A CHARACTERIZATION OF CENTRAL EXTENSIONS IN THE VARIETY OF QUANDLES VALÉRIAN EVEN, MARINO GRAN AND ANDREA MONTOLI

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1 Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES VLÉRIN EVEN, MRINO GRN ND NDRE MONTOLI bstract: The category o symmetric quandles is a Mal tsev variety whose subvariety o abelian symmetric quandles is the category o abelian algebras. We give an algebraic description o the quandle extensions that are central or the adjunction between the variety o quandles and its subvariety o abelian symmetric quandles. Keywords: Quandle, symmetric quandle, abelian object, Mal tsev variety, central extension, categorical Galois theory. MS Subject Classiication (2010): 57M27, 08B05, 1820, 13B Introduction quandle [19] is a set equipped with two binary operations and 1 such that the ollowing identities hold (or all a, b, c ): (1) a a = a = a 1 a (idempotency); (2) (a b) 1 b = a = (a 1 b) b (right invertibility); (3) (a b) c = (a c) (b c) and (a 1 b) 1 c = (a 1 c) 1 (b 1 c) (sel-distributivity). This structure is o interest in knot theory, since the three axioms above correspond to the Reidemeister moves on oriented link diagrams. From a purely algebraic viewpoint, quandles capture the properties o group conjugation: given a group (G,, 1), by deining the operations a b = b a b 1 and a 1 b = b 1 a b on the underlying set G one gets a quandle structure. Quandles and quandle homomorphisms orm a category denoted Qnd. This category, being a variety in the sense o universal algebra [7], is an exact category (in the sense o Barr [1]). The variety Qnd has some interesting categorical properties, as recently observed in [9, 10, 2]. The present work continues Received pril 9, Research partially supported by the Centro de Matemática da Universidade de Coimbra (CMUC), unded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT Fundação para a Ciência e a Tecnologia under the projects PTDC/MT/120222/2010 and PEst-C/MT/UI0324/2013 and the grant number SFRH/BPD/69661/2010. The third author is a Postdoctoral Researcher o the Fonds de la Recherche Scientiique-FNRS. 1

2 2 V. EVEN, M. GRN ND. MONTOLI this line o research, by investigating the properties o the adjunction between the variety o quandles and its subvariety bsymqnd o abelian symmetric quandles, in particular rom the viewpoint o the categorical theory o central extensions [16]. The variety bsymqnd o abelian symmetric quandles is the subvariety o Qnd determined by the two additional identities and a b = b a (a b) (c d) = (a c) (b d). bsymqnd is a Mal tsev variety (actually even a naturally Mal tsev one [18], see Section 2), and it turns out to be an admissible subvariety o Qnd: this act guarantees the validity o a Galois theorem o classiication o the corresponding central extensions (see [15, 16]). This is particularly interesting by keeping in mind that the variety Qnd is not congruence modular, since it contains the variety o sets as a subvariety. However, the subvariety bsymqnd o abelian symmetric quandles yields an adjunction that is similar to the classical one I Qnd bsymqnd (1) H V I V ab (2) U where V is any modular variety and V ab its subvariety o abelian algebras in the sense o commutator theory [12]. Many interesting results in the categorical theory o central extensions discovered in the last years actually concern subvarieties o Mal tsev varieties (see [11], or instance, and the reerences therein). The example investigated in the present paper is then o a rather dierent nature, and will be useul to establish some new connections between algebraic quandle theory and categorical algebra. To explain the main result o this paper more precisely, let us briely recall how the categorical notions o trivial extension and o central extension are deined in any variety V with respect to a chosen subvariety X o V. surjective homomorphism : B in V is a trivial extension i the commutative

