Geometrical Methods in Goursat Categories

The main aim of the paper is to show that the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3 , hold in any regular Mal’tsev categories. We prove that Mal’tsev categories may be characterized through variations of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3 , that is classically expressed in terms of four congruences R, S 1 , S 2 and T , and characterizes congruence modular varieties. The proof of this result in a varietal context may be obtained exclusively through the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3 . This was shown by H.P. Gumm in Geometric Methods in Congruence Modular Algebras. We prove that for any 2 n + 1 -permutable category E , the category Equiv ( E ) of equivalence relations in E is also a 2 n + 1 -permutable category.


Introduction
The study of Mal'tsev categories originates from a classical theorem of Mal'tsev in 1954 [5]. For a variety of universal algebras V (i.e., a category of models of a finitary one-sorted algebraic theory), he proved that the following conditions are equivalent: (i) for any pair of congruences R and S on the same algebra X in V, the equality RS = SR holds; (ii) the algebraic theory of V contains a ternary operation p satisfying the equations p(x, y, y) = x, p(x, x, y) = y.

Structure of the paper
In Section 2 we introduce the basic categorical notions and results we shall need in the following section. In particular we recall the relationship between some epimorphisms such as strong epimorphisms, regular epimorphisms and split epimorphisms. We then recall some basics facts about regular categories and relations in regular categories. We also recall the notion Barr's Metatheorem concerning the internal logic of regular categories.
In Sections 3 and 4 the main results of the present paper show that stronger versions of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, characterize regular categories that are Mal'tsev categories and that are Goursat categories.
In Section 5 we prove that for any 2n + 1-permutable category E, the category Equiv(E) of equivalence relations in E is also a 2n + 1-permutable category.

Preliminaries
In this section, we recall some elementary categorical notions and properties needed in the subsequent sections.

Regular epimorphisms
In this subsection, we examine various types of epimorphisms in order to understand the notions of regular categories. In the category Set of sets, the epimorphisms are precisely the surjective maps. In the category Grp and Ab of abelian groups, the epimorphisms are the surjective homomorphisms. The notion of monomorphism is defined dually: In any category E, the following properties hold.
(i) every split epimorphism is regular; (ii) every regular epimorphism is strong.

Regular categories and relations
Recall that a kernel pair of a morphism f : A → B is a pair of parallel morphisms Definition 2.6. [13] A finitely complete category E is regular when (a) E has coequalisers of kernel pairs; (b) regular epimorphisms are stable under pullbacks in E: That is, given a pullback where f is regular epimorphism, then g is a regular epimorphism.
Example 2.7. Regular categories, even complete ones are abundant. The category of models of any equational theory is complete and regular. This includes the categories of sets, lattices, groups, rings, etc. The category of compact Hausdorff spaces is complete regular as is every abelian category. Also the category opposite to the category of sets is regular.
This definition of regular category is equivalent with the following one: any morphism f admits a (unique up to isomorphism) factorization f = mr, where r is a regular epimorphism and m(= Imf ) is a monomorphism, and these factorizations are pullback stable.
If R is a relation from A to B, namely a subobject (r 1 , r 2 ) : R A × B, its opposite relation, denoted R • , is a relation from B to A, the subobject (r 2 , r 1 ) : Given another relation (s 1 , s 2 ) : S B × C from B to C, On first builds the pullback We obtain a composite relation Im(r 1 π 1 , s 2 π 2 ) : SR → A × C.
Definition 2.8. [26] Let E be a category with finite limits.
An (internal) equivalence relation is a reflexive, symmetric and transitive relation. In particular, the kernel pair f 1 , f 2 : Eq(f ) ⇒ A × A of a morphism f : A → B (obtained by building the pullback of f along f ) is an equivalence relation. The equivalence relations that occur as kernel pairs of some morphism in a category E are called effective.
We denoted by Equiv(E) the category whose objects are equivalence relations in E and arrows from r 1 , r 2 : R A × A to s 1 , s 2 : S B × B are pairs (f, g) of arrows in E making the following diagram commute Note that the regular image f (R) can be obtained as the relational composite When R is an equivalence relation, f (R) is also reflexive and symmetric.

