Triangular Scheme Revisited in the Light of n-permutable Categories

The first diagrammatic scheme was developed by H.P. Gumm under the name Shifting Lemma in case to characterize congruence modularity. A diagrammatic scheme is developed for the generalized semi distributive law in Mal’tsev categories. In this paper we study this diagrammatic scheme in the context of n-permutable, and of Mal’tsev categories in particular. Several remarks concerning the Triangular scheme case are included.


Introduction
We are going to establish a diagrammatic scheme for equivalence distributivity in equivalence n-permutable category. In this section we recall some basic definitions and results from the literature, needed throughout the article.

n-permutable varieties
A variety of universal algebras is called n-permutable, n ≥ 2, when any pair of congruences R and S on a same algebra n-permutes: (R, S) n = (S, R) n . Where (R, S) n denotes the composite RSRS · · · of R and S, n times. This notion determines a sequence of families of varieties, whose first two instances are well known: for n ≥ 2, we regain 2-permutable varieties [15], better known as Mal'tsev varieties; for n ≥ 3, Equivalently, L is distributive if and only if it satisfies the Horn sentence Hence, (s) is a Horn sentence and a lattice L satisfies (d) if and only if it satisfies (s). Thus (s) is another characterization of distributivity. A variety V of universal algebras is called congruence distributive when the lattice Cong(A) of congruences on any algebra The ∧-semidistributive law above is often denoted by SD ∧ . More general (in fact, weaker) Horn sentences have been investigated in Geyer [12]. For n ≥ 2 put n = {0, 1, · · · , n − 1} and let l 2 (n) denote the set {I : I ⊆ n and |I| ≥ 2}. For ∅ = K ⊆ l 2 (n) we define the generalized meet semidistributive law SD ∧ (n, K) for lattices as follows: for all x, y 0 , · · · , y n−1 As a particular case, when K = {J : J ⊆ n and |J| = 2} is denoted by SD ∧ (n, 2). Notice that SD ∧ (n, 2) is the following lattice Horn sentence: In particular, SD ∧ (2, 2) is the ∧-semidistributivity law defined in by: (SD ∧ ) 1.2 n-schemes and weak n-scheme Recall from [16] that a sublattice L of an equivalence lattice EqA satisfies the Triangular scheme if for each R, S, T ∈ L with R ∧ S ⊆ T and for x, y, z ∈ A such that x, y ∈ T, x, z ∈ S, z, y ∈ R we have z, y ∈ T .
This can be visualized as follows: The following assertions are proved in [17]: (a) if ConA is distributive, then it satisfies the Triangular scheme; (b) if A is a congruence permutable algebra, then ConA is distributive if and only if ConA satisfies the Triangular scheme.
We are going to establish a diagrammatic scheme for equivalence distributivity in equivalence n-permutable category. The scheme is similar to that of Gumm [26] for congruence modularity.
A sublattice L of EqA satisfies the n-scheme (the weak n-scheme) if for each R, S, T ∈ L with R ∧ S ⊆ T (or R ∧ S = R ∧ T , respectively) and for x, y, z 1 , · · · , z n ∈ A such that x, y ∈ R, x, z 1 ∈ S, z 1 , z 2 ∈ T, z 2 , z 3 ∈ S, · · · , z n−1 , y ∈ S for n odd and z n−1 , y ∈ T for n even we have x, y ∈ T . These schemes can be also visualized but, contrary to the previous cases, classes of the same congruence fail to be parallel:

Regular categories and relations
Our categories will always be regular, in the sense of Barr [1]; we recall that a category is regular if it has finite limits, each arrow factors as a regular epi followed by a mono, and regular epis are pull-back stable. (It turns out that in a regular category the kernel pair of an arrow always has a coequalizer, given by the regular epi part of the factorization of the arrow). In a regular category, it is possible to compose relations. If (R, r 1 , r 2 ) is a relation from X to Y and (S, s 1 , s 2 ) a relation from Y to Z, their composite SR is a relation from X to Z obtained as the regular image of the arrow where (R × Y S, π 1 , π 2 ) is the pullback of r 2 along s 1 . The composition of relations is then associative, thanks to the fact that regular epimorphisms are assumed to be pullback stable.

