyingΣis 5-permutable.

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S' S

R i ₀

i ₁

i ₂ i' ₂

i' ₁ i' ₀

S' S

R i ₀

i ₁

i ₂ i' ₂

i' ₁ i' ₀

FIGURE3. Two-step tilted ladders in representation(R,S∪S⁰)

Proof. The last statement of the theorem immediately follows from the first statement of the theorem by Theorem 3.1 of Hagemann and Mitchke. So we prove the first statement.

Let us assume to the contrary that(Y₀,Z₀)∈ρ and there is no directed path of length 4 fromZ₀toY₀in the digraph(B,ρ). Our aim is to get a contradiction. We mention that ifBwas finite, we could basically describeBby a (finite) representation, and we could give a much simpler argument than what follows. However, in general,Bmay be infinite, not even locally finite, which causes technical difficul- ties in the proof. As we earlier remarked, we may assume without loss of generality thatBis a subuniverse ofP^Hfor some setH.

We define two subalgebras of P^H. LetC₀ equal the subalgebra generated by {Y₀,Z₀}inP^H, and letD₀equal the subalgebra generated by

{m(W₁,W2,W3)|Wi∈C₀, i=1,2,3}

inP^H. We note that

{Y₀,Z₀} ⊆C₀⊆D₀⊆B,

and both C₀ andD₀ are finite. Indeed, D₀ is a finitely generated algebra in the variety generated by a finite algebra, namely byP. SinceD₀⊆P^His finite, there exist finitely many elements inH, saynof them, such that the projection map from D₀to those coordinates is bijective. LetY,Z,D,C,ρ_Dbe the images ofY₀,Z₀,D₀, C0 andρ|_D₀ under such a bijective projection. LetρC=ρD|C. Since the bijective projection fromD₀toDis an isomorphism of algebras, there is no directed path of length 4 fromZtoY in the digraph(D,ρ_D).

Let (T,S∪S⁰) and (R,S∪S⁰) be representations for ρC and ρD, respectively, where

S={s₁, . . . ,s_n}andS⁰={s⁰₁, . . . ,s⁰_n}.

Then the mapi7→i⁰ is an isomorphism between the restrictions ofT toSandS⁰, and also between the restrictions ofRtoSandS⁰. As we remarked earlier, we may assume thatR⊆T.

We define two colored digraphs related toCandD, respectively. First, we define the colored digraphW_CforC. The base digraph ofW_Cis obtained from four copies

(T₀,S₀∪S⁰₀), . . . ,(T₃,S₃∪S⁰₃) of(T,S∪S⁰)by the natural identifications

S₀⁰ =S₁, S⁰₁=S₂, S⁰₂=S₃

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where natural identification is meant to identify the relevant copies ofs⁰ andsfor eachs∈S. The coloring ofW_Cis obtained by coloring the elements ofS₀byZand coloring the elements ofS₃⁰ byY. The colored digraphW_Dis defined similarly by using four copies

(R₀,S₀∪S⁰₀), . . . ,(R₃,S₃∪S⁰₃)

of the representation (R,S∪S⁰) of ρ_D. Observe that neither W_C norW_D is extendible, for otherwise(Y,Z)would lie in a 5-cycle of(C,ρC)or(D,ρD)and hence (Y₀,Z₀)would lie in a 5-cycle of(B,ρ).

Let ˆW_Dbe the colored quasiordered set obtained fromW_Dby taking the transitive closure of the relation ofW_Dwhile preserving the coloring ofW_D. Then ˆW_Dis a non-extendible colored quasiordered set. So it contains a twisted edge or a tightX.

Now, our aim is to prove the following.

Claim: There is a non-crossed two-step tilted ladder that is contained in both of the representations(T,S∪S⁰)and(R,S∪S⁰).