Barr's embedding Metatheorems
The following celebrated lemma is crucial for the whole of category theory. All this allows defining the "Yoneda embedding of E" [6] which is the functor This functor Y E is a full and faithful embedding of E in the category [E op , Set] of contravariant functors from E to Set and natural transformations between them.
These well-known facts allow at once a powerful set-theoretical-like technique for proving various results in a category by just proving them in the category of sets.  (i) some finite diagram is commutative; (ii) some morphism is a monomorphism; (iii) some morphism is an isomorphism; (iv) some finite diagram is a limit diagram; (v) some arrow f : A −→ B factors (of course, uniquely) through some specified If this statement P is valid in the category of sets, it is valid in every category.

Shifting Lemma, Little Desarguesian Theorem, Little Pappian
Theorem, Escher Cube, Closure Lemma 1 and 3 For a variety V of universal algebras, Gumm's Shifting Lemma [16] is stated as follows. Given congruences R, S 1 and T on the same algebra X in V such that R ∧ S 1 T , whenever x, y, t, z are element in X with (x, y) ∈ R ∧ T, (x, t) ∈ S 1 , (y, z) ∈ S 1 and (t, z) ∈ R, it then follows that (t, z) ∈ T : A variety V of universal algebras satisfies the Closure Lemma 1, [27] if given congruences R, S i and T on the same algebra Mainly we are interested in the following two special case. Firstly, letting x = x and z = z we obtain: A variety V of universal algebras satisfies the Little Desarguesian Theorem [27] if given congruences R, S i and T on the same algebra Escher Cube or Closure Lemma 2 : A variety V of universal algebras satisfies the Closure Lemma 2, [27] if given congruences R, S i and T on the same algebra X in V such that R ∧ S i T , whenever x, y, , z, x , y , z , u, u are element in X with (y, y ), (u , u), The little Pappian Theorem A variety V of universal algebras satisfies the little Pappian Theorem, [27] if given congruences R, S i and T on the same algebra Similarly, on identifying S 2 with T and u with z we obtain: A variety V of universal algebras satisfies the Closure Lemma 3 if given congruences R, S and T on the same algebra X in V such that R ∧ S T , one has

Regular Mal'tsev Categories
A finitely complete category E is called a Mal'tsev category if every reflexive relation in E is an equivalence relation. (2) abelian categories are Mal'tcev categories; (4) any slice of a Malcev category.
These categories are also characterized by other properties on relations, as follows: [15] Let E be a regular category. The following statements are equivalent: (v) every reflexive relation T in E is transitive: T T = T . (2) the Escher Cube holds true in E; (3) the Closure Lemma 1 holds true in E; (4) the Closure Lemma 3 holds true in E; (5) the Little Pappian Theorem holds true in E.
Proof. In a regular context, it suffices to give a proof in set-theoretical terms (see Metatheorem 2.12 and [27], for instance). We show that the Little Desarguesian Theorem holds true in E.
Let R, S 1 , S 2 and T be equivalence relations on the same object X in E such that R ∧ S i T and assume that x, y, z, x , y , z , are related as in (2.2). The pairs (y, y), (y , y ), (x, z), (x , z ) are elements of S 1 . Defining equivalence relations P, Q and F on S 1 by: We obtain If we want to apply the Shifting Lemma we have to check that We may apply the Shifting Lemma to We conclude that (x , z )Q(y , y ) and, consequently, that (x , y ) ∈ T . Similarly we show that the Closure Lemma 1 holds true in E; Let R, S 1 , S 2 and T be equivalence relations on the same object X in E such that R ∧ S i T and assume that x, y, z, x , y , z , are related as in (2.1). The pairs Similarly we can show (3), · · · , (5).
Using the fact that in a Mal'tsev category reflexive relations coincide with equivalence relations, or with symmetric relations, we are now going to show that regular Mal'tsev categories may be characterized through a stronger version of the diagram (2.2) where, in the assumption, the equivalence relations are replaced by reflexive relations. Note that, for diagram such as (2.2) where R, 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. (1) E is a Mal'tsev category; (2) the Little Desarguesian Theorem holds in E when R, T, S 1 , S 2 are reflexive relations on X; (3) the Escher Cube holds in E when R, T, S 1 , S 2 are reflexive relations on X; (4) the Closure Lemma 1 holds in E when R, T, S 1 , S 2 are reflexive relations on X. (2) ⇒ (1) : We shall prove that every reflexive relation e 1 , e 2 : E X × X is symmetric (Theorem 3.2(iii)). Suppose that (x, y) ∈ E, and consider the reflexive relations T and R on E defined by the following pullbacks: The reflexive relation on E we consider are the kernel pair Eq(e 1 ), Eq(e 2 ) of e 1 and e 2 respectively.
Eq(e 1 ), Eq(e 2 ), are equivalence relation, with the property that Eq(e i ) R and Eq(e i ) T, so that R ∧ Eq(e i ) = Eq(e i ) T, i = 1, 2. We may apply the assumption to the following relations given in solid lines  (1) E is a Mal'tsev category; (2) the Little Desarguesian Theorem holds in E when R, T, are reflexive relations and S 1 , S 2 are equivalence relations on X; (3) the Escher Cube holds in E when R, T, are reflexive relations and S 1 , S 2 are equivalence relations on X; (4) the Closure Lemma 1 holds in E when R, T, are reflexive relations and S 1 , S 2 are equivalence relations on X; (5) the Little Pappian Theorem holds true in E when R, T, are reflexive relations and S 1 , S 2 are equivalence relations on X; (6) the Closure Lemma 3 holds true in E when R, T, are reflexive relations and S 1 , S 2 are equivalence relations on X.
A variety of algebras V is a majority category if and only if it admits a majority term, i.e., a ternary term m(x, y, z) satisfying the equations: For a proof of this statement, we refer the reader to [23].

