SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS MARIA MANUEL CLEMENTINO, ANDREA MONTOLI AND LURDES SOUSA

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1 Pré-Publicaçõe do Departamento de Matemática Univeridade de Coimbra Preprint Number SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS MARIA MANUEL CLEMENTINO, ANDREA MONTOLI AND LURDES SOUSA Abtract: We give an explicit decription o the emidirect product in any emiabelian variety. Moreover, we ue thi decription to characterize the topology o the emidirect product in the topological model o any emi-abelian theory. 1. Introduction The emidirect product i a claical contruction in group theory, which i ued to obtain an equivalence between group action and plit extenion. D. Bourn and G. Janelidze gave in [5] a categorical deinition o emidirect product, and proved that it till give an equivalence between plit extenion and internal action in the context o emi-abelian categorie, i.e. pointed Barr-exact protomodular categorie with inite colimit. In the category o group, the categorical emidirect product coincide with the claical one. Moreover, it i known that the emidirect product o two group, with repect to a given action, i, a a et, the carteian product o the two group. Thi i not true in all emi-abelian varietie. E.B. Inyangala proved in [8] that it i true in varietie o right Ω-loop, howing that, given two right Ω-loop X and B and an action ξ o B on X there exit bijection ϕ and ψ making the diagram X 1,0 X B ϕ ψ X k X ξ B 0,1 π B B B Received October 28, Thi work wa partially upported by the Centro de Matemática da Univeridade de Coimbra (CMUC), unded by the European Regional Development Fund through the program COMPETE and by the Portuguee Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEt-C/MAT/UI0324/2013 and grant number PTDC/MAT/120222/2010 and SFRH/BPD/69661/

2 2 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA commutative, where the bottom row i the plit extenion correponding to ξ. J.R.A. Gray and N. Martin-Ferreira howed in [7] that right Ω-loop are the unique varietie where thi property i valid. On the otherhand, F. Borceuxand M.M. Clementinoprovedin [2] thatthe equivalence between internal action and plit extenion, obtained via the categorical emidirect product, hold in the categorie o topological model o any emi-abelian algebraic theory. In the preent paper, we give an explicit decription o the emidirect product in any emi-abelian variety, howing that the emidirect product correpondingtoaninternalactionoanobjectb onanobjectx canbedecribed a a ubet o the carteian product o B and a uitable number o copie o X, extending the reult o [8]. Moreover, we ue thi act to prove that, in the cae o topological model o a emi-abelian theory, the emidirect product i alway a retract o the topological product o B and ome copie o X. The paper iorganizedaollow: in Section2we recallthe categoricaldeinition o emidirect product. In Section 3 we decribe emidirect product in the context o emi-abelian varietie, howing that a emidirect product can bealwayeenaaubetoacarteianproduct. InSection4wecharacterize the emi-abelian varietie in which the incluion o the emidirect product into the correponding carteian product i a bijection, howing that any emi-abelian variety uch that all emidirect product o object B and X are in bijection with carteian product o the orm X n B i a variety o right Ω-loop; in thi way we generalize Inyangala reult. Moreover, we tudyanexampleoavarietythatcanbe decribedaemi-abelianuingtwo dierent et o operation, with dierent cardinality, howing that they give rie to dierent incluion o the emidirect product into the correponding carteian product. In Section 5 we tudy explicitly other concrete example. In Section 6 we conider the cae o topological model o emi-abelian theorie. Acknowledgement. We are very grateul to Franci Borceux or uggeting u the tudy o emidirect product in topological emi-abelian algebra.

3 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 3 2. The categorical notion o emidirect product In thi ection we recall rom [5] the categorical notion o emidirect product. Let C be a initely complete category. For any morphim p: E B in C, we can deine the pullback unctor p : Pt(B) Pt(E), where the category Pt(B), called the category o point over B, i the category o point o the comma category C over B, i.e. the cocomma category 1 B over C/B. Thi amount to the category whoe object are the plit epimorphim with codomain B. In act a morphim rom the terminal object 1 B : B B to an object : A B i preciely an arrow : B A uch that = 1 B. Deinition 2.1. A initely complete category C i aid to be a category with emidirect product i, or any arrow p: E B in C, the pullback unctor p (ha a let adjoint and) i monadic. In thi cae, denoting by T p the monad deined by thi adjunction, given a T p -algebra (D,ξ) the emidirect product (D,ξ) (B,p)i an object in Pt(B) correponding to (D, ξ) via the canonical equivalence K: Pt(B) K p! p [Pt(E)] Tp Pt(E) Let u recall rom [5] that, being C initely complete, the pullback unctor p have let adjoint p! (or any p in C) i and only i C ha puhout o plit monomorphim. For Barr-exact categorie [1], i, moreover, the unctor p are conervative, that i i C i protomodular [4], the exitence o emidirect product i guaranteed. In act: Theorem 2.2 ([5], Theorem 3.4). A initely complete Barr-exact category i a category with emidirect product i and only i it i protomodular and ha puhout o plit monomorphim.