S0 S = S'₁ ₀ S = S'2 1 S = S'₃ ₂ S'₃

i i i

1

2 3

R₀ R₂

R₁ R₃

S' S

R i ₀

i ₁ i ₂ i ₃

i ₄ i' ₄

i' ₃ i' ₂ i' ₁ i' ₀

colored byZ Y

i ₀

i₄

colored by

FIGURE 4. A twisted path inW_Dcoming from a twisted edge in Wˆ_Dand the corresponding four-step tilted ladder inR. The vertices of the path u₀→u₁→u₂→u₃→u₄ in the upper picture are labeled by the corresponding elements inSin order to better see the connection with the lower picture. The black nodes are colored vertices. The one labeled byi₀is colored byqand the one labeled byi₄is colored bypwhereq6≤p.

First, let us look at the case, when ˆW_D contains a twisted edge. ThenW_D must contain a directed path of length 4, say

u₀→u₁→u₂→u₃→u₄

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whereuj∈Sjfor 0≤j≤3,u₄∈S₃⁰,u₀is colored byqandu₄is colored bypsuch thatq6≤p.We call this path atwisted path. Suppose thatuj is a copy ofij ∈Sif 0≤ j≤3 andu₄is a copy ofi⁰₄∈S⁰. We depicted the situation in Figure 4.

Clearly, the edges in the twisted path yield thatij≤i⁰_j+1, 0≤ j≤3 inR. Hence, by item (1) of Lemma 3.3, we also have thati₀≤ · · · ≤i₄inSandi⁰₀≤ · · · ≤i⁰₄in S⁰. So the elementsi_jandi⁰_jwhere 0≤ j≤4 constitute a four-step tilted ladder in R. SinceR⊆T, the same four-step tilted ladder is contained inT as well.

S S'

T i ₀

i ₁ i ₂ i ₃

i ₄ i' ₄

i' ₃ i' ₂ i' ₁ i' ₀

Y

colored by colored byZ

FIGURE5. The four-step tilted ladder in(T,Y∪Z)with the cross- edge(i⁰₀,i4)wherei₄is colored bypandi⁰₀is colored byq.

We shall prove that this ladder is non-crossed, that is,i⁰₀6≤i₄inT. Suppose to the contrary thati⁰₀≤i₄inT. Since(Y,Z)∈ρC, the colored quasiordered set(T,Y∪Z) whereY andZare considered as partial maps with domainsSandS⁰, respectively, is extendible, see Figure 5. It follows thatZ(i⁰₀)≤Y(i₄)which contradicts the fact that

Z(i⁰₀) =q6≤p=Y(i₄).

Thus(T,S∪S⁰)contains a non-crossed four-step tilted ladder. Observe then that i₀,i₁,i₂,i⁰₀,i⁰₁andi⁰₂form a non-crossed two-step tilted ladder in(T,S∪S⁰). More- over, this is a two-step tilted ladder also contained and non-crossed in(R,S∪S⁰).

In the case when ˆW_Dcontains a tight X, our aim is to prove the same conclu- sion, namely, that there is a non-crossed two-step tilted ladder contained in both of (T,S∪S⁰)and(R,S∪S⁰). The 4 colored elements of the tight X fall in the union ofS₀andS⁰₃. So either one of S₀andS₃⁰ contains 1 colored element and the other contains 3 of them, or both ofS₀andS⁰₃contain 2 colored elements. In either case, the midpoint of the tightX is inRt for some 0≤t≤3 and is denoted byrt. Then inW_D,r_tis connected to the colored elements of the tightXvia four directed paths whose vertices—possibly apart fromrt—are in the union of theSv, 0≤v≤3, and S⁰₃, and each of these paths has at most one vertex in each of theS_v, 0≤v≤3, and S⁰₃. We call the subdigraph formed by these four directed paths andrt atight X of paths inW_D. An example of a tightX of paths is depicted in the first picture of Figure 6.