3 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 3 square induced by the units o the relection η HI() HI() B ηb HI(B) is a pullback. surjective homomorphism : B is a central extension when there exists a surjective homomorphism p: E B such that the extension π 1 : E B E in the pullback E B π 2 π 1 E p B o along p is a trivial extension. In any modular variety V the central extensions deined in this way, relatively to the adjunction (2), are precisely the surjective homomorphisms : B whose kernel congruence Eq() = {(a 1, a 2 ) (a 1 ) = (a 2 )} is central in the sense o commutator theory: [Eq(), ] =, where is the smallest congruence on (see [14, 17]). In the present paper we characterize the central extensions corresponding to the adjunction (1) as those surjective quandle homomorphisms : B such that (a condition equivalent to) [Eq(), ] = holds and, moreover, each iber 1 (b) = {a (a) = b} is an abelian symmetric quandle, or any b B (Theorem (3.13)). 2. Symmetric quandles and abelian symmetric quandles quandle is symmetric i it satisies the additional identity: a b = b a, (3) or all a, b. We write SymQnd or the corresponding category o symmetric quandles, which is then a subvariety o the variety Qnd o all quandles. Here below we observe that the category SymQnd is a Mal tsev variety [20], which will be shown to be an admissible subcategory o Qnd or the categorical theory o central extensions [16]. Proposition 2.1. [2] The category SymQnd is a Mal tsev variety.

4 4 V. EVEN, M. GRN ND. MONTOLI Proo : Let p be the ternary term deined by We then have the identities p(a, b, c) = (a c) 1 b. p(a, a, b) = (a b) 1 a = (b a) 1 a = b, p(a, b, b) = (a b) 1 b = a. Recall that a quandle is abelian [19] i it satisies the additional axiom (a b) (c d) = (a c) (b d) or all a, b, c, d. Note that this axiom is equivalent to the ollowing one: (a b) 1 (c d) = (a 1 c) (b 1 d). (4) Remark 2.2. Not all abelian quandles are symmetric. Indeed, recall that a quandle is trivial i a b = a = a 1 b or all a, b. ny trivial quandle is abelian, but it is not symmetric (as long as it has at least two elements). Let us write bsymqnd or the category o abelian symmetric quandles, U : bsymqnd SymQnd and V : SymQnd Qnd or the inclusion unctors. Since bsymqnd is a subvariety o SymQnd and SymQnd is a subvariety o Qnd, both these unctors have let adjoints, denoted by ab: SymQnd bsymqnd and sym: Qnd SymQnd, respectively: Qnd sym V SymQnd ab U bsymqnd. We are now going to show that abelian symmetric quandles are the internal Mal tsev algebras in SymQnd. Deinition 2.3. n internal Mal tsev algebra in a variety V is an algebra V with a homomorphism p : such that p (a, a, b) = b and p (a, b, b) = a. Let us write Mal(V) or the category o internal Mal tsev algebras in C. In a Mal tsev category, thus in particular in the category SymQnd, any morphism preserves the Mal tsev operation (see Corollary 4.1 in [13], or instance): this means that the subcategory Mal(SymQnd) is ull in SymQnd. The ollowing observation has been ound independently by Bourn [2]:

5 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 5 Theorem 2.4. bsymqnd = Mal(SymQnd). Proo : Let bsymqnd, and let p : be the Mal tsev operation on deined by p (a, b, c) = (a c) 1 b. We have to check that it is a quandle homomorphism. For any a, b, c, x, y, z we have p ((a, b, c) (x, y, z)) = p (a x, b y, c z) = ((a x) (c z)) 1 (b y) = ((a c) (x z)) 1 (b y) = ( (a c) 1 b ) ((x z) 1 y) = p (a, b, c) p (x, y, z). This shows that belongs to Mal(SymQnd). Conversely, when Mal(SymQnd), the unique internal Mal tsev operation on is necessarily given by (any o) the Mal tsev operations o the theory o the variety SymQnd. ccordingly, it is deined by p (a, b, c) = (a c) 1 b, and it is such that p (a, b, a) = a 1 b. Moreover, p : preserves the binary operation, so that the equality gives p ((a, b, a) (x, y, x)) = p (a, b, a) p (x, y, x) (a x) 1 (b y) = (a 1 b) (x 1 y). This is precisely the identity (4), and the quandle belongs to bsymqnd. We now recall the deinition o two classes o morphisms in Qnd, irst investigated by Bourn, that will be important or our work: Deinition 2.5. [2] Let Σ be the class o split epimorphisms : B with a given section s: B (i.e. s = 1 B ) in the category Qnd such that the map s(b) : 1 (b) 1 (b) is surjective, or any b B. In other words, the split epimorphism with section s is in Σ i, or any b B and a 1 (b), there is a k a 1 (b) such that s(b) k a = a. Remark 2.6. This element k a also depends on b, so that one should write k b,a, instead. We shall simply write k a, however, to simpliy the notations.