4 4 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA I C i initely complete, o that we can deine p or every morphim p, ha puhout o plit monomorphim, o that the unctor p have let adjoint p!, and an initial object 0, then it i enough to conider the unctor i B or the unique morphim i B : 0 B: Propoition 2.3 ([11], Corollary 3). Let C be a category with inite limit, puhout o plit monomorphim and initial object. Then the ollowing tatement are equivalent: (i) all pullback unctor i B deined by the initial arrow are monadic; (ii) or any morphim p in C, the pullback unctor p i monadic, i.e. C admit emidirect product. When the category C i pointed, the algebra or the monad (T i B,η,µ) are called internal action in [3] and the endounctor T i B i uually denoted by B ( ). We recall that η X and µ X are the unique morphim uch that k 0 η X = ι X and k 0 µ X = [k 0,ι B ]k 0, a diplayed in the diagram B X k0 X +B η X X ι X, B (B X) µ X B X k 0 (B X)+B k 0 [k 0,ι B ] X +B, where k 0 and k 0 denote the kernel o [0,1]: (B X)+B B and o [0,1]: X +B B, repectively. The algebra or thi monad are pair (X,ξ: B X X) atiying the uual condition: ξη X = 1 X, and ξµ X = ξ(1 ξ). Conequently, or C a above, aying that C ha emidirect product mean that, or each internal action ξ : B X X, there exit (up to iomorphim) a unique plit epimorphim A B ollowing diagram commute: uch that X = Ker and making the B X ξ X k 0 X +B k A [k,] ι B B [0,1] B, ( )

5 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 5 Then A B i the emidirect product o X and B with repect to ξ. Sometime we will identiy thi emidirect product with the object A or with the plit extenion X k A B. ( ) When C i the category o group, B X i the ubgroup o the ree product X + B generated by the element o the orm bxb 1, with b B and x X. Hence an internal action ξ i nothing but the realization in X o the conjugation in the claical emidirect product X B. 3. Semidirect product in emi-abelian varietie An immediate conequence o Theorem 2.2 i that every emi-abelian category [9] ha emidirect product in the categorical ene recalled above. Thi i the cae, in particular, o emi-abelian varietie, which were characterized by D. Bourn and G. Janelidze in [6]. Theorem 3.1. A variety o univeral algebra i emi-abelian i and only i it ha, among it operation, a unique contant 0, n binary operation α i, i = 1,...,n, and an (n+1)-ary operation θ atiying the ollowing equation: α i (x,x) = 0 or any x (I) θ(α(x,y),y) = x or any x,y, (II) where α(x,y) denote (α 1 (x,y),...,α n (x,y)). The aim o thi ection i to give an explicit decription o the emidirect product in emi-abelian varietie. We tart by recalling a reult o E.B. Inyangala [8]. Let C be a variety o right Ω-loop, i.e. a (emi-abelian) variety which ha, among it operation, a unique contant 0, a binary + and a binary atiying the ollowing equation: (a) x+0 = x; (b) 0+x = x; (c) (x y)+y = x; (d) (x+y) y = x.