We call the four directed paths thebranchesof the tightXof paths. We make this definition more precise: ifr_tis inS_vfor some 0≤v≤3 we countr_t to the branches, otherwise the branches are meant withoutrt. Clearly, the branches with endpoints in S₀ have the same length. The same is true for the branches with endpoints in

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S⁰₃. We say that a branch is long if it has length at least 2. So either all of the branches with endpoints inS₀ are long or all of the branches with endpoints inS⁰₃ are long. Without loss of generality we assume that the branches with endpoints in S⁰₃are long. By the use of item (1) of Lemma 3.3, similarly as we saw it for twisted paths, each of these long branches induces a tilted ladder of at least two steps inR.

However, some of the induced tilted ladders might not be non-crossed.

Suppose thatr_t ∈W_Dis a copy ofr^∗∈Rand the endpoints of the branches inR_t are the copies ofi^∗,j^∗,k^∗andl^∗inR, respectively, where

i^∗∈ {i,i⁰}, j^∗∈ {j,j⁰}, k^∗∈ {k,k⁰}, l^∗∈ {l,l⁰}for somei,j,k,l∈S.

Clearly, two of i^∗,j^∗,k^∗ andl^∗ are less than or equal to r^∗ and two of them are greater than or equal tor^∗. We may assume thati^∗, j^∗≤r^∗≤k^∗, l^∗. By applying item (2) of Lemma 3.3, there exists anr⁺∈Rsuch thati⁰, j⁰≤r⁺≤k⁰, l⁰.

Letu∈W_Dbe one of the colored elements inS₀. Souis an endpoint of a branch of the tightX of paths. Suppose thatuis a copy ofh⁰∈S⁰ and the other endpoint of the branch inRt is a copy of one of the elementsi⁰, j⁰, k⁰, l⁰, sayl⁰, inS⁰. By item (1) of Lemma 3.3, the edges of this branch yield a chain inS⁰— theS⁰ part of the tilted ladder related to this branch— with maximal elementh⁰ and minimal elementl⁰. Byr⁺≤l⁰, we obtainr⁺≤h⁰. Thus, all of the colored vertices of the tightX of paths inS₀are copies of elements ofS⁰that are comparable withr⁺inR.

l

r

S0 S = S'₁ ₀ S = S'2 1 S = S'3 2 S'₃

R₀ R₂

R₁ R₃

j

l k

i

Y

colored byZ

S S'

R

l'

i

k'

i'

r r

k i

l j

k'

i' j'

k

k'₂

1 0 1

0

k2

0 0

l₁ l'₁

0 0

1

i₂ i'¹₂

j'0₁ 0

j1

colored by 0

0 1

j₁ ⁰

0

k2

k₁

i¹ i₂

*

+

*

FIGURE6. A tightXof paths inW_Dcoming from a tightX in ˆW_D and the corresponding edge structure ofR. The vertices of the tight X of paths in the upper picture are labeled by the corresponding elements inSandr^∗∈Rin order to better see the connection with the lower picture. The black nodes are colored vertices. The ones labeled byi₂, j₁,k₂,l₁are colored bya,b, c,d, respectively.

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Before continuing the main line of our proof, we supply an example in Figure 6 to illustrate the notions introduced in the preceding few paragraphs.The tightX of paths in the first picture of Figure 6 has two long branches of length 2: the paths that are labeled by theiand thek, respectively. The paths that are labeled by the j and thel, respectively, are short branches of length 1. The tightXof paths induces some edges inRas shown in the lower picture of the figure. The edges betweenS andS⁰and the edges with endpointr^∗are directly obtained from edges of the tight X of paths. The vertical edges come from the so obtained edges which connect S andS⁰ by the use of item (1) of Lemma 3.3. The vertexr⁺ and the edges with endpoint r⁺ are obtained from the edges with endpoint labeled byr^∗ by item (2) of Lemma 3.3. Observe that the four branches of the tightX of paths induce four tilted ladders in the lower picture: the ones given by thei, the j, thek and thel, respectively. ByR⊆T,(T,S∪S⁰)also contains the edges induced by the tightX of paths inR.

