The triangular principle is equivalent to the triangularschemeKeith A. Kearnes and Emil W. KissAbstract. The triangular scheme is a diagrammatic characterization of congruence join-semidistributivity. The triangular principle is a variant of this condition, where one of thecongruences is replaced by a tolerance. This paper contains two proofs showing that thetriangular principle and the triangular scheme are equivalent for varieties. The first one is aroutine argument using tame congruence theory, and works only for locally finite varieties.The second proof is an elementary, but nontrivial calculation with terms, which worksfor arbitrary varieties. This yields a stronger Mal’tsev-condition characterizing congruencejoin-semidistributivity than the one obtained from the triangular scheme.1. The history of the triangular schemeA celebrated result of H. P. Gumm is the following characterization of congruencemodularity. An algebra A satisfies the shifting scheme (sometimes called the shift-ing lemma), if for every α, β, γ ∈ Con(A) such that α ∩ β ⊆ γ, and any elementsa, b, c, d ∈ A, the implicationabcdβ βααγ =⇒ γholds. Writing a β b to mean (a, b) ∈ β, this diagram asserts that, if a β b,c β d, a α c, b α d and a γ b, then also c γ d. If we replace α by an arbitrarytolerance (reflexive, symmetric, compatible binary relation), then we get the shiftingprinciple. Gumm proved in [8] that a variety is congruence modular if and only ifevery algebra in the variety satisfies the shifting scheme, or equivalently, if everyalgebra in the variety satisfies the shifting principle. This proved to be a useful toolfor building up the theory of the modular commutator.Is it possible to characterize congruence distributivity in a similar way? A naturalcandidate for such a diagram was proposed by I. Chajda and E. Horv´ath in [2].1991 Mathematics Subject Classification: 08B05, 08B10.Key words and phrases: Mal’tsev-condition, tolerance, tame congruence theory.This work was supported by the Hungarian National Foundation for Scientific Research(OTKA), grant no. T043671 and T043034.12 KEITH A. KEARNES AND EMIL W. KISSDefinition 1.1. An algebra A satisfies the triangular scheme if for every choiceof congruences α, β, γ ∈ Con(A) such that α ∩ β ⊆ γ, and every choice of elementsa, b, c ∈ A, the implicationbacβαγ=⇒γFigure 1. The triangular schemeholds. That is, b α c, a β c and a γ b imply that a γ c. The triangular principle isthe same statement with the congruence α replaced by an arbitrary tolerance.In a congruence distributive variety, the triangular scheme clearly holds. Indeed,if the congruence lattice of A is distributive, then, as noticed in [2], we have(a, c) ∈ β ∧(α ∨ γ) = (β ∧ α) ∨(β ∧ γ) ⊆ γ .In [2] it is also shown, using J´onsson-terms, that the triangular principle is satisfiedin every congruence-distributive variety. In the papers [1, 2, 3, 4, 7] various sufficientconditions are provided, which ensure that a variety satisfying the triangular schemeis congruence distributive. Such sufficient conditions are congruence permutabil-ity, the shifting scheme (hence congruence modularity), or the so-called trapezoidlemma (which characterizes congruence distributivity by the results in [4]). How-ever, the relationship between the triangular scheme, the triangular principle andcongruence distributivity has remained open even for locally finite varieties.The real meaning of the triangular scheme for locally finite varieties is revealedby a result of David Hobby and Ralph McKenzie. Theorem 9.11 of [9] states, amongother things, the following.Theorem 1.2. For any locally finite variety V, the following are equivalent (thenumbering of the statements below follows that of [9]).(1) V omits types 1, 2, 5 of tame congruence theory.(3) If A ∈ V and α, β, γ ∈ Con(A), thenβ ∩(α ◦ γ) ⊆¡α ∨(β ∧ γ)¢∩¡γ ∨(α ∧ β)¢.(4) There exists an integer n and ternary terms d0, . . . , d2nsuch that V satisfiesthe following identities for 1 ≤ i ≤ n:d0(x, y, z) = xd2i−2(x, x, y) = d2i−1(x, x, y) ,d2i−1(x, y, x) = d2i(x, y, x) , d2i−1(x, y, y) = d2i(x, y, y) ,d2n(x, y, z) = z .(6) The congruence lattice of every finite algebra in V is join-semidistributive.THE TRIANGULAR PRINCIPLE IS EQUIVALENT TO THE TRIANGULAR SCHEME 3Condition (3) of this theorem clearly implies that the triangular scheme holdsin A. In fact it is easy to see the following statement for any (not necessarily locallyfinite) variety.Proposition 1.3. For any variety V, the following are equivalent.(1) Every algebra in V satisfies the triangular scheme.(2) If A ∈ V and α, β, γ ∈ Con(A), then we haveβ ∩(α ◦ γ) ⊆ γ ∨(α ∧ β) .(3) V satisfies the congruence-inclusion in condition (3) of Theorem 1.2.(4) V satisfies the Mal’tsev-condition in (4) of Theorem 1.2.Proof. To prove the implication (1) =⇒ (2) suppose that α, β, γ ∈ Con(A) arearbitrary, and (c, a) ∈ β ∩(α ◦ γ). Then there exists an element b ∈ A such thata γ b α c. Apply the triangular scheme for these elements with γ replaced byγ0= γ ∨(α ∧ β). We get thatbacβαγ=⇒γ0= γ ∨(α ∧ β)holds, proving (2).Now if (2) is true, then reversing the roles of α and γ we get thatβ ∩(γ ◦ α) ⊆ α ∨ (γ ∧ β) .By taking the converse of this inclusion and combining with (2) we obtain condi-tion (3) of Theorem 1.2. Therefore the implication (2) =⇒ (3) is proved.To show that (3) =⇒ (4) consider the free algebra in V generated by x, y, z,and define the congruencesα = Cg(y, z) , β = Cg(x, z) , γ = Cg(x, y) .Then (the converse of) (3) implies that(x, z) ∈ γ ∨(α ∧ β) .Using the standard technique for producing Mal’tsev-conditions it is trivial to seethat we get the terms diof (4) above.Finally to prove (4) =⇒ (1) suppose that an algebra A has terms disatis-fying (4). To prove that A satisfies the triangular scheme consider the elementsci= di(a, b, c). The identities in (4) imply that cjγ cj+1when j is even, andcj(α ∧ β) cj+1when j is odd. Since c0= a, c2n= c and α ∧ β ⊆ γ, we get that(a, c) ∈ γ by transitivity. ¤This observation shows immediately that the triangular scheme does not implycongruence distributivity, not even for locally finite varieties. Indeed, there arelocally finite varieties satisfying the conditions of Theorem 1.2 that are not congru-ence distributive, for example, Polin’s