6 6 V. EVEN, M. GRN ND. MONTOLI Given an internal equivalence relation (R, r 1, r 2 ) on, i.e. a congruence on, we write δ R : R or the homomorphism deined by δ R (a) = (a, a). n equivalence relation (R, r 1, r 2 ) is said to be a Σ-equivalence relation i the split epimorphism r 1 : R with section δ R : R belongs to the class Σ. Given a quandle homomorphism : B, we write (Eq(), 1, 2 ) or the kernel pair o, where 1 : Eq() and 2 : Eq() are the canonical projections: in a variety o universal algebras Eq() is simply the kernel congruence on deined by Eq() = {(a 1, a 2 ) (a 1 ) = (a 2 )}. Deinition 2.7. [5, 6, 2] morphism : B in Qnd is Σ-special i (Eq(), 1, 2 ) is a Σ-equivalence relation. The ollowing result is a direct consequence o Theorem 3.9 in [2], and will be useul later on: Theorem 2.8. Let : B be a Σ-special homomorphism in Qnd. Then any congruence R on permutes with Eq() in the sense o the composition o relations: R Eq() = Eq() R. Corollary 2.9. Given a pushout o surjective homomorphisms g B C l D h where is Σ-special, the induced homomorphism (g,) C D B to the pullback is surjective. Proo : The proo is essentially the same as the one given in [9], Lemma 1.7 (which is adapted rom [8]). 3. Central extensions in the category o quandles I C is a initely complete category, a double equivalence relation C in C is an equivalence relation internal in the category o equivalence relations in C.

7 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 7 It can be represented by a diagram π 1 C p 1 π 2 p 2 s 1 S R r1 r 2, where r 1 π 1 = s 1 p 1, r 1 π 2 = s 2 p 1, r 2 π 1 = s 1 p 2 and r 2 π 2 = s 2 p 2. In this case one usually says that C is a double equivalence relation on the equivalence relations R and S. Deinition 3.1. Given equivalence relations R and S on, a double equivalence relation C on R and S (as in (5)) is called a centralizing relation when the square s 2 (5) C p 2 S is a pullback. π 1 R r2 s 1 Deinition 3.2. connector between R and S is an arrow p: R S such that 1. p(x, x, y) = y 1. p(x, y, y) = x 2. xsp(x, y, z) 2. zrp(x, y, z) 3. p(x, y, p(y, u, v)) = p(x, u, v) 3. p(p(x, y, u), u, v) = p(x, u, v) In the Mal tsev context [3] the existence o a connector between R and S is already guaranteed by the existence o a partial Mal tsev operation p: R S, i.e. when the identities p(x, x, y) = y and p(x, y, y) = x in Deinition 3.2 are satisied. ccordingly, in a Mal tsev category the existence o a double centralizing relation on R and S is equivalent to the existence o a partial Mal tsev operation. Moreover, a connector is unique, when it exists: accordingly, or two given equivalence relations, having a connector becomes a property. In a Mal tsev variety a congruence R on an algebra is called algebraically central i there is a centralizing double relation on R and, this latter being the largest equivalence relation on. In terms o commutators, this act is expressed by the condition [R, ] =.