S S'

T

l'

i

k'

i'

r r

k i

l j

k'

i' j'

k k'₂

1 0 1

0

k2

0 0

l₁ l'₁

0 0

i₂1 i'¹₂

j'0₁ 0

j1

Y colored byZ

colored by

*

+

FIGURE7. The shape of(T,Y∪Z)with tilted ladders and cross- edges (i₂,i⁰₀) and(k⁰₀,k₂) indicated ifW_D contains the tight X of paths in Figure 6. The verticesi₂,j⁰₁,k₂,l⁰₁are colored bya,b,c,d, respectively.

It turns out that in the example, one of tilted ladders labeled by theior thek is non-crossed. Suppose to the contrary thati₂≤i⁰₀andk⁰₀≤k₂, see Figure 7. Hence

i₂, j⁰₁≤r⁺≤k₂, l₁⁰.

Moreover, because of (Y,Z)∈ρC, the colored quasiordered set (T,Y∪Z) is extendible. This is impossible, sincei₂, j⁰₁,k₂, l₁⁰ are colored bya, b, c,d, respectively, in(T,Y∪Z)and soi₂, j⁰₁,r⁺,k₂,l₁⁰ would constitute a tightXin(T,Y∪Z).

We return to the proof of the general case. ByR⊆T, the edges induced by the tightXof paths inRare also contained inT. We prove that at least one of the tilted ladders corresponding to a long branch ending inS⁰₃ is non-crossed. Suppose that all of them were crossed.

Letu∈W_Dbe one of the colored endpoints of a branch inS⁰₃. Suppose thatuis a copy ofh∈Sand the other endpoint of the branch is a copy of one of the elements i⁰, j⁰,k⁰,l⁰, sayk⁰, inS⁰. Since the tilted ladder corresponding to this long branch is crossed,k⁰≤hinT. Byr⁺≤k⁰, we haver⁺≤h. Thus, all of the colored vertices of the tight X of paths inS⁰₃are copies of elements inSthat are comparable with r⁺inT.

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Now we can finish off the proof like in the example. Since (Y,Z)∈ρ_C, the colored quasiordered set(T,Y∪Z)is extendible. The four colored elements of the tight X of paths are copies of four elements colored by a, b, c, d, respectively, (T,Y∪Z). We already verified above that these four colored elements of(T,Y∪Z) are comparable with r⁺. Actually, the proof of this fact also gives that these five elements constitute anX. ThisX clearly is a tightXin(T,Y∪Z), a contradiction.

Hence there is a non-crossed tilted ladder in T which corresponds to a long branch, so this tilted ladder must have at least two-steps. This implies, just as we saw it in the case of a twisted edge, that there is a non-crossed two-step tilted ladder in(T,S∪S⁰)and hence in(R,S∪S⁰). This concludes the proof of the claim.

Thus, the non-extendibility of W_D yields that there is a non-crossed two-step tilted ladder contained in(R,S∪S⁰)and(T,S∪S⁰). We shall define four elements Qa,Qb,Qc and Q_d inC such that Qa,Qb

ρC

−→Qc,Qd. Let us take a non-crossed two-step tilted ladder in(T,S∪S⁰)that is determined by the verticesi≤ j≤kofS.

First, we defineQaas a map fromStoP:

Q_a(t):=







a, ifi≤t≤k, 1, ifi≤t6≤k, 0, otherwise.

The mapQbfromStoPis defined similarly toQaby usingbinstead ofa. We repeat the preceding definition of Q_a andQ_b for Q_c andQ_d, except that we use c,d,i⁰,j⁰,k⁰ andS⁰ instead ofa,b,i,j,kandS, respectively. In this way we obtain two maps Q_c andQ_d from S⁰ toP. The so defined maps Q_a,Qb,Qc and Q_d are clearly monotone partial maps fromT toP. Let us color the elements ofSandS⁰ byQaandQc, respectively, inT.

