A pretorsion theory for right groups

A PRETORSION THEOR Y FOR RIGHT GR OUPS ALBER TO F A CCHINI AND CARMELO ANTONIO FINOCCHIAR O A B S T R AC T . Let S be a right group. Then there e xist two congruences ∼ and ≡ on S such that S is the product of its quotient semigroups S / ∼ and S / ≡ , where S / ∼ is a group and S / ≡ is a right zero semigroup. If E is the set of all idempotents of S and we fix an element e 0 ∈ E , then the pointed right group ( S , e 0 ) is the coproduct of its pointed subsemigroups ( Se 0 , e 0 ) and ( E , e 0 ) in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups. 2020 Mathematics Subject Classification. Primary 18B40. Secondary 18E40, 20M07. 1. I N T RO D U C T I O N Right groups form a widely studied class of semigroups [2, Section 1.11]. Every right group S is, up to isomorphism, the product of a non-empty right zero semigroup E and a group G . In applications, it is often con venient to also consider pointed right groups, instead of right groups. This occurs, for instance, in the study of digroups ([11, Section 4] and [6]). A pointed right gr oup is a pair ( S , e 0 ) , where S is a right group and e 0 is an idempotent element of S . The category of right groups and that of pointed right groups exhibit rather different behavior . Some of the differences are obvious. For ex- ample, the category RGrp of right groups has no initial object, whereas the category RGrp ∗ of pointed right groups has a null object. The main differ - ence between the tw o cate gories, ho wev er, lies in the representation of an object S as a product of a right zero semigroup E and a group G . In the category RGrp ∗ of pointed right groups, this representation S ∼ = E × G is si- multaneously a product and a coproduct. By contrast, in the category RGrp of right groups we are faced with a pretorsion theory [5, 7]: for e very right K e y wor ds and phr ases. Right group, right zero semigroup, pointed right group, pretor- sion theory . The second author was partially supported by GNSA GA, by the research project PIAC- ERI “ A CIV A - Anelli commutativi, loro ideali e variet ` a algebriche” and by the research project PRIN 2022 “Unirationality , Hilbert schemes, and singularities”. 1 2 ALBER TO F A CCHINI AND CARMELO ANTONIO FINOCCHIAR O group S there is a pre-exact sequence E ( S ) i − → S π − → S / ∼ in RGrp , where E ( S ) is the subsemigroup of all idempotent elements of S , ∼ is the smallest congruence on S in which all elements of E ( S ) are pair -wise congruent, and the quotient semigroup S / ∼ is a group (Section 5). Thus, this is a pretorsion theory in the category RGrp of right groups in which the pretorsion objects are the nonempty right zero semigroups, and the torsion-free objects are the groups. Note that this pretorsion theory is not a torsion theory in the cate- gory RGrp of right groups, since on the one hand the category RGrp has no null object but only terminal objects, and on the other hand because in this pretorsion theory there are se veral tri vial morphisms between two objects S and S ′ , one for each idempotent element of S ′ . That is, there is exactly one tri vial morphism from S to S ′ for each idempotent element e ′ of S ′ , namely the morphism constantly equal to e ′ . The tri vial morphisms S → S induce modulo ∼ all and only the morphisms S / ∼ → S in the cate gory RGrp that are right in verses of the canonical projection π : S → S / ∼ . In the cate gory RGrp ∗ of pointed right groups, on the contrary , the canonical projection π : S → S / ∼ has a unique right in verse. The category RGrp ∗ of pointed right groups is equi valent to the product category of the category Set ∗ of pointed sets and the category Grp of groups (Theorem 4.2), and the category RGrp of right groups is equiv alent to the product category of the category Set  = / 0 of non-empty sets and the category Grp (Theorem 4.5). Right groups and pointed right groups form v arieties in the sense of Uni- versal Algebra, of signature (2,2) and ( 2 , 0 , 1 ) respectiv ely (Section 3). T riv- ially , the classes of semigroups that are groups or right groups do not form subv arieties of the variety of signature ( 2 ) of semigroups, since multiplica- ti vely closed subsets of groups or right groups are not, respectiv ely , groups or right groups. 