Example 3.6. (i) A subvariety V of the variety of rings is a majority category if and only
if V satisfies the equation x n = x for some n > 2. In particular, the category BoRg of Boolean rings is a majority category.Then the polynomial is a majority term for V.
(ii) The category NReg of von Neumann regular rings (see [21] ) is the class of all rings R such that for any a ∈ R there exists x ∈ R such that a = axa . The category NReg is a majority category.
Suppose that A, B, C are rings and that R is a subring of A × B × C which is a von Neumann regular ring. Let a = (a, b, c ), b = (a, b , c), c = (a , b, c) be any elements of R. Then since R is von Neumann regular, there exists x = (x 1 , x 2 , x 3 ) ∈ R such that Then it is easy to see that For a regular category E, the property of being a majority category can be equivalently defined as follows (see [21]): for any reflexive relations R, S and T on the same object X in E, the inequality holds.
Lemma 3.7. Let E be a majority category E. Then Proof. The proof of this result is based on the fact that a majority category satisfies the Shifting Lemma.

Regular Goursat Categories
A regular category E is called a Goursat category [14] when it is 3-permutable, i.e. for any pair of equivalence relations R and S on the same object in E one has RSR = SRS. (i) E is a Goursat category; (ii) ∀R, S ∈ Equiv(X), R ∨ S = RSR(= SRS) ∈ Equiv(X); (iii) every relation P → X × Y in E ,P P • P P • = P P • ; (iv) every reflexive relation F in E , F • F = F F • ∈ Equiv(X); (v) every reflexive and positive relation R = U • U in E is an equivalence relation; (vi) E • EE , for any reflexive relation E;  (1) The little desarguesian theorem (2.2) holds in E when S 1 is a reflexive relation and S 2 , R and T are equivalence relations.
(2) The Closure Lemma 1 (2.1) holds in E when S 1 is a reflexive relation and S 2 , R and T are equivalence relations. Proof. The proof is based on Proposition 3.4 in [15] which stipule that a Goursat category satisfies the Shifting Lemma when S 1 is a reflexive relation and R and T are equivalence relations.
In a regular context, it suffices to give a proof in set-theoretical terms (see Metatheorem 2.12 and [27], for instance). We show that the Little Desarguesian Theorem holds true in E.
Let R, S 1 , S 2 and T be equivalence relations on the same object X in E such that R ∧ S i T and assume that x, y, z, x , y , z , are related as in (2.2). The pairs We obtain If we want to apply the Shifting Lemma we have to check that F ∧ P ≤ Q. Thus, let We may apply the Shifting Lemma to We conclude that (x , z )Q(y , y ) and, consequently, that (x , y ) ∈ T .
Theorem 4.4. Let E be a regular category and consider the following assertions.
(1) E is a Goursat category; (2) the little desarguesian theorem (2.2) holds in E when S 1 , S 2 are reflexive relations and R and T are reflexive and positive relations; Then Proof.
(2) ⇒ (1) We shall prove that for any reflexive relation E on X in E, EE • = E • E (see Theorem 4.2 (iv)). Suppose that (x, y) ∈ EE • . Then, for some z in X, one has that (z, x) ∈ E and (z, y) ∈ E. Consider the reflexive and positive relations R = EE • and T = E • E, and the reflexive relation E on X. Then we have: With the assumption of previous proof, we obtain A variety V of universal algebras is called 3-permutable when the strictly weaker equality RSR = SRS holds. Such varieties are characterized by the existence of two quaternary operations p and q satisfying the identities p(x, y, y, z) = x, p(u, u, v, v) = q(u, u, v, v), q(x, y, y, z) = z. We start by giving a direct proof of Little Desarguesian theorem in the varietal context. We give an alternative proof which is suitable to be extended to the categorical context of regular categories.