Obviously, the colored quasiordered set(T,Qa∪Q_c)contains no tightX. Since i≤ j≤kdetermine a non-crossed tilted ladder contained in(T,S∪S⁰), there is no edge inT from the interval[i⁰,k⁰]to the interval[i,k]. Hence(T,Q_a∪Q_c)contains no twisted edges as well. Thus, the colored quasiordered set (T,Qa∪Qc)is extendible. SoQ_a−→^ρ^C Q_c, and henceQ_a andQ_care inC. We obtainQ_a−→^ρ^C Q_d and Q_b−→^ρ^C Qc,Q_d similarly, henceQ_b,Qd∈C. ThusQa,Qb

ρD

−→Qc,Qd.

For the preimages Q⁰_a,Q⁰_b,Q⁰_c,Q⁰_d of Q_a,Qb,Qc,Qd in C₀ we have Q⁰_a,Q⁰_b −→^ρ Q⁰_c,Q⁰_d. Moreover the majority operation m is compatible with ρ on B hence Q⁰_a,Q⁰_b−→^ρ m(Q⁰_a,Q⁰_b,Q⁰_c)−→^ρ Q⁰_c,Q⁰_dwherem(Q⁰_a,Q⁰_b,Q⁰_c)is inD₀. Therefore, there is an elementM∈Dsuch thatQa,Qb

ρD

−→M−→^ρ^D Qc,Qd. From this fact we derive a contradiction.

We define a colored digraph (U,f). The digraphU is formed by four copies (Rv,Sv∪S⁰_v), 0≤v≤3, of (R,S∪S⁰) with the natural identifications S⁰₀=S₁⁰ = S₂=S₃.The partial map f is defined by coloring the elements inS₀, S₁, S₂⁰ and S⁰₃by Qa,Qb,Qc andQ_d, respectively. The existence ofM guarantees that(U,f) is extendible. On the other hand, the two-step tilted ladderi≤ j≤k, i⁰≤ j⁰≤k⁰ in (R,S∪S⁰) from the definition of Qa has a copy iv ≤ jv ≤kv, i⁰_v ≤ j⁰_v≤k⁰_v in (Rv,Sv∪S⁰_v)for each 0≤v≤3. We sketched the situation in Figure 8. Notice then thati₀,i₁≤ j₀⁰ ≤k⁰₂,k₃⁰ form a non-extendible colored digraph contained in(U,f) wherei₀,i1,k⁰₂,k₃⁰ are colored bya,b,c,drespectively, a contradiction.

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i₀ j₀

k₀ k₁

j₁ i₁

3

j'₃ k'₃ k'₂

j'₂

i'₂

S'₂ S'₃

S₀ S₁

S'₀

S'=₀ S'₁=S₂=S₃ j'0

i'₀ k'₀

i'

FIGURE8. A part of digraphU with two-step tilted ladders. The vertices i₀,i₁,j⁰₀,k₂⁰,k⁰₃ form a non-extendible colored digraph in (U,f).

As we explained at the beginning of this section, Theorem 3.4 yields the following consequence.

Corollary 3.5. For anyn≥5, n-permutability is not join-prime in the lattice of interpretability types of varieties.

We do not know if the corollary holds whenn=3 orn=4. Next we are going to prove that 4-permutability cannot be achieved in Theorem 3.4. In order to do this we need to define the notion of aG-obstruction for a digraphG. AG-colored digraph is a pair(H,f) where H is a digraph and f is a partial map fromH to G. AG-colored digraph isextendibleif there is an edge-preserving total map from H toG that extends f. A G-colored digraph (H,f) is aG-obstructionif (H,f) is non-extendible but any(H⁰,f⁰)properly contained in(H,f)is extendible. It is immediate from the definition that if Gis reflexive, then the base digraph of any G-obstruction is a connected irreflexive digraph. We make use of the following fact, cf. Theorem 3.8 in [13], in the proof of the next proposition: a digraph G admits a majority operation if and only if the number of colored elements in any G-obstruction is at most 2.

Proposition 3.6. Let P be the 6-element poset in Figure 1,Pan order primal algebra of finite type related to P, andΣa finite basis for the varietyVPgenerated byP.