8 8 V. EVEN, M. GRN ND. MONTOLI lso, in the variety Qnd o quandles we shall say that a surjective homomorphism : B in Qnd is an algebraically central extension i its kernel congruence Eq() is algebraically central: there is a connector between Eq() and. Given a homomorphism : B in Qnd, each iber 1 (b) (or b B) is a subquandle o. We shall say that has abelian symmetric ibers i 1 (b) bsymqnd. Lemma 3.3. Consider the ollowing pullback E B π 2 π 1 E p B. I : B has abelian symmetric ibers then so does π 1 : E B E. Moreover, i p: E B is a surjective homomorphism, then : B has abelian symmetric ibers i π 1 : E B E has abelian symmetric ibers. Proo : The irst assertion ollows rom the act that i (e, a) E B then the ibers π1 1 (e) and 1 ((a)) are isomorphic. The proo o the second assertion is similar, the surjectivity o p guaranteeing that, or any a, there exists e E such that (e, a) E B. Lemma 3.4. [2] Let : B be a split epimorphism, with section s: B, in Σ. Consider the ollowing pullback o along a split epimorphism p: E B (t,1 ) E B π2 π 1 (1 E,s p) E t p B. Then (1 E, s p) and (t, 1 ) are jointly epimorphic. Proo : Let (e, a) E B ; we shall show that (e, a) can be rewritten as a product o two elements in the images o (1 E, s p) and (t, 1 ), respectively. Since the split epimorphism is in Σ, there exists an element k a 1 ((a)) such that s(a) k a = a. lso, we always have e = (e 1 tp(e)) tp(e). ccordingly, by using the act that (a) = (k a ) and p(e) = (a), we see s

9 that CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 9 (e, a) = ((e 1 tp(e)) tp(e), s(a) k a ) = (e 1 tp(e), s(a)) (tp(e), k a ) = (e 1 tp(e), sp(e)) (t(k a ), k a ) = (e 1 tp(e), sp(e 1 tp(e))) (t(k a ), k a ) = (1 E, s p)(e 1 tp(e)) (t, 1 )(k a ). Corollary 3.5. Let R be an equivalence relation and S be a Σ-equivalence relation on the same quandle in Qnd. I there is a connector on R and S, then it is unique. Proo : This ollows directly rom Lemma 3.4. Lemma 3.6. Let R be an equivalence relation and S be a Σ-equivalence relation on the same quandle. For a homomorphism p: R S, the ollowing conditions are equivalent : (1) p is a partial Mal tsev operation: p(x, y, y) = x and p(x, x, y) = y; (2) p is a connector between R and S. Proo : We only have to prove that 1. implies 2. Remark that in any variety, in particular in Qnd, the equivalence relation R is the kernel pair o the canonical quotient r : /R, and S the kernel pair o s: /S. R S p 1 i S p 2 π 1 i R π 2 p S s 1 δ S s 2 r 1 R δ R. r 2

10 10 V. EVEN, M. GRN ND. MONTOLI By assumption we have that p i S = s 2 and p i R = r 1. To see that (x, p(x, y, z)) S, we have to prove that s p = s r 1 π 1. The equalities s p i S = s s 2 = s s 1 = s s 1 p 1 i S (p 1 i S = 1 S ) = s r 1 π 1 i S (s 1 p 1 = r 1 π 1 ) and s p i R = s r 1 = s r 1 π 1 i R, imply that s p = s r 1 π 1 by Lemma 3.4. similar argument shows that (z, p(x, y, z)) R. Now, to see that p(x, y, p(y, u, v)) = p(x, u, v), let us consider (a, b, c, d) R R S and write φ(a, b, c, d) = p(a, b, p(b, c, d)) and ψ(a, b, c, d) = p(a, c, d). Observe that φ(a, b, c, c) = p(a, b, p(b, c, c)) = p(a, b, b) = a = p(a, c, c) = ψ(a, b, c, c) or all (a, b, c, c) R R S, and φ(e, e, e, ) = p(e, e, p(e, e, )) = p(e, e, ) = ψ(e, e, e, ) or all (e, e, e, ) R R S. Now, let (x, y, u, v) R R S: since the split epimorphism s 1 : S with section δ S : S is in Σ, there exists k (u,v) = (u, k v ) s 1 1 (u) such that (u, v) = (u, u) (u, k v). Then one can write (x, y, u, v) = (x 1 u, y 1 u, u, u) (u, u, u, k v ) or all (x, y, u, v) R R S. It ollows that φ(x, y, u, v) = ψ(x, y, u, v). similar argument shows that p(p(x, y, u), u, v) = p(x, u, v). Lemma 3.7. Let : B be an algebraically central extension with abelian symmetric ibers, then Eq() is isomorphic to a product Q, where Q is an abelian symmetric quandle.