Regular Mal'tsev Categories and Triangular Scheme
A finitely complete category E is called a Mal'tsev category if every reflexive relation in E is an equivalence relation. These categories are also characterized by other properties on relations, as follows: Let E be a regular category. Then the following statements are equivalent: (i) E is a Mal'tsev category; (ii) ∀F, E ∈ Equiv(X), E ∨ F = F E(= EF ) ∈ Equiv(X), for any object X; Theorem 2.2. Let E be a regular Mal'tsev category. Then the following conditions are equivalent: (1) the Triangular scheme holds in E; (2) ∀E, F, P ∈ Equiv(X) we have Proof. To prove the implication (1) ⇒ (2) suppose E, F, P ∈ Equiv(X) are arbitrary, and (c, a) ∈ F ∧ (EP ). Then there exists an element b such that aP bEc. Apply the triangular scheme for these elements with P replaced by P = P ∨ (E ∧ F ). We get that (2) is true, then reversing the roles of E and P we get that By taking the converse of this inclusion and combining with (2) we obtain condition (3).
(2) ⇒ (1) Let R, S and T be equivalence relations on an object X such that R ∧ S ≤ T we show that R ∧ ST ≤ T .
We have (1) Equiv(X) satisfies SD ∧ (n, 2) for X in E; (2) the scheme depicted in Figure 1 holds for S, R 0 , · · · , R n−1 in Equiv(X) and x 0 , · · · , x k , y, z, are related as in Figure 1 where k = n(n−1) 2 − 1 and T stands for Figure 1 Proof. Suppose SD ∧ (n, 2) holds. Using the premise of SD ∧ (n, 2) we obtain whence Equiv(X) satisfies the Horn sentence This implies the scheme, for the situation on the left hand side in Figure 1 then gives To show the converse, suppose that the scheme given by Figure 1 holds, S, R 0 , · · · , R n−1 in Equiv(X) with S ∧ R 0 = S ∧ R 1 = · · · = S∧R n−1 , and suppose that (y, z) ∈ S∧ 0≤i<j<n (R i •R j ), there exist x 0 , x 1 , · · · , x k of X such that for each j(1 ≤ j ≤ k) there exist u, v such that (z, x j ) ∈ R u and (x j , y) ∈ R v (according to the left hand side of Figure 1). Then the scheme applies and we conclude (y, z) ∈ T . Since T ⊆ R 0 , (y, z) ∈ R 0 . Hence (y, z) ∈ S ∧ R 0 . This proves the " ≤ " part of SD ∧ (n, 2). The reverse part is simpler and does not need the scheme: proving the theorem.
Note that, for diagram such as Figure 1 where R i , S or T are not symmetric, the relations are always to be considered from left to right and from top to bottom. To avoid ambiguity with the interpretation of such diagrams, from now on we will write a E −→ b to mean that (a, b) ∈ E whenever E is a non-symmetric relation.
Theorem 2.4. Let E be a regular category. Then the following conditions are equivalent: (i) E is an equivalence SD ∧ (n, 2) Mal'tsev category; (ii) the scheme depicted in Figure 1 holds when S, R 0 , · · · , R n−1 and T are reflexive relations.
Proof. (i) ⇒ (ii) Since E is a Mal'tsev category, reflexive relations are necessarily equivalence relations. Since E is also equivalence SD ∧ (n, 2), by Proposition 2.3, the the scheme depicted in Figure 1 holds for any reflexive relations in E.
(ii) ⇒ (i) To prove that E is a Mal'tsev category, we show that any reflexive relation e 1 , e 2 : E X × X in E is also symmetric (Theorem 2.1 (iii)). Suppose that (x, y) ∈ E, and consider the reflexive relations T and S on E defined as follows: (aEb, cEd) ∈ S ⇔ (a, d) ∈ E, and (aEb, cEd) ∈ T ⇔ (c, b) ∈ E.
yEy (xEx and yEy by the reflexivity of the relation E). We conclude that (xEx, yEy) ∈ T and, consequently, that (y, x) ∈ E, so that E is a Mal'tsev category. Since the Figure   1 holds in E, by Proposition 2.3 the category E is equivalence SD ∧ (n, 2).
3 n-permutable category Definition 3.1. [23] A regular category E is an a n permutable category when the composition of (effective) equivalence relations on a given object is n-permutable: for two (effective) equivalence relations R and S on the same object, we have (R, S) n = (S, R) n . Where the composition of n alternating factors R and S is denoted by (R, S) n = RSRS · · · Theorem 3.2. ( [23], Theorem 3.5 of [5]) Let n ≥ 2 and let E be a regular category.
Then the following statements are equivalent: (i) E is n-permutable category; (ii) (P, P • ) n+1 ≤ (P, P • ) n−1 for any relation P ; (iii) (R, S) n is an equivalence relation and is therefore R ∨ S; (iv) (1 X ∧ T )T • (1 X ∧ T ) ≤ T n−1 , for any relation T on an objet X; (v) for any reflexive endorelation E X × X in E, the relation (E, E • ) n−1 is an equivalence relation; (vi) for any such reflexive endorelation E, the relation (E, E • ) n−1 is transitive; (vii) for any such reflexive endorelation E we have (E, E • ) n−1 = (E • , E) n−1 ; (viii) E • ≤ E n−1 for any reflexive relation E. Proof. For (i), suppose R, S, T in Equiv(X) with R ∧ S = R ∧ T and x, y, z 1 , · · · , z n−1 ∈ X such that the assumptions of the weak scheme are satisfied. Then x, y ∈ R ∧ (S, T ) n = R ∧ (S ∨ T ) = R ∧ T due to ∧-semidistributivity, i.e. it satisfies the weak n-scheme for each n ≥ 2.
Conversely, let Equiv(X) satisfy the weak n-scheme, let R, S, T in Equiv(X) with R ∧ S = R ∧ T . Let x, y ∈ R ∧ (S ∨ T ). Due to equivalence n-permutability, we have x, y ∈ R ∧ (S, T ) n with n factors. Thus it is almost evident that the assumptions of the weak n-scheme are satisfied. Applying this scheme, we conclude x, y ∈ R ∧ T . We Let n be an odd number (n ≥ 3). A relation P X × X on X is called positive when it is of the form P = (E • , E) n−1 , for some relation E X × X. In set-theoretic terms, P is positive when there exists a relation E and x 1 , x 2 , · · · , x n−1 such that (x, x ) ∈ P if (x, x 1 ) ∈ E, (x 2 , x 1 ) ∈ E · · · (x , x n−2 ) ∈ E. Proof. Suppose that E is an n-permutable category and consider a reflexive and positive relation P with 1 ≤ P = (E • , E) n−1 . Then P is symmetric. Since And As for the transitivity of P , we have (E • , E) n−1 (E • , E) n−1 = (E • , E) 2n−2 = (E • , E) n−1 .
By Theorem 3.2 (i) ⇒ (ii). Conversely, let U be a reflexive relation on X. Then P = (U • , U ) n−1 is a reflexive and positive relation, thus an equivalence relation by assumption. It follows that E is an n-permutable category by Theorem 3.2 (v).