6 6 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA Then, given a plit extenion ( ) in C, it i poible to deine two ettheoretical map ϕ: X B A ψ: A X B (x,b) x+(b) a (a (a),(a)), treating k a an incluion. Propoition 3.2 ([8]). The two map ϕ and ψ are invere to each other. In other term, given any plit extenion ( ), A i in bijection with the carteian product o B and Ker. Moreover, E.B. Inyangala proved in [8] that, i a emi-abelian variety C ha a binary + and a binary, and the map ϕ and ψ are deined uing + and a above, then ϕ and ψ are bijection, whoe retriction to X and B are identitie, i and only i the equation (a)-(d) are atiied, i.e. i and only i C i a variety o right Ω-loop. Later, in [7] J.R.A. Gray and N. Martin-Ferreira extended thi reult, howing that the exitence o uch ϕ and ψ induce binary operation + and making the algebra right Ω-loop. In order to achieve thi reult, they made a thorough tudy o the map ϕ and ψ decribed above, and o their generalization in emi-abelian varietie, een a natural tranormation between uitable unctor. We are now going to expre the natural tranormation tudied in [7] in term o the operation o the emi-abelian variety, in order to get an explicit decription o emidirect product. Thi way we will obtain ome reult already contained in [7], but we will give dierent proo, that will be ueul later in the tudy o emidirect product o topological algebra. Let C be a emi-abelian variety. For each plit extenion ( ) in C, uing the operation α 1,...,α n and θ introduced at the beginning o the ection, we can deine two et-theoretical map ϕ: X n B A ψ: A X n B (x,b) θ(x,(b)) a (α(a,(a)),(a)), wherexdenote(x 1,...,x n ),andtreatingagaink aanincluion. Toimpliy our calculation, or any map h : Z W and z = (z 1,...,z n ) Z n, h n (z) denote (h(z 1 ),...,h(z n )).

7 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 7 Propoition 3.3. For any plit epimorphim A B in C, we have that: 1. ϕψ = id A, and thereore A i a retract o X n B. 2. A i in bijection with the ubet Y = {(x,b) X n B α(θ(x,(b)),(b))= x}. Proo: 1. For any a A we have: ϕψ(a) = ϕ(α(a,(a)),(a))= θ(α(a,(a)),(a)) = a, where the lat equality ollow rom equation (II). 2. Let u irt prove that ψ(a) Y: or any a A, we have ψ(a) = (α(a,(a)),(a)), hence: α(θ(α(a,(a)),(a)),(a))= α(a,(a)). It remain to prove that ψϕ Y = id Y. Let u oberve that, or any algebra Z and any z Z, we have θ(0,z) = θ(α(z,z),z) = z. Then, or any (x,b) Y: ψϕ(x,b) = ψθ(x,(b)) = (α(θ(x,(b)),θ(x,(b))),θ(x,(b))) = (α(θ(x,(b)),θ( n n (x),(b))),θ( n (x),(b))) = (α(θ(x,(b)),θ(0,(b))),θ(0,b)) = (α(θ(x,(b)),(b)),b)= (x,b), where the lat equality hold becaue (x,b) Y. Propoition 3.3 allow u to give an explicit decription o emidirect product in any emi-abelian variety. Theorem 3.4. Given a emi-abelian variety C, object B,X C and an internal action ξ: B X X o B on X, the emidirect product X ξ B o X and B w.r.t. the action ξ i the et Y decribed in the previou propoition: where A B Y = {(x,b) X n B α(θ(x,(b)),(b))= x}, i the plit epimorphim in C correponding to the action ξ, equipped with the ollowing tructure: i ω i an m-ary operation o the

8 8 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA variety, then in Y we have: ω Y ((x 1,b 1 ),...,(x m,b m )) = = (ξ n α B X (ω B X (θ B X (x 1,b 1 ),...,θ B X (x m,b m )),ω B X (b)),ω B (b)). Proo: Being A B the plit epimorphim in C correponding to the emidirect product X ξ B, diagram ( ) ay that ξ i the retriction o the morphim [k,]. We know that A i in bijection with Y via the map ϕ and ψ tudied in Propoition 3.3. Given (x i,b i ) Y, or i = 1,...,m, let u = ω A (θ A (x 1,(b 1 )),...,θ A (x m,(b m ))). Then ω Y ((x 1,b 1 ),...,(x m,b m )) = ψω A (ϕ(x 1,b 1 ),...,ϕ(x m,b m )) = ψ(u) = (α A (u,(u)),(u)). Since and are morphim (and o they preerve the operation): (u) = ω B (θ B ( n (x 1 ),(b 1 )),...,θ B ( n (x m ),(b m ))), and, ince = id B and X = Ker, (u) = ω B (θ B (0,b 1 ),...,θ B (0,b m )) = ω B (b). Thu, (u) = ω A ( m (b)), and we obtain: ω Y ((x 1,b 1 ),...,(x m,b m )) = (α A (u,ω A ( m (b))),ω B (b)) = (α A (ω A (θ A (x 1,(b 1 )),...,θ A (x m,(b m ))),ω A ( m (b))),ω B (b)) = ([k,] n α B X (ω B X (θ B X (x 1,b 1 ),...,θ B X (x m,b m )),ω B X (b)),ω B (b)), and, ince ξ i the retriction o [k,], we inally obtain the claimed equality. 4.A detailed decription o ϕ and ψ The aim o thi ection i to make explicit the relationhip between the propertie o the map ϕ and ψ deined in Section 3 and the equation o the variety. Let C be a variety which ha, among it operation, a unique contant 0, n binary operation α i, i = 1,...,n, atiying the equation (I), and an (n+1)-ary operation θ. Given a plit epimorphim with kernel X a in ( ), let ϕ: X n B A and ψ: A X n B be the map deined in Section 3.