11 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 11 Proo : Let C be the centralizing relation on Eq() and ; consider the ollowing diagram C c 1 c 2 Eq() q Q where q is the coequalizer o c 1 and c 2. By the Barr-Kock theorem [1, 4], the lower squares are pullbacks. By Lemma 3.3, the homomorphisms Q 1 have abelian symmetric ibers, hence Q is an abelian symmetric quandle. s a consequence, any algebraically central extension : B has its kernel pair Eq() isomorphic to a product o an abelian algebra and. Proposition 3.8. I : B has symmetric ibers, then it is Σ-special. Proo : Consider the kernel pair o Eq() 1 δ One has to check that ( 1, δ ) is in Σ. Let a and (a, a ) 1 1 (a), then in particular (a) = (a ), so that a 1 a is such that (a) = (a 1 a). It ollows that (a, a 1 a) Eq(), and then (a, a) (a, a 1 a) = (a a, a (a 1 a)) = (a, (a 1 a) a) = (a, a ). Remark 3.9. Observe that when a split epimorphism : B with section s: B has symmetric ibers, then s(b) : 1 (b) 1 (b) is always injective: i x 1 (b) and y 1 (b) are such that s(b) x = s(b) y, since s(b) 1 (b), we get x s(b) = y s(b), and hence x = y by right invertibility. The results in [2] will be useul to show that the category o abelian symmetric quandles is admissible with respect to surjective homomorphisms in the category o quandles. In the ollowing we shall characterize categorically 1 2 δ B B

12 12 V. EVEN, M. GRN ND. MONTOLI central and normal extensions in Qnd with respect to the adjunction between the category o quandles and the category o abelian symmetric quandles: Qnd H I bsymqnd The ollowing theorem shows that the unctor I preserves a certain type o pullbacks. This is equivalent to the admissibility condition o the subvariety bsymqnd o Qnd. Theorem In the previous adjunction, the relector I : Qnd bsymqnd preserves all pullbacks in Qnd o the orm P p 1 p 2 H(X) φ H(Y ) (6) where φ: H(X) H(Y ) is a surjective homomorphism lying in the subcategory bsymqnd and : H(Y ) is a surjective homomorphism. Proo : Consider the ollowing commutative diagram where: the square on the back is the given pullback, where φ: H(X) H(Y ) is a surjective homomorphism in the subcategory bsymqnd; the universal property o the unit η P : P HI(P ) induces a unique arrow HI(p 2 ): HI(P ) H(X) with HI(p 2 ) η P = p 2 ; the universal property o the unit η : HI() induces a unique arrow HI(): HI() H(Y ) with HI() η = ; (P, π 1, π 2 ) is the pullback o HI(p 1 ) along η. p 1 P η P γ p 2 P π 2 π 1 η HI(P ) HI(p 2 ) HI(p 1 ) HI() HI() H(X) φ H(Y )