2. P R E L I M I N A RY N O T I O N S A N D T E R M I N O L O G Y 2.1. Direct-product decompositions of semigr oups. F or any pair S , T of semigroups, the external dir ect pr oduct S × T of S and T is the cartesian product endo wed with the componentwise multiplication. As far as internal direct-product decompositions are concerned, notice that in category theory , the fact that S ∼ = A × B ( S is isomorphic to the product of two objects A and B ) is equi v alent to the e xistence of two morphisms ϕ : S → A and ψ : S → B whose product ϕ × ψ : S → A × B is an isomorphism. In this case, ϕ and ψ are necessarily epimorphisms. For semigroups, this means that we must not talk of direct-product decompositions of a semigroup S as a direct product A PRETORSION THEOR Y FOR RIGHT GR OUPS 3 of two subsemigroups A and B of S , b ut of a direct product of S as a direct- product decomposition of two quotients A and B of S . Equi valently , a direct product of S corresponds to (is) a pair ( ∼ , ≡ ) of congruences of S . Remark 2.1. This explains why when we deal with direct-product decom- positions of a group G , we don’t deal with a product-decomposition of G as a direct product of two subgr oups of G , b ut as a direct product of two normal subgroups of G . Similarly , when we deal with direct-product de- compositions of a right group S in Theorem 2.6, we will see that S is the internal direct product of two quotients E and G of S and we will ha ve two projections S → E and S → G . Another example is giv en by the variety of rings with identity . If we construct the external direct product R × S of two rings R and S with identity , then R and S are homomorphic images of R × S , they are not subrings of R × S , because the identities are different. Clearly , the set of all congruences of a semigroup S is a bounded lattice under inclusion with least element the identity congruence = and greatest element the tri vial congruence ω . No w if S is an y set and ∼ , ≡ are two relations on S , it is possible to define their composite r elation ∼ ◦ ≡ , that is { ( a , b ) ∈ S × S | there e xists c ∈ S with a ∼ c and c ≡ b } . The follo wing fact is straightforward (see also [1, Theorem 5.9]). Lemma 2.2. Let S be a semigr oup and ∼ , ≡ be two congruences on S. If ∼ and ≡ ar e permutable (that is, ∼ ◦ ≡ and ≡ ◦ ∼ coincide), then the least upper bound ∼ ∨ ≡ of ∼ and ≡ in the lattice of all congruences of S coincides with ∼ ◦ ≡ . The standard necessary and sufficient properties for internal direct-prod- uct decompositions of a group G as a direct product of two normal sub- groups A and B (that the intersection A ∩ B is trivial and that AB = G ) for a semigroup S become: Definition 2.3. Let S be a semigroup and ∼ , ≡ be two congruences on S . W e say that S = ( S / ∼ ) × ( S / ≡ ) is the dir ect pr oduct of its homomorphic images S / ∼ and S / ≡ if ∼ and ≡ are complementary permutable conguences, that is ∼ ◦ ≡ is equal to ≡ ◦ ∼ and is equal to the trivial congruence ω , and ∼ ∧ ≡ is the identity congruence = . See [1, Theorem 5.9]. The fact that the two congruences ∼ and ≡ must be permutable in Definition 2.3 follo ws from that fact that if A and B are semi- groups, then the e xternal direct products A × B and B × A are isomorphic via the switch s : A × B → B × A , so that there is an isomorphism ϕ : S → A × B 4 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O if and only if there is an isomorphism ψ = e ϕ : S → B × A . Recall that semigroups are not congruence permutable in general, while groups and rings are ([1, Examples on p. 87] and [10]). The condition “ ∼ ◦ ≡ is equal to the tri vial congruence ω ” in Definition 2.3 is equi valent to the fact that the product morphism S → ( S / ∼ ) × ( S / ≡ ) is a surjective mapping, and the condition “ ∼ ∧ ≡ is the identity = ” is equiv- alent to the fact that the product morphism S → S / ∼ × S / ≡ is an injectiv e mapping. Of course, Definition 2.3 applies to any algebra in the sense of Uni versal Algebra and can be extended from the case of two congruences to any finite number of congruences. Definition 2.4. Let A be any algebra and ∼ 1 , . . . , ∼ n be n congruences on A . Then A = A / ∼ 1 × · · · × A / ∼ n is the dir ect pr oduct of its homomorphic images A / ∼ 1 , . . . , A / ∼ n if ∼ 1 , . . . , ∼ n are pairwise permutable conguences that are coindependent , that is ∼ i ∨  ∧ j  = i ∼ j  is equal to the tri vial congru- ence ω on A for e very i = 1 , . . . , n , and ∧ n i = 1 ∼ i is the identity congruence = . For related results, see [8, Proposition 6.4], [3, Sections 2.6 and 2.8] and [4, p. 94]. Also, compare Definition 2.4 with the notion of semidirect- product decomposition of an algebra A studied in [9]. Gi ven an y algebra A , any congruence ∼ on A and any subalgebra B of A , then A = ∼ ⋊ B is the semidir ect pr oduct of the congruence ∼ and the subalgebra B if the composite morphism π ι B : B → A / ∼ is an isomorphism, where ι B : B → A is the inclusion and π : A → A / ∼ is the canonical projection. 