9 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 9 Oberve that the map ψ i well-deined (i.e. it take value in X n B) thank to the equation (I). We have thereore the ollowing diagram: ϕ X X n X ψ X 1,0 X n B k ϕ A ψ 0,1 π B B ϕ B B where ϕ X,ψ X are obtained via the univeral property o kernel, and ψ B ( ) ϕ B (b) = (ϕ(0,b)) = θ(0,(b)) = θ(0,b), ψ B (b) = π B (ψ((b))) = id B, o that ϕ = ϕ B π B and ψ = 0,1 ψ B. The ollowing propoition i a reormulation o ome reult in [7]. Propoition 4.1. We have that: 1. ϕ B = id B or any plit epimorphim ( ) i and only i the ollowing equation i atiied in the variety: θ(0,x) = x or all x; (III) 2. ϕψ = id A or any plit epimorphim ( ) i and only i equation (II) i atiied in the variety; 3. i equation (III) hold in the variety, then ψϕ = id Xn B or any plit epimorphim ( ) i and only i the equation i atiied in the variety. α(θ(x,y),y) = x or all x,y (IV) Proo: 1. Suppoe that equation (III) hold. Then ϕ B (b) = θ(0,b) = b, or all b B. Converely, uppoe that ϕ B = id B or any plit epimorphim ( ). Applying thi act to the plit epimorphim A 1A A A, we eaily get equation (III). 1 A or any algebra 2. The act that equation (II) implie that ϕψ = id A wa already proved (ee Propoition 3.3). Converely, uppoe that ϕψ = id A or any plit epi- 1,1 morphim o the orm A A π2 A. Then, or any x,y A, we have (x,y) = θ(α((x,y),(y,y)),(y,y))= (θ(α(x,y),y),θ(α(y,y),y)),

10 10 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA and hence θ(α(x,y),y) = x or all x,y. 3. We have that ψϕ(x,b) = ψ(θ(x,(b)) = (α(θ(x,(b)),θ(x,(b))),θ(x,(b))) = (α(θ(x,(b)),θ( n n (x),(b))),θ( n (x),(b))) = (α(θ(x,(b)),θ(0,(b))),θ(0,b)) = (α(θ(x,(b)),(b)),b). Hence, i equation (IV) hold, we get ψϕ = id Xn B. Converely, uppoe 1,1 that ψϕ = id or any plit epimorphim o the orm A A π2 A or any x,y, we have (x,y) = (α(θ(x,y),y),y), and the irt component o thi equality give equation (IV).. Then, Theorem For each emi-abelian variety C, and each n binary operation α i and (n+1)-ary operation θ atiying equation (I)-(II), or any plit epimorphim A B the map ϕ and ψ are bijection between A and the carteian product X n B i and only i equation (IV) i atiied in C. 2. I a emi-abelian variety C atiie equation (IV), then it i poible to deine in C binary operation + and atiying the condition or right Ω-loop. Proo: 1. Thank to the previou propoition, it uice to oberve that equation (I) and (II) imply equation (III); indeed: θ(0,x) = θ(α(x,x),x)= x. 2. The operation + and can be deined in the ollowing way: x+y = θ(α(x,0),y), They atiy the equation o a right Ω-loop: (a) x+0 = θ(α(x,0),0)= x, by equation (II); (b) 0+x = θ(α(0,0),x)= θ(0,x) = x; x y = θ(α(x,y),0).