LetB≤P⁷be the subalgebra defined by the representation(Q,S)in Figure 9 andρ the reflexive binary relation on B defined by the representation(R,S∪S⁰)in Figure 9. Then the digraph(B,ρ)admits a majority operation m and operations satisfy- ingΣ. Moreover,((a,b,c,c,c,1,1),(a,b,d,1,1,d,d))is an edge of(B,ρ), but no directed cycle of length 4 contains this edge in(B,ρ). Hence there exists a variety that is not 4-permutable and supports a majority term m and terms satisfyingΣ.

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1 2 4

5 7

6

1 4 5

6

7 7'

6' 5'

4'

2 1' 2'

3 3 3'

FIGURE 9. A representation (Q,S) of B ≤ P⁷ and a representation (R,S∪S⁰) of the binary relation ρ on B where S = {1,2,3,4,5,6,7}andS⁰={1⁰,2⁰,3⁰,4⁰,5⁰,6⁰,7⁰}.

Proof. Let ˜B denote the digraph (B,ρ). It should be clear that ρ is a reflexive relation onBand that the digraph ˜Badmits operations satisfyingΣ. Let

fc= (a,b,c,c,c,1,1)and fd= (a,b,d,1,1,d,d).

It is obvious now that(fc,fd)∈ρ.In order to prove that there is no directed path of length 3 from f_d to f_c in ˜Bwe define a colored digraphU_B. We put together U_Bfrom three separate copies(Rj,Sj∪S⁰_j), 0≤ j≤2, of(R,S∪S⁰) with the natural identifications S⁰₀=S₁,S⁰₁=S₂ and with coloring the elements ofS₀ by the components of f_d and the elements of S⁰₂ by the components of f_c in an orderly manner. Let ˆU_B be the colored quasiordered set obtained fromU_B by taking the transitive closure of the relation ofU_B.Observe now that there is a directed path of length 3 from f_d to f_c in (B,ρ) if and only ifU_B is extendible if and only if Uˆ_Bis extendible. Nevertheless, the latter condition does not hold since—with the notationSj={1_j, . . . ,7j}andS⁰_j={1⁰_j, . . . ,7⁰_j}for 0≤j≤2—the colored vertices 1₀,2₀,5₀,7⁰₂ and the midpoint (the vertex distinct from the elements ofS₁andS⁰₁) ofR₁form a tightX in ˆU_B.

The main part of the proof is to verify that the digraph ˜B admits a majority operation. As we mentioned preceding the present proposition, it suffices to prove that every ˜B-obstruction has at most two colored elements. Let us suppose to the contrary that(H,f)is a ˜B-obstruction with at least three colored elements. SoH is a digraph, and f is a partial map from H to B⊆P⁷. We define a P-colored digraphW as follows. For every edgee inH we take a copy(Re,Se∪S⁰_e) of the representation(R,S∪S⁰)in Figure 9. If two edgeseandghave a common vertex, we make one of the natural identifications S_e=S_g, S_e=S⁰_g, S⁰_e=S_gandS⁰_e=S⁰_g accordingly to the type of the incidence of the edges e and g (for example, we makeS_e=S_gif the edgeseandghave a common tail-vertex). Finally, to complete the definition of W, for any colored vertex h of(H,f) we color the subdigraph (Re,Se∪S⁰_e)ofWby f(h)at the elements ofSeifhis the tail-vertex ofeand at the elements ofS⁰_eifhis the head-vertex ofe. This is obviously a consistent definition.

Let ˆW be the colored quasiordered set obtained fromW by taking the transitive closure of the relation of W. We gather some facts about ˆW before continuing the main line of our proof. Let α denote the largest equivalence contained in the quasiorder relation of ˆW. Let ˆW/α be the quotient poset ofW byα. Notice that as H is connected, ˆW/α has two minimal elements: one of them is theα-block that contains the elements 1eand 1⁰_efor alle, and the other is theα-block that contains the elements 2eand 2⁰_efor alle. The non-minimalα-blocks are of one element. The quotient poset ˆW/α has height 3. Each of the minimalα-blocks contains colored