2.2. Right groups. W e will make use of the notation, the terminology and se veral results in [11, Section 4] and [2]. A semigroup ( S , · ) is a right zer o semigr oup [2, p. 4] if a · b = b for all a , b ∈ S . For these semigroups we will usually write the operation · as π 2 , because it corresponds to the second canonical projection π 2 : S × S → S . Thus a π 2 b = b . Similarly , a π 1 b = a . Hence right zero semigroups are those of the form ( S , π 2 ) for some set S . The full subcate gory of the category of semigroups whose objects are all right zero semigroups is clearly isomorphic to the category Set of sets, be- cause ev ery mapping between two right zero semigroups is a semigroup morphism. For an arbitrary semigroup S , let L be the set of all left identities of S , that is, L : = { e ∈ S | e x = x for e very x ∈ S } . Then L is a subsemigroup of S , the operation induced on it by the operation · of S is the operation π 2 , and we hav e an embedding ( L , π 2 ) → ( S , · ) . For ev ery e ∈ L , we can consider the centralizer C S ( e ) : = { x ∈ S | ex = xe } of e in S . Then each C S ( e ) is also a subsemigroup of S , which contains e as a two-sided identity , so that C S ( e ) is A PRETORSION THEOR Y FOR RIGHT GR OUPS 5 a monoid. Moreover , each C S ( e ) is a left ideal for S , that is y C S ( e ) ⊆ C S ( e ) for e very y ∈ S . If S is either left cancellati ve or right cancellativ e, then the monoids C S ( e ) are pair-wise disjoint, because if x ∈ C S ( e ) ∩ C S ( e ′ ) , then ex = xe = x and e ′ x = xe ′ = x , so e = e ′ by one-sided cancellati vity . The union of all these left ideals C S ( e ) is a left ideal of S , hence in particular a subsemigroup of S . This prov es that if the union of all these left ideals C S ( e ) is the semigroup S itself and S is either left cancellativ e or right cancellati ve, then the monoids C S ( e ) form a partition of S , and the embedding ( L , π 2 ) → ( S , · ) has a left in verse ( S , · ) → ( L , π 2 ) . This left in verse of the embedding is the semigroup morphism that maps all elements of the block C S ( e ) of the partition to e . The class of right groups, which we will introduce with Theorem 2.6, is a natural class of left cancellativ e semigroups S that hav e these properties, that is, such that S is the union of all centralizers of the left identities of S . Gi ven a semigroup ( S , · ) , we will denote by E ( S , · ) the set of all idempo- tents of S . Recall the follo wing important lemma. Lemma 2.5. [2, Lemma 1.26] Every idempotent of a right simple semi- gr oup S is a left identity for S . W e will no w collect sev eral equi valent conditions for a semigroup to be a right group, most of them well known, and we will provide a sketch of the proof for con venience of the reader . W e will emphasize some machinery regarding the algebraic-theoretic features of right groups that will be helpful in the rest of the paper . Theorem 2.6. [2, Section 1.11, Theorem 1.27] The following assertions on a semigr oup S  = / 0 ar e equivalent: (a) S is right simple (that is, aS = S for all a ∈ S ) and left cancellative. (b) F or every a , b ∈ S ther e e xists a unique element x ∈ S such that ax = b. (c) S is right simple and contains an idempotent. (d) S is isomorphic to the e xternal dir ect pr oduct of a non-empty right zer o semigr oup E and a gr oup G. (e) Ther e exists an element e ∈ S suc h that: (1) e is a left identity for S, and (2) every element of S has a right in verse with r espect to e. (f) (1) S has a left identity , and (2) for every left identity e of S and every element a ∈ S, a has a right in verse with r espect to e. The semigroups satisfying the pre vious equi valent conditions are called right gr oups. Pr oof. (a) ⇔ (b) ⇔ (c) ⇐ (d). See [2, Section 1.11, Theorem 1.27]. (c) ⇒ (d). This immediately follows from Lemma 2.5. Let E be the set of all idempotents of S . By Lemma 2.5 ev ery element of E is a left identity 6 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O for S , that is, ea = a for ev ery e ∈ E and e very a ∈ S . In particular , E is a non-empty subsemigroup of S and is a right zero semigroup. Then one fixes any element e 0 ∈ E and takes