11 (c) (d) SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 11 (x y)+y = θ(α(x,y),0)+y = θ(α(θ(α(x,y),0),0),y) (by (IV)) = θ(α(x,y),y) = x (by (II)); (x+y) y = θ(α(x,0),y) y = θ(α(θ(α(x,0),y),y),0) (by (IV)) = θ(α(x,0),0)= x (by (II)). Let u oberve that J.R.A. Gray and N. Martin-Ferreira proved in [7] that the exitence o uitable map ϕ and ψ or any plit epimorphim in a variety (giving natural tranormation o uitable unctor between categorie o point) allow to deine operation θ and α i atiying equation (I). Thi mean that, in a emi-abelian variety, or any et o operation (α i,θ) there exit exactly one pair o map (ϕ, ψ) or any plit epimorphim. Theorem 4.2 extend then the reult o [8] and [7], giving a complete characterization o thoe emi-abelian varietie in which the emidirect product o two object X and B (w.r.t. an action o B on X) naturally underlie the carteian product o B and a certain number o copie o X, and, moreover, we how that one ingle copy o X uice. Let u oberve, moreover, that, i a emi-abelian variety C atiie the condition o Theorem 4.2, then the map ϕ and ψ induce bijection between X and X n or any plit epimorphim ( ). I n 2, thi implie that all the algebra o the variety, except the trivial one, are ininite. The ollowing i an example o a variety, with n 2, which atiie equation (IV). Example 4.3. Let C be the variety deined by the ollowing operation: a uniquecontant0, twobinaryoperationα 1 and α 2 and aternaryoperationθ atiying the equation (I), (II) and (IV). A concrete example o an algebra belongingtothivarietyigivenbytheetr N orealequence(butrcanbe replaced by any non-trivial, not necearily ininite, right Ω-loop) equipped with the operation deined by α 1 (x,y) = (x 2n 1 y 2n 1 ) n N = (x 1 y 1,x 3 y 3,...), α 2 (x,y) = (x 2n y 2n ) n N = (x 2 y 2,x 4 y 4,...), θ(x,y,z) = (x 1 +z 1,y 1 +z 2,x 2 +z 3,y 2 +z 4,...),

12 12 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA or any x = (x n ) n N, y = (y n ) n N and z = (z n ) n N in R N. It i immediate to ee that the equation (I) are atiied. Concerning equation (II) we have: θ(α 1 (x,y),α 2 (x,y),y) = θ((x 2n 1 y 2n 1 ) n N,(x 2n y 2n ) n N,y) = (x 1 y 1 +y 1,x 2 y 2 +y 2,x 3 y 3 +y 3,x 4 y 4 +y 4,...) = x. Finally, concerning equation (IV), we have: and α 1 (θ(x,y,z),z) = α 1 ((x 1 +z 1,y 1 +z 2,x 2 +z 3,y 2 +z 4,...),z) = (x 1 +z 1 z 1,x 2 +z 3 z 3,...) = x, α 2 (θ(x,y,z),z) = α 2 ((x 1 +z 1,y 1 +z 2,x 2 +z 3,y 2 +z 4,...),z) = (y 1 +z 2 z 2,y 2 +z 4 z 4,...) = y. Then C atiie the condition o Theorem 4.2, and hence, or any plit extenion ( ) in C, we have that A i in bijection with the carteian product X 2 B. Thi example can be eaily generalized to the cae o any n 2. Oberve that, in thi cae, the operation + and which give the tructure o right Ω-loop are the uual um and ubtraction o equence, repectively, i.e.: x+y = (x n +y n ) n N, x y = (x n y n ) n N. Then C can be een a a emi-abelian variety uing both et o operation {0,+, } and {0,α 1,α 2,θ}. Uing the irt one, we have that, or any plit extenion ( ), the object A i in bijection with X B, while, uing the econd one, A i in bijection with X 2 B. The previou example how that, i a variety can be decribed a emiabelian uing two dierent et o operation, then the two correponding decription o the emidirect product may be dierent. 5. Example In thi ection we preent example that illutrate the reult o Section 3 in the abence o equation (IV). Given a plit epimorphim A B