Theorem 4.5. Let E be a majority and 3-permutable varieties.Let x, y, z, x , y , z , be elements of X and R, T, S 1 , S 2 congruences on X with R ∧ S i T , for i = 1, 2. Then the diagram (2.2) holds in E.
Proof. Let R, S 1 , S 2 and T be congruence on the same algebra X in E such that R∧S i T and assume that x, y, z, x , y , z , are related as in (2.2). From the relations xT xRx S 1 S 2 x xT xRx S 1 S 2 y xT yRy S 1 S 2 y yT yRy S 1 S 2 y xT xRxS 1 S 2 x xT xRxS 1 S 2 y xT xRyS 1 S 2 y yT yRyS 1 S 2 y we may deduce the following ones by applying the quaternary operations p and q, respectively: p(x, x, x, y)T p(x, x, y, y)Rp(x , x , y , y )Sp(x , y , y , y ) and q(x, x, x, y)T q(x, x, y, y)Rq(x , x , y , y )Sq(x , y , y , y ) one has xT p(x, x, y, y)Rp(x , x , y , y )Sx and yT q(x, x, y, y)Rq(x , x , y, y)Sy ; one has xT xRp(x, x, y, y)Sx and yT yRq(x, x, y, y)Sy.
Since E is majority variety, then R ∧ S 1 S 2 (R ∧ S 1 )(R ∧ S 2 ) T T = T . We apply the Shifting Lemma to Again, we apply the Shifting Lemma to we obtain x T p(x , x , y , y ) = q(x , x , y , y )T y ; it follows that (x , y ) ∈ T .
We adapt this varietal proof into a categorical one using an appropriate matrix and the corresponding relations which may be deduced from it (see [16] for more details). The kind of matrix we use translates the quaternary identities into the property on relations given in Theorem 4.2(iii): x y y z x z u u v v α α Thus, the introduction of a new element α, to represent the identity p(u, u, v, v) = q(u, u, v, v)(= α).
For this matrix, the interpretation gives: for any binary relation P , if xP u, yP u, yP v and zP v , then xP α and zP α, for some α; this gives the property P P o P P o ≤ P P o Since P P o ≤ P P o P P o is always true, we get precisely P P o P P o = P P o from Theorem 4.2(iii).
Theorem 4.6. Let E be a regular majority and Goursat categories. Let x, y, z, x , y , z , be elements of X and R, T, S 1 , S 2 are equivalence relations on X with R ∧ S i T , for i = 1, 2. Then the diagram (2.2) holds in E.
Proof. We extend the proof of Theorem 4.5 to a categorical context by constructing an appropriate matrix as follows x x x y x y x y y y x y x y y y x y x x y y θ θ x x y y ω ω We define a relation P from X 3 to X 2 by: (a, b, c)P (d, e) ⇔ ∃l such that aT dRlS 1 S 2 b, lRe, eS 1 S 2 c and aS 1 S 2 d.
From the matrix we see that (x, x, x )P (x, x ), (x, y, y )P (x, x ), (x, y, y )P (y, y ) and (y, y, y )P (y, y ), we also see that (x, x, x )P P • P P • (y, y, y ) from which we conclude that (x, x, x )P P • (y, y, y ). It then follows that (x, x, x ) P (θ, ω) and (y, y, y )P (θ, ω), for some (θ, ω). Let S = S 1 S 2 , i.e., there exist α and β such that xT θRαSx, αRω, ωSx and xSθ yT θRβSy, βRω, ωSy and ySθ Since E is majority category, R ∧ S 1 S 2 (R ∧ S 1 )(R ∧ S 2 ) T T = T . We apply the Shifting Lemma to Again, we apply the Shifting Lemma to we obtain x T ωT y ; it follows that (x , y ) ∈ T .