for G the left ideal S e 0 , which turns out to be a group with identity e 0 . Then it is easy to check that the mapping G × E → S defined by ( a , e ) 7→ ae for e very ( a , e ) ∈ G × E is a semigroup isomorphism between the external direct product G × E and the semigroup S . That is, e very Se 0 is a complement of E in S . This pro ves (d). (c) ⇒ (f). Assume that (c) holds for the semigroup S . By condition (c) and [2, Lemma 1.26], S has a left identity . Suppose no w that f is an y left identity of S and let a ∈ S . By condition (b), the equation ax = f has a unique solution and, by definition, such a solution is a right in verse of a with respect to f . This prov es that (c) ⇒ (f). (f) ⇒ (e) is tri vial. (e) ⇒ (c). Let us suppose that (e) holds, let e be a left identity of S and, for e very a ∈ S , let a − 1 ∈ S denote a right in verse of a with respect to e . Given elements a , s ∈ S , we hav e s = es = aa − 1 s . This proves that aS = S for all a ∈ S , that is, S is right simple. Furthermore, e is trivially an idempotent of S . The conclusion is no w clear . □ Remark 2.7. The pr ojection π G : S → G, its kernel ∼ , and the quotient gr oup S / ∼ . In Theorem 2.6(d), the projections π G : S → G and π E : S → E are defined by x 7→ ( x − 1 ) − 1 and x 7→ ( x − 1 ) x respecti vely . Here x − 1 denotes the right in verse of x ∈ S with respect to a fixed element e 0 ∈ E (i.e., x − 1 is the unique element in S such that x ( x − 1 ) = e 0 .) By associativity and con- sidering x ( x − 1 )( x − 1 ) − 1 , one sees that e 0 ( x − 1 ) − 1 = xe 0 . Since all elements of E are left identities in S , it follows that ( x − 1 ) − 1 = xe 0 . The projec- tion π G is a semigroup morphism, because, for e very x , y ∈ S , we have that xe 0 ye 0 = xye 0 . The kernel of π G is the congruence ∼ on S defined, for ev- ery x , y ∈ S , by x ∼ y if xe 0 = ye 0 . Notice that this congruence ∼ does not depend on the choice of the idempotent element e 0 ∈ E , because for any other f ∈ E one has that, for all x , y ∈ S , xe 0 = ye 0 if and only if x f = y f (because multiplying xe 0 = ye 0 by f on the right we get that xe 0 f = ye 0 f , that is x f = y f ; similarly , multiplying by e 0 on the right, x f = y f implies xe 0 = ye 0 ). The congruence class modulo ∼ of ev ery element x ∈ S is xE . The pr ojection π E : S → E , its kernel ≡ , and the quotient right zer o semi- gr oup S / ≡ . As we hav e said in the previous paragraph, the projection π E : S → E is defined by x 7→ ( x − 1 ) x . The image π E ( x ) = ( x − 1 ) x of x ∈ S is an idempotent element of S , because (( x − 1 ) x ) 2 = x − 1 xx − 1 x = x − 1 e 0 x = x − 1 x . W e ha ve that π E ( x ) = ( x − 1 ) x is the unique element f ∈ E such that x f = x . (This characterization also immediately sho ws that f = π E ( s ) be- longs to E , because s f f = s f = s . It also sho ws that the projection π E , like the congruence ∼ , does not depend on the choice of the fixed element e 0 .) A PRETORSION THEOR Y FOR RIGHT GR OUPS 7 The projection π E is a semigroup morphism, because, for ev ery x , y ∈ S , if f is the unique idempotent such that y f = y , then also xy f = xy , so that π E ( xy ) = f = π E ( x ) π E ( y ) . The kernel of the projection S → E is the congru- ence relation ≡ on S for which the quotient semigroup S / ≡ is { Se | e ∈ E } . Notice that Ge = Se 0 e = S e for e very e ∈ E . Therefore ev ery right group S has a partition { Se | e ∈ E } , where ev ery block Se of the partition is a group with identity e , so that E is the set of all the identities of these groups Se . Moreov er all the groups Se are pair-wise isomorphic via the group isomor- phism r f : Se → S f , r f ( x ) = x f for all e , f ∈ E , gi ven by right multiplication by f . No w that we have the two congruences ∼ and ≡ on an y right group S , it is clear that the product decomposition of S corresponding to the pair ( ∼ , ≡ ) in the sense of Definition 2.3 is the product decomposition of S as a right zero semigroup E and a group G of Theorem 2.6.(d) Example 2.8. Let us gi ve an e xample concerning the equi valent statements of Theorem 2.6. Let S be any nonempty set, and consider the semigroup ( S , π 2 ) , which is ob viously a right zero semigroup. Then S is right simple, because a π 2 S = S for all a ∈ S ; and S is left cancellativ e, because a π 2 b = a π 2 c is b = c . For e very a , b ∈ S there e xists a unique element x ∈ S such that ax = b , it is b . Every element of S is idempotent. Every element of S is a left identity . For any two elements e , a ∈ S , the right in verse of a with respect to e is e . 