13 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 13 in a emi-abelian variety C with binary operation α i, i = 1,...,n, and an (n+1)-operation θ atiying equation (I) and (II), diagram ( ): ϕ X X n X ψ X 1,0 X n B k ϕ A ψ 0,1 π B B ϕ B B give an incluion 1,0 ψ X o X into the carteian product X n B. In the ollowing example we how that thi incluion can be o dierent orm. Example 5.1. Let C be the variety o Heyting emilattice, which i deined by a contant and two binary operation and atiying the ollowing equation: x = x, x x = x, x y = y x, x (y z) = (x y) z, (x x) =, x (x y) = x y, y (x y) = y, ψ B x (y z) = (x y) (x z). P.T. Johntone proved in [10] that the variety o Heyting emilattice i emiabelian, with the ollowing operation: α 1 (x,y) = (x y), α 2 (x,y) = (((x y) y) x), θ(x,y,z) = (x z) y. In thi variety, given a plit extenion ( ), we have, or x X: ψ(x) = (α 1 (x,(x)),α 2 (x,(x)),(x))= (α 1 (x, ),α 2 (x, ), ) = = ((x ),(((x ) ) x), ) = (,x, ), hence the incluion o X into X X B i given by the econd incluion o X into the product: X 0,1,0 X X B. Example 5.2. Let C be the variety deined by the ollowing operation: a unique contant 0, a binary ubtraction α (which mean a binary operation α uch that α(x,x) = 0 and α(x,0) = x or any x) and a ternary operation θ atiying the ollowing equation: θ(α(x,y),α(x,y),y)= x.

14 14 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA It i a emi-abelianvariety, with n = 2 and α 1 = α 2 = α. A concrete example o thi ituation i given by the diviible abelian group (Q,+) with α and θ given by: x+y +2z α(x,y) = x y, θ(x,y,z) =. 2 In thi variety, given a plit extenion ( ), we have, or x X: ψ(x) = (α(x,(x)),α(x,(x)),(x))= (α(x,0),α(x,0),0)= (x,x,0), hence the incluion o X into X X B i given by the diagonal o X: X 1,1,0 X X B. Example 5.3. Let C be the emi-abelian variety having a unique contant 0, a binary ubtraction α, a binary um ρ and a ternary operation θ, uch that ρ and α atiy the uual group equation, and, moreover, the ollowing equation i atiied: θ(α(x,y),α(x,y),y)= x. A concrete example o thi ituation i the diviible abelian group (Q, +) conidered in the example above, with ρ, α and θ given by: x+y +2z ρ(x,y) = x+y, α(x,y) = x y, θ(x,y,z) =. 2 There are two way o decribing C a a emi-abelian variety. One i with n = 1, uing the group operation ρ and α. From thi point o view, given a plit extenion ( ), we have that A i in bijection with the carteian product X B. The econd way i with n = 2, uing α 1 = α 2 = α and the ternary operation θ. From thi econd point o view, given a plit extenion ( ), we have that A i in bijection with a ubet o the carteian product X X B. The two point o view are not in contradiction, becaue, a oberved in the previou example, the act that α i a ubtraction implie that the incluion o X into X X B i given by the diagonal o X. 6. The emidirect product o topological algebra F. Borceux and M.M. Clementino proved in [2] that, given a emi-abelian theory T, the category T op(t) o it topological model ha emidirect product in the categorical ene, although it i not a emi-abelian category, becaue it ail to be Barr-exact in general. The reult preented in Section 3 allow u to give an explicit decription o the emidirect product in T op(t),