13 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 13 The quandle homomorphism p 1 is Σ-special by Lemma 3.3 since φ has abelian symmetric ibers, thus the homomorphism γ is surjective by Corollary 2.9. The act that π 1 γ = p 1 and HI(p 2 ) π 2 γ = p 2 implies that γ is also injective. Indeed, this latter property ollows rom the act that the pullback projections p 1 and p 2 are jointly monomorphic. ccordingly, the arrow γ is bijective, thus an isomorphism. Since η is a surjective homomorphism it ollows that the right ace o the diagram is a pullback (see Proposition 2.7 in [16], or instance), and the pullback 6 is preserved by the unctor I, as desired. Corollary The unctor I preserves products o the type Q where Q is an abelian symmetric quandle and is any quandle. Proo : Remark that Q is the ollowing pullback Q p2 Q p 1 1 where 1 is the terminal object in Qnd, i.e. element. the trivial quandle with one Lemma Consider the ollowing pullback E B π 2 π 1 E p B. (7) I is an algebraically central extension with abelian symmetric ibers, then π 1 is an algebraically central extension with abelian symmetric ibers. Moreover, i p: E B is a surjective homomorphism, then is an algebraically central extension with abelian symmetric ibers i π 1 is an algebraically central extension with abelian symmetric ibers. Proo : First remark that we already know that the property o having abelian symmetric ibers is preserved and relected by pullbacks along surjective homomorphisms by Lemma 3.3. Let : B be an algebraically central extension with abelian symmetric ibers. Write p : Eq() or the connector between and Eq().

14 14 V. EVEN, M. GRN ND. MONTOLI Deine the quandle homomorphism p π1 : (E B ) Eq(π 1 ) E B as p π1 ((e, a), (e, b), (e, c)) = (e, p (a, b, c)). We have and p π1 ((e, a), (e, b), (e, b)) = (e, p (a, b, b)) = (e, a) p π1 ((e, a), (e, a), (e, b)) = (e, p (a, a, b)) = (e, b). It is then a connector by Lemma 3.6. Now let π 1 : E B E be an algebraically central extension with abelian symmetric ibers. Write p π1 : (E B ) Eq(π 1 ) E B or the connector between (E B ) (E B ) and Eq(π 1 ). The surjectivity o p: E B implies the surjectivity o the homomorphism π 2 : (E B ) Eq(π 1 ) Eq() deined by π 2 ((e, a), (e, b), (e, c)) = (a, b, c). First let us show that Eq( π 2 ) Eq(π 2 p π1 ). Let (((e 0, a), (e 0, b), (e 0, c)), ((e 1, a), (e 1, b), (e 1, c))) Eq( π 2 ). Since has abelian symmetric ibers by Lemma 3.3, it is Σ-special by Proposition 3.8. This means that the split epimorphism Eq() δ 1 is in Σ. In other terms, or all b and all (b, c) 1 1 (b) there exists k (b,c) 1 1 (b), where k (b,c) = (b, k c ), such that (b, b) k (b,c) = (b, c). Such a k (b,c) = (b, k c ) is unique by Remark 3.9: it ollows that, or any (b, c) Eq(), the element k c such that (k c ) = (b) = (c) and b k c = c is unique. Then, or i {0, 1}, we have ((e i, a), (e i, b), (e i, c)) = ((e i, a) 1 (e i, b), (e i, b), (e i, b)) ((e i, b), (e i, b), (e i, k c )). Consequently we remark that π 2 p π1 ((e i, a), (e i, b), (e i, c)) = π 2 p π1 (((e i, a) 1 (e i, b), (e i, b), (e i, b)) ((e i, b), (e i, b), (e i, k c ))) = π 2 (p π1 ((e i, a) 1 (e i, b), (e i, b), (e i, b)) p π1 ((e i, b), (e i, b), (e i, k c )) = π 2 (((e i, a) 1 (e i, b)) (e i, k c )) = π 2 ((e i 1 e i) e i, (a 1 b) k c ) = (a 1 b) k c or both i {0, 1}. This implies the existence o a quandle homomorphism p : Eq() such that p π 2 = π 2 p π1, i.e. p (a, b, c) = (a 1 b) k c