3. C A T E G O R I E S O F R I G H T G RO U P S The category of gr oup actions on sets has, as objects, all triplets ( G , X , ϕ ) , where G is a group, X is a set, and ϕ : G → Aut Set ( X ) , g 7→ ϕ g , is a group morphism. Clearly Aut Set ( X ) is the symmetric group Sym X . A morphism ( G , X , ϕ ) → ( H , Y , ψ ) is a pair ( Φ , f ) , where Φ : G → H is a group mor- phism, f : X → Y is a mapping, and ψ Φ ( g ) ( f ( x )) = f ( ϕ g ( x )) for e very g ∈ G and x ∈ X . It is immediately seen that if ( Φ , f ) : ( G , X , ϕ ) → ( H , Y , ψ ) is a morphism in the category of group actions on sets, then ( Φ , f ) is an iso- morphism if and only if Φ is an isomorphism of groups and f is a bijection. If 1 : G → Aut Set ( X ) is the trivial group morphism, we will say that ( G , X , 1 ) is the tri vial group action of G on X . Similarly , the cate gory of gr oup actions on pointed sets has, as objects, all 4-tuples ( G , X , x 0 , ϕ ) , where G is a group, X is a set, x 0 is a fixed element of X , (so that ( X , x 0 ) is a pointed set), and ϕ : G → Aut Set ∗ ( X , x 0 ) is a group morphism. Here Aut Set ∗ ( X , x 0 ) is simply the group of all permutations of X that fix x 0 , that is, all bijections f : X → X such that f ( x 0 ) = x 0 . A morphism ( G , X , x 0 , ϕ ) → ( H , Y , y 0 , ψ ) is a pair ( Φ , f ) , where Φ : G → H is a group 8 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O morphism, f : X → Y is a mapping, f ( x 0 ) = y 0 , and ψ Φ ( g ) ( f ( x )) = f ( ϕ g ( x )) for e very g ∈ G and x ∈ X . The cate gory of pointed right gr oups has as objects the pairs ( S , e 0 ) , where S is a right group and e 0 is an idempotent element of S , and as mor- phisms f : ( S , e 0 ) → ( S ′ , e ′ 0 ) all semigroup morphisms f : S → S ′ such that f ( e 0 ) = e ′ 0 . Notice that pointed right groups ( S , · , e 0 , − 1 ) do form a v ariety of alge- bras, where · is binary and associativ e, e 0 is 0-ary , − 1 is unary , e 0 x = x for e very x ∈ S , and xx − 1 = e 0 for ev ery x ∈ S (Theorem 2.6(e)). Also right groups, viewed as algebras ( S , · , \ ) with two binary operations · and \ and the three identities x · ( y · z ) = ( x · y ) · z , x · ( x \ y ) = y and x \ ( x · y ) = y , form a v ariety in the sense of Univ ersal Algebra. Proposition 3.1. If S is a right gr oup, e ∈ E : = E ( S ) , ι e : Se → S and ι E : E → S are the inclusions, r e : S → Se is defined by r e ( s ) = se for ev- ery s ∈ S , and π E : S → E is defined by “ π E ( s ) is the unique element of S such that s π E ( s ) = s”, then the decomposition S = Se × E shows that: (1) ( S , r e , π E ) is the pr oduct of Se and E in the cate gory of semigr oups. (2) (( S , e ) , ι e , ι E ) is the copr oduct of ( S e , e ) and ( E , e ) in the category of pointed right gr oups. (3) ( S , ι e , ι E ) is not the copr oduct of Se and E in the cate gory of semi- gr oups. Pr oof. The mapping ϕ : S → Se × E in the external direct product Se × E of the homomorphic images Se and E of S , defined by ϕ ( s ) = ( se , π E ( s )) for e very s ∈ S , is a semigroup isomorphism, because ϕ ( ss ′ ) = ( ss ′ e , π E ( ss ′ )) = ( ses ′ e , π E ( s ′ )) = = ( se , π E ( s ))( s ′ e , π E ( s ′ )) = ϕ ( s ) ϕ ( s ′ ) , and the in verse of ϕ is the mapping ϕ − 1 : Se × E → S defined by ϕ − 1 ( x , y ) = xy for e very ( x , y ) ∈ Se × E . In order to prove (1), fix any semigroup T and semigroup morphisms f e : T → Se and f E : T → E . Define a mapping f : T → S setting f ( t ) = f e ( t ) f E ( t ) for ev ery t ∈ T . Then f is a semigroup morphism, because, for e very t , t ′ ∈ T , we hav e that f ( t ) f ( t ′ ) = f e ( t ) f E ( t ) f e ( t ′ ) f E ( t ′ ) = = f e ( t ) f e ( t ′ ) f E ( t ′ ) = f e ( t t ′ ) f E ( t t ′ ) = f ( t t ′ ) . Also r e f = f e and π E f = f E , because, for e very t ∈ T , we hav e that r e f ( t ) = f e ( t ) f E ( t ) e = f e ( t ) e = f e ( t ) and π E f ( t ) = π E ( f e ( t ) f E ( t )) = f E ( t ) . As far as uniqueness of f is concerned, let g : T → S be any other semi- group morphism such that r e g = f e and π E g = f E . Then, for e very t ∈ T we A PRETORSION THEOR Y FOR RIGHT GR OUPS 9 hav e that g ( t ) = ϕ − 1 ϕ g ( t ) = ϕ − 1 ( g ( t ) e , π E ( g ( t ))) = ϕ − 1 ( r e g ( t ) , π E g ( t )) = ϕ − 1 ( f e ( t ) , f E ( t )) = f e ( t ) f E ( t ) = f ( t ) , so that g = f . As far as (2) is concerned, let ( T , e T ) be a pointed right group and f e : S e → T and