15 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 15 a we are going to how. Let T be a emi-abelian theory determined by a contant 0, n binary operationα i and an (n+1)-aryoperation θ atiying equation (I) and (II), and let E be a initely complete category. We will denote by E(T) the category o modelo T in E. When E = Set, C i the emi-abelianvarietycorreponding to the theory T. Let U: E(T) E be the (orgetul) unctor which orget the algebraic tructure o any object in E(T). Given a plit epimorphim A B in E(T), we can repeat internally in E the contruction o the map ϕ and ψ tudied in Propoition 3.3, obtaining thu two morphim in E: ϕ: U(X) n U(B) U(A), ψ: U(A) U(X) n U(B), where X i the kernel o. The proo o Propoition 3.3 ue only inite limit, hence it i Yoneda-invariant. Thi mean that, or any initely complete category E and or any plit extenion ( ) in E(T), U(A) i a ubobject o U(X) n U(B). In particular, when E = Top, or any plit extenion ( ) in Top(T) we have that, a a topological pace, A i a ubpace, actually a retract, o the topological product X n B. More explicitly: Propoition 6.1. Given a emi-abelian theory T and a plit extenion ( ) in Top(T), the map ϕ: X n B A and ψ: A X n B, o Propoition 3.3 are continuou. Proo: ϕ and ψ are deined uing only the operation θ and α i, that are continuou becaue they are morphim in T op, and the canonical morphim induced by the product in Top, hence they are continuou. From Propoition 6.1 and Theorem 3.4 it ollow that: Theorem 6.2. Let T be an algebraic theory. Given object B,X and an internal action ξ: B X X on X in Top(T), the emidirect product X ξ B o X and B w.r.t the action ξ i the algebra Y X n B decribed in Theorem 3.4 equipped with the product topology. In particular, i the theory T deine

16 16 M.M. CLEMENTINO, A. MONTOLI AND L. SOUSA a variety o right Ω-loop, then X ξ B i iomorphic, a a topological pace, to the topological product X B. The reult above can be applied not only or E = Top, but or any ubcategory o Top which i cloed under inite product, like the ubcategorie o compact, Haudor, connected, and totally diconnected pace. Moreover, we immediately get that, given a plit extenion ( ) in Top(T), i both X and B are compact, then A alo i, and the ame hold or the propertie o being Haudor, connected and totally diconnected. We conclude by oberving that the lat tatement o Theorem 6.2 actually holdiwe replace Top with anyinitely completecategorye: i the algebraic theory T deine a variety o right Ω-loop, let u conider the category E(T) o it model in E. Given a plit extenion ( ) in E(T), we have that U(A) i iomorphic to U(X) U(B), where U i, a above, the orgetul unctor U: E(T) E. I the category E(T) ha puhout o plit monomorphim, thiact can be ue to provethatit haemidirectproductin the categorical ene recalled in Section 2, uing the ame argument that wa ued in [11] or the cae o internal group and internal groupoid with a ixed object o object. Reerence [1] M. Barr, Exact categorie, in: Lecture Note in Mathematic, vol. 236 (1971), Springer-Verlag, [2] F. Borceux, M.M. Clementino, Topological emi-abelian algebra, Adv. Math. 190 (2005), no. 2, [3] F. Borceux, G. Janelidze, G.M. Kelly, Internal object action, Commentatione Mathematicae Univeritati Carolinae 46 (2005), no. 2, [4] D. Bourn, Normalization equivalence, kernel equivalence and aine categorie, in Lecture Note in Mathematic, vol (1991), Springer-Verlag, [5] D. Bourn, G. Janelidze, Protomodularity, decent, and emidirect product, Theory Appl. Categorie 4 (1998), [6] D. Bourn, G. Janelidze, Characterization o protomodular varietie o univeral algebra, Theory Appl. Categorie 11 (2003), [7] J.R.A. Gray, N. Martin-Ferreira, On algebraic and more general categorie whoe plit epimorphim have underlying product projection, Appl. Categorical Structure, to appear, arxiv: v1. [8] E.B. Inyangala, Semidirect product and croed module in varietie o right Ω-loop, Theory Appl. Categorie 25 (2011), [9] G. Janelidze, L. Márki, W. Tholen, Semi-abelian categorie, J. Pure Appl. Algebra 168 (2002), no. 2-3, [10] P.T. Johntone, A note on the emiabelian variety o Heyting emilattice, Field Intitute Communication 43 (2004),

17 SEMIDIRECT PRODUCTS OF (TOPOLOGICAL) SEMI-ABELIAN ALGEBRAS 17 [11] G. Metere, A. Montoli, Semidirect product o internal groupoid, J. Pure Appl. Algebra 214 (2010), Maria Manuel Clementino CMUC, Department o Mathematic, Univerity o Coimbra, Coimbra, Portugal addre: mmc@mat.uc.pt Andrea Montoli CMUC, Department o Mathematic, Univerity o Coimbra, Coimbra, Portugal addre: montoli@mat.uc.pt Lurde Soua Polytechnic Intitute o Vieu, Portugal & CMUC, Centre or Mathematic o the Univerity o Coimbra, Portugal addre: oua@etv.ipv.pt