15 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES 15 where k c is the unique element such that b k c = c as above. Moreover, we have p (a, b, b) = (a 1 b) b = a or (a, b, b) Eq() and p (a, a, b) = (a 1 a) k b = a k b = b or (a, a, b) Eq(), so p is a connector by Lemma 3.6. Beore stating our main result, we recall that a surjective homomorphism : B is a normal extension when the homomorphism 1 in the pullback o along itsel is a trivial extension: Eq() 2 1 B Theorem Given a surjective homomorphism : B in Qnd, the ollowing conditions are equivalent: (1) is an algebraically central extension with abelian symmetric ibers; (2) is a normal extension; (3) is a central extension. Proo : Let : B be an algebraically central extension with abelian symmetric ibers, then its kernel pair Eq() is isomorphic to a product Q with Q an abelian symmetric quandle by Lemma 3.7. Corollary 3.11 shows that is then a normal extension. Every normal extension is a central extension. Let : B be a central extension. Then there is a surjective homomorphism p: E B such that the irst projection π 1 : E B E in the pullback (7) is a trivial extension. Then : B is an algebraically central extension with abelian symmetric ibers by Lemma 3.12, because π 1 is the pullback o a morphism lying in bsymqnd. Reerences [1] M. Barr, Exact Categories, in: Lecture Notes in Math. 236, Springer, (1971). [2] D. Bourn, structural aspect o the category o quandles, preprint, [3] D. Bourn, M. Gran Centrality and connectors in Maltsev categories, lgebra Universalis, 48 (3), 2002, [4] D. Bourn, M. Gran, Regular, protomodular and abelian categories, in Categorical Foundations -Special Topics in Order, Topology, lgebra and Shea Theory, Cambridge Univ. Press, Encycl.

16 16 V. EVEN, M. GRN ND. MONTOLI Mathem. ppl. 97, 2004, [5] D. Bourn, N. Martins-Ferreira,. Montoli, M. Sobral, Schreier split epimorphisms in monoids and in semirings, Textos de Matemática (Série B), Departamento de Matemática da Universidade de Coimbra, vol. 45 (2013). [6] D. Bourn, N. Martins-Ferreira,. Montoli, M. Sobral, Monoids and pointed S-protomodular categories, accepted or publication in Homology, Homotopy and pplications, [7] S. Burris, H.P. Sankappannavar, course in universal algebra, Graduate Texts in Mathematics 78, Springer-Verlag (1981). [8]. Carboni, G.M. Kelly, M.C. Pedicchio, Some remarks on Maltsev and Goursat categories, ppl. Categ. Struct. 1 (1993) [9] V. Even, M. Gran, On actorization systems or surjective quandle homomorphisms, J. Knot Theory Ramiications, 23, 11, (2014) [10] V. Even, M. Gran, Closure operators in the category o quandles, preprint, arxiv: , [11] T. Everaert, Higher central extensions in Mal tsev categories, ppl. Categ. Structures 22 (2014) [12] R. Freese, R. Mc Kenzie, Commutator theory or congruence modular varieties, London Math. Soc. Lect. Note Series 125, Cambridge Univ. press, [13] M. Gran, Central extensions and internal groupoids in Maltsev categories, J.Pure ppl. lgebra 155 (2001) [14] M. Gran, Central extensions in modular varieties, J. Pure ppl. lgebra 174 (2002) [15] G. Janelidze, Pure Galois Theory in Categories, J. lgebra 132, (1991) [16] G. Janelidze, G. M. Kelly, Galois theory and a general notion o central extension, J. Pure ppl. lgebra 97 (1994) [17] G. Janelidze, G.M. Kelly, Central extensions in universal algebra: a unication o three notions, lgebra Universalis 44 (2000) [18] P.T. Johnstone, ine categories and naturally Mal cev categories, J. Pure ppl. lgebra 61 (1989) [19] D. Joyce, classiying invariant o knots, the knot quandle, J. Pure ppl. lgebra 23 (1982) [20] J.D.H. Smith, Mal cev Varieties, Lecture Notes in Math. 554 (1976). Valérian Even Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium address: valerian.even@uclouvain.be Marino Gran Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium address: marino.gran@uclouvain.be ndrea Montoli CMUC, Department o Mathematics, University o Coimbra, Coimbra, Portugal, and Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium address: montoli@mat.uc.pt