f E : E → T be two semigroup morphisms with f e ( e ) = f E ( e ) = e T . Define a mapping ψ : S → T setting ψ ( s ) = f e ( se ) f E ( π E ( s )) for ev ery s ∈ S . Notice that all elements of E are idempotent, so that ev ery element of the image f E ( E ) is an idempotent element of T . By Lemma 2.5, e very element of f E ( E ) is a left identity for T . The mapping ψ is a semi- group morphism, because for e very s , s ′ ∈ S we hav e that ψ ( ss ′ ) = f e ( ss ′ e ) f E ( π E ( ss ′ )) = = f e ( ses ′ e ) f E ( π E ( s ′ )) = f e ( se ) f e ( s ′ e ) f E ( π E ( s ′ )) = = f e ( se ) f E ( π E ( s )) f e ( s ′ e ) f E ( π E ( s ′ )) = ψ ( s ) ψ ( s ′ ) . Moreov er , for ev ery s ∈ S we hav e that ψ ι e ( se ) = ψ ( se ) = f e ( se ) f E ( e ) = f e ( se ) f e ( e ) = f e ( se ) , so that ψ ι e = f e . Similarly , ψ ι E ( e ′ ) = ψ ( e ′ ) = f e ( e ′ e ) f E ( π E ( e ′ )) = f e ( e ) f E ( e ′ ) = e T f E ( e ′ ) = f E ( e ′ ) , hence ψ ι E = f E . For uniqueness, let ψ ′ : S → T be any other semigroup morphism such that ψ ′ ι e = f e and ψ ′ ι E = f E . Then, for every s ∈ S we hav e that s = ( se ) π E ( s ) with se ∈ S e and π E ( s ) ∈ E , so ψ ′ ( s ) = ψ ′ ( se ) ψ ′ ( π E ( s )) = ψ ′ ι e ( se ) ψ ′ ι E ( π E ( s )) = f e ( se ) f E ( π E ( s )) = ψ ( s ) . Therefore ψ ′ = ψ . For (3), let A = { x , y } be a set of two elements and W the free semi- group freely generated by the set A , so that W is the semigroup of all words of length ≥ 1 in the alphabet A with respect to justapposition. Let ≃ be the congruence on W generated by the set of the two pairs ( x , xx ) and ( y , yy ) . Set T : = W / ≃ , so that its two elements [ x ] ≃ and [ y ] ≃ are two dis- tinct idempotents. Consider the constant semigroup morphisms f e : Se → T and f E : E → T defined by f e ( se ) = [ x ] ≃ and f e ( e ′ ) = [ y ] ≃ for e very se ∈ Se and e ′ ∈ E . Then there is no mapping ω : S → T such that ω ι e = f e and ω ι E = f E , because ω ( e ) = ω ι e ( e ) = f e ( e ) = [ x ] ≃ and ω ( e ) = ω ι E ( e ) = f E ( e ) = [ y ] ≃ , a contradiction. □ The category of pointed right groups is a category with a zero object and, by Proposition 3.1, ( S , r e , π E , ι e , ι E ) is a biproduct. Corollary 3.2. If S is a right gr oup, π ∼ : S → S / ∼ is defined by π E ( s ) = sE for e very s ∈ S, and π E : S → E is defined by “ π E ( s ) is the unique element of S such that s π E ( s ) = s”, then ( S , π ∼ , π E ) is the pr oduct of S / ∼ and E in the cate gory of semigr oups. 10 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O The proof follows from the isomorphism r e : S / ∼ → Se and Proposi- tion 3.1(1). The dif ference between right groups and pointed right groups is not stated so explicitly in [11]. Theorem 3.3. Ther e is a faithful, essentially surjective functor fr om the cate gory of gr oup actions on pointed sets to the category of pointed right gr oups. Pr oof. Associate to any group action ( G , X , x 0 , ϕ ) on the pointed set ( X , x 0 ) the semigroup ( X × G ) ϕ : = X × G with the operation defined by ( x , g )( x ′ , g ′ ) = ( ϕ g ( x ′ ) , gg ′ ) for all ( x , g ) , ( x ′ , g ′ ) ∈ X × G . It is v ery easy to check that this semigroup X × G is a right group. It is pointed relativ ely to its idempotent element ( x 0 , 1 G ) . Gi ven any morphism ( G , X , x 0 , ϕ ) → ( H , Y , y 0 , ψ ) in the category of left group actions on pointed sets, we ha ve that the morphism is a pair ( Φ , f ) with Φ : G → H a group morphism, f : X → Y a mapping, f ( x 0 ) = y 0 and ψ Φ ( g ) ( f ( x )) = f ( ϕ g ( x )) for e very g ∈ G and ev ery x ∈ X . It is possi- ble to associate to ( Φ , f ) : ( G , X , ϕ ) → ( H , Y , ψ ) the semigroup morphism F ( Φ , f ) = f × Φ : X × G → Y × H defined by ( f × Φ )( x , g ) = ( f ( x ) , Φ ( g )) for all x ∈ X , g ∈ G . Clearly , the mapping ( Φ , f ) → F ( Φ , f ) = f × Φ is injecti ve. That is, we have a faithful functor F defined by F ( G , X , x 0 , ϕ ) = ( X × G , ( x 0 , 1 G )) and F ( Φ , f ) = f × Φ . Gi ven any pointed right group ( S , e 0 ) , associate to it the 4-tuple ( Se 0 , E ( S ) , x 0 , 1 ) , where 1 : Se 0 → Sym E ( S ) is the tri vial action, for which 1 se 0 : E ( S ) → E ( S ) is the identity mapping of E ( S ) for every se 0 ∈ Se 0 . Then F ( S e 0 , E ( S ) , x 0 , 1 ) = E ( S ) × S e 0 , the semigroup with operation ( x , se 0 )( x ′ , s ′ e 0 ) = ( x ′ , ss ′ e ′ 0 ) , so that F ( Se 0 , E ( S ) , x 0 , 1 ) ∼ = S . This prov es that F is essentially surjecti ve. □ Theorem 3.4. Ther e