Equivalence Relations in n-permutable Categories
In this section we investigate the category Equiv(E) of internal equivalence relations in a regular category E. We show that Equiv(E) is a 2n + 1-permutable category whenever E is. For n = 1 (see [17]) E be a Goursat category. Then the category Equiv(E) is a also a Goursat category.
Following [1], given any pair (R, S) of reflexive relations on an object X in a regular category E, let us denote by (R; S) n ; n ≥ 2 the alternate composition RSRS · · · of length n which is a reflexive relation as well. Clearly we have: (R, S) n ≤ (R, S) n+1 and (S, R) n ≤ (R, S) n+1 . Then call n-permutable a regular category satisfying (R, S) n = (S, R) n for all pairs (R, S) of equivalence relations. (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.
[4] Given any 2n-permutable or (2n + 1)-permutable regular category E, any regular epimorphism f : X Y and any equivalence relation S on X, then the reflexive relation f (S) n is an equivalence relation in E.
Another characterization of (2n + 1)-permutable categories in terms of (equivalence) relations is given by the preservation of equivalence relations through the regular image by a regular epimorphism as follows: (i) the regular category E is (2n + 1)-permutable.
(ii) for any regular epimorphism f : X Y in E and any equivalence relation S on X, then f (S) n is an equivalence relation.
The kernel pair of a morphism (f, g) in Equiv(E) is given by the kernel pairs Eq(f ) of f and Eq(g) of g in E Eq(g) Consequently, a morphism (f, g) is a monomorphism in Equiv(E) if and only if both f and g are monomorphisms in E. When E is (2n + 1)-permutable category, a similar property holds with respect to regular epimorphisms: Lemma 5.4. Let R and S be two equivalence relations in a (2n+1)-permutable category E and (f, g) : R → S be a morphism in Equiv(E). Then (f, g) is a regular epimorphism in Equiv(E) if and only if both f and g are regular epimorphisms in E.
Proof. When f and g are regular epimorphisms in E, it is not difficult to check that (f, g) is necessarily the coequalizer of its kernel pair in Equiv(E) given in (5.1) (one uses the fact that g = coeq(g 1 , g 2 ) and f = coeq(f 1 , f 2 ) in E).
Conversely, let (f, g) be a morphism in Equiv(E) as in (5.1) that is a regular epimorphism in Equiv(E). Consider the kernel pairs of f and g, the (regular epimorphism, monomorphism) factorization f = iq of f , and the regular image (q(R) n , t 1 , t 2 ) of (R, r 1 , r 2 ) along q. We obtain the following commutative diagram Eq(g) here (q(R) n , t 1 , t 2 ) ∈ Equiv(E) (by Theorem 5.3) and (i, j) is the morphism in Equiv(E) such that (i, j)(q, α) = (f, g). Note that j is induced by the fact that (i × i) t 1 , t 2 α is the (regular epimorphism, monomorphism) factorization of s 1 , s 2 g, thus it is a monomorphism.
From the fact that (f, g) is the coequalizer of its kernel pair in Equiv(E) and that (q, α) is the coequalizer of (f 1 , g 1 ) and (f 2 , g 2 ) in Equiv(E) (since (i, j) is a monomorphism), it easily follows that (i, j) is an isomorphism in Equiv(E ). This implies that f and g are regular epimorphisms in E.
Corollary 5.5. Let E be a (2n + 1)-permutable category. Then the category Equiv(E) is a regular category.
Proof. Let (R, (p 1 , q 1 ), (p 2 , q 2 )) be an equivalence relation on (S, s 1 , s 2 ) in the category Equiv(E) and f = (f 1 , f 2 ) a regular epimorphism in Equiv(E). We must prove that f (R) n is an equivalence relation in Equiv(E). The relation f (R) n is obtained through the following diagram

C~~d~c
where U = f 2 (R) n and V = f 1 (T ) n are in Equiv(E).
One has: akr 1 = cα 1 t and bkr 1 = cα 2 t, thus the following diagram commutes: Since t is a strong epimorphism and a, b is a monomorphism, there exists β 1 : U → V such that a, b β 1 = cα 1 , dα 2 . The existence of β 2 is obtained similarly. Since it follows that β 1 , β 2 : U V × V is a monomorphism and thus (U, β 1 , β 2 ) is a relation on V . All parallel morphisms of the left face represent equivalence relations and all horizontal morphisms are regular epimorphisms, so that all parallel morphisms of the right face also represent equivalence relations (Theorem 5.3), and then f (R) n is an equivalence relation in Equiv(E).