is a faithful, essentially surjective functor fr om the cate gory of gr oup actions on non-empty sets to the cate gory of right gr oups. Pr oof. The proof is similar to that of Theorem 3.3. Define a functor F of the category of left group actions on non-empty sets to the category of right groups as follows. Associate to any left group action ( G , X , ϕ ) on a non-empty set X the semigroup ( X × G ) ϕ : = X × G with the operation defined by ( x , g )( x ′ , g ′ ) = ( ϕ g ( x ′ ) , gg ′ ) for all ( x , g ) , ( x ′ , g ′ ) ∈ X × G . It is very easy to check that X × G with this operation is a right group. Set F ( G , X , ϕ ) = ( X × G ) ϕ . Gi ven any morphism ( G , X , ϕ ) → ( H , Y , ψ ) in the A PRETORSION THEOR Y FOR RIGHT GR OUPS 11 category of left group actions on non-empty sets, we hav e that the mor- phism is a pair ( Φ , f ) with Φ : G → H a group morphism, f : X → Y a mapping, and ψ Φ ( g ) ( f ( x )) = f ( ϕ g ( x )) for ev ery g ∈ G and every x ∈ X . It is possible to associate to ( Φ , f ) : ( G , X , ϕ ) → ( H , Y , ψ ) the semigroup mor- phism F ( Φ , f ) = f × Φ : ( X × G ) ϕ → ( Y × H ) ψ defined by ( f × Φ )( x , g ) = ( f ( x ) , Φ ( g )) for all x ∈ X , g ∈ G . In this way we get a faithful functor from the category of group actions on non-empty sets to the category of right groups. Gi ven any right group S , consider the triplet ( S / ∼ , E ( S ) , 1 ) , where ∼ is the congruence on S defined by x ∼ y if xe = ye for some idempotent ele- ment e (see Remark 2.7; the congruence class of x ∈ S modulo the congru- ence ∼ is xE ), and 1 : S / ∼ → Aut Set ( E ( S )) is the trivial group morphism. Then F ( S / ∼ , E ( S ) , 1 ) = ( E ( S ) × S / ∼ ) 1 = E ( S ) × S / ∼ ∼ = S , in view of Theorem 2.6 and Remark 2.7. This pro ves that F is essentially surjecti ve. □ Proposition 3.5. Let F be the functor defined in Theor em 3.4, let G be a gr oup and X any nonempty set. Then, for every gr oup morphism ϕ : G → Aut Set ( X ) , the right gr oups F ( G , X , ϕ ) and F ( G , X , 1 ) ar e isomorphic, wher e 1 is the trivial gr oup morphism. Pr oof. Let η : F ( G , X , 1 ) → F ( G , X , ϕ ) be the mapping defined by setting η ( x , g ) : = ( ϕ g ( x ) , g ) for all x ∈ X , g ∈ G . It is straightforward to see that η is an isomorphism of right groups. □ Remark 3.6. Observe that the functor F defined in Theorem 3.4 is not full. As a matter of fact, consider a group G and a nonempty set X in such a way there exists a nontrivial group morphism ϕ : G → Aut Set ( X ) . By Proposition, 3.5, the right groups F ( G , X , ϕ ) and F ( G , X , 1 ) are isomorphic. On the other hand, the group actions ( G , X , ϕ ) , ( G , X , 1 ) are not isomorphic: indeed, if there e xists an isomorphism ( Φ , f ) : ( G , X , ϕ ) → ( G , X , 1 ) , for some group automorphism Φ : G → G and some bijection f : X → X , then it would follow (from the definition of morphism in the category of group actions) that f ( x ) = f ( ϕ g ( x )) , for all x ∈ X , g ∈ G , and this would force (since f is bijecti ve) each ϕ g to be tri vial, that is, ϕ = 1 , a contradiction. This sho ws that F is not full. Similarly , it can be seen that the functor defined in Theorem 3.3 is not full either . 4. H O M O M O R P H I S M S O F R I G H T G RO U P S The results in Subsection 2.2 and Section 3 suggest to in vestigate semi- group morphisms ϕ : S → S ′ between two right groups S , S ′ . Let 12 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O Hom SGrp ( S , S ′ ) denote the set of all such morphisms. Since images of idem- potents via semigroup morphisms are idempotents, we ha ve that ϕ maps the set E = E ( S ) of all idempotents of S into the set E ′ = E ( S ′ ) of all idempo- tents of S ′ . Thus it is possible to consider the restriction ε : E → E ′ of ϕ to E . Fix an element e 0 ∈ E . Then ϕ maps the group Se 0 to the group S ′ ε ( e 0 ) , so that it is possible to consider the restriction ϕ | Se 0 : Se 0 → S ′ ε ( e 0 ) , which is a group morphism. Proposition 4.1. Let S , S ′ be right gr oups. F or every semigr oup morphism ϕ : S → S ′ consider the triplet ( ε : E → E ′ , e 0 , ϕ | Se 0 : S e 0 → S ′ ε ( e 0 )) de- scribed in the pr evious para graph. Then: (a) The triplet ( ε : E → E ′ , e 0 , ϕ | Se 0 : Se 0 → S ′ ε ( e 0 )) completely deter- mines the semigr oup morphisms ϕ : S → S ′ . (b) F or every mapping ε : E → E ′ , any element e 0 ∈ E and any gr oup morphism ψ : S e 0 → S ′ ε ( e 0 ) , the triplet ( ε , e 0 , ψ ) corresponds to a semi- gr oup morphism ϕ : S → S ′ , that is, ther e e xists a semigr oup morphism ϕ : S → S ′ such that ε ( e ) = ϕ ( e ) for every e ∈ E and ψ ( x ) = ϕ ( x ) for every x ∈ Se 0 . (c) T wo such triplets ( ε 1 , e 1 , ψ 1 ) , ( ε 2 , e 2 , ψ 2 ) correspond to the same semi- gr oup morphism ϕ : S → S ′ if and only if ε 1 = ε 2 and the diagr am (1) Se 1 ψ 1 / / r e 2   S ′ ε 1 ( e 1 ) r ε 2 ( e 2 )   Se 2 ψ 2 / / S ′ ε 2 ( e 2 ) commutes. Pr oof. (a) For a giv en semigroup morphism ϕ : S → S ′ consider the triplet ( ε : E → E ′ , e 0 , ϕ | Se 0 : Se 0 → S ′ ε ( e 0 )) . If we consider the pointed right groups ( S , e 0 ) and ( S ′ , ε ( e 0 )) , ϕ : S → S ′ becomes a morphism ϕ : ( S , e 0 ) → ( S ′ , ε ( e 0 )) of pointed right groups. In vie w of Proposition 3.1((1) and (2)), S is both a product and a coproduct of Se 0 and E . Similarly , S ′ is both a product and a coproduct of S ′ ε ( e 0 ) and E ′ . Now e very element s ∈ S can be written in a unique way as a product of the element se 0 of Se 0 and the element π E ( s ) of E . Therefore (2) ϕ ( s ) = ϕ (( se 0 )( π E ( s ))) = ϕ ( se 0 ) ϕ ( π E ( s )) = ϕ | Se 0 ( se 0 ) ε ( π E ( s )) . Hence the triplet ( ε : E → E ′ , e 0 , ϕ | Se 0 : Se 0 → S ′ ε ( e 0 )) completely deter- mines ϕ : S → S ′ . A PRETORSION THEOR Y FOR RIGHT GR OUPS 13 (b) Gi ven a triplet ( ε , e 0 , ψ ) as in (b), define ϕ : S → S ′ setting, for ev ery s ∈ S , ϕ ( s ) = ψ ( se 0 ) ε ( π E ( s )) . Then ϕ is a semigroup morphism, because ϕ ( s 1 ) ϕ ( s 2 ) = ψ ( s 1 e 0 ) ε ( π E ( s 1 )) ψ ( s 2 e 0 ) ε ( π E ( s 2 )) = = ψ ( s 1 e 0 ) ψ ( s 2 e 0 ) ε ( π E ( s 2 )) = ψ ( s 1 s 2 e 0 ) ε ( π E ( s 1 ) π E ( s 2 )) = = ϕ ( s 1 s 2 ) . (c) T wo triplets ( ε 1 , e 1 , ψ 1 ) , ( ε 2 , e 2 , ψ 2 ) correspond to the same semigroup morphism ϕ : S → S ′ if and only if ϕ ( s ) = ψ 1 ( se 1 ) ε 1 ( π E ( s )) = ψ 2 ( se 2 ) ε 2 ( π E ( s )) for e very s ∈ S . Hence if ( ε 1 , e 1 , ψ 1 ) , ( ε 2 , e 2 , ψ 2 ) correspond to the same semigroup morphism ϕ : S → S ′ , then ε 1 = ε 2 , because the y are both the re- striction of ϕ to E . Moreover Diagram (1) commutes because, for e very s ∈ S , r ε 2 ( e 2 ) ψ 1 ( se 1 ) = r ϕ ( e 2 ) ϕ ( se 1 ) = ϕ ( se 1 ) ϕ ( e 2 ) = ϕ ( se 1 e 2 ) = ϕ ( se 2 ) and ψ 2 r e 2 ( se 1 ) = ψ 2 ( se 1 e 2 ) = ψ 2 ( se 2 ) = ϕ ( se 2 ) . Therefore r ε 2 ( e 2 ) ψ 1 = ψ 2 r e 2 , and the diagram commutes. For the con verse, we must prov e that if ε 1 = ε 2 and Diagram (1) com- mutes, then ψ 1 ( se 1 ) ε 1 ( π E ( s )) = ψ 2 ( se 2 ) ε 2 ( π E ( s )) for every s ∈ S . Now ε 1 = ε 2 and the commutati vity of the diagram imply that r ε 1 ( e 2 ) ψ 1 ( se 1 ) = ψ 2 r e 2 ( se 1 ) , that is, ψ 1 ( se 1 ) ε 1 ( e 2 ) = ψ 2 ( se 2 ) for ev ery s ∈ S . Multiplying on the right by ε 1 ( π E ( s )) = ε 2 ( π E ( s )) , we get that ψ 1 ( se 1 ) ε 1 ( π E ( s )) = ψ 2 ( se 2 ) ε 2 ( π E ( s )) , as desired. □ Notice that the vertical arro ws in Diagram (1) are group morphisms. More generally , for any two elements e , f ∈ E = E ( S ) , where S is right group, the mapping r f : Se → S f , r f ( s ) = s f for all s ∈ Se , is a group iso- morphism, because r f ( se ) r f ( s ′ e ) = se f s ′ e f = ss ′ e f = ses ′ e f = r f (( se )( s ′ e )) . Its in verse is the mapping r e : S f → Se . This is the reason why the restriction ϕ | Se 0 determines the behavior of ϕ on all the blocks of the partition { Se | e ∈ E } of S . From Proposition 4.1 it can be easily sho wn that: Theorem 4.2. The cate gory RGrp ∗ of pointed right gr oups is equivalent to the pr oduct cate gory Set ∗ × Grp of the cate gory Set ∗ of pointed sets and the cate gory Grp of gr oups. 4.1. Right inv erses ϕ : S / ∼ → S to the canonical projection π : S → S / ∼ . In order to try to limit as much as possible the need of introducing the artificial concept of pointed right group, it is con venient to consider the right in verses of the canonical projection π : S → S / ∼ . Here S is a right group and ∼ is the semigroup congruence on S considered in Remark 2.7. For any e ∈ E , ∼ is the kernel of the semigroup morphism r e : S → S defined 14 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O by r e ( x ) = xe for every x ∈ S . The congruence class modulo ∼ of ev ery element x ∈ S is xE = { xe | e ∈ E } . For e very right group S , the quotient semigroup S / ∼ is a group. If ψ : S → S ′ is any semigroup morphism between two right groups S and S ′ , then ψ induces a group morphism e ψ : S / ∼ → S ′ / ∼ . T o see it, fix an element e 0 ∈ E . If s , t ∈ S and s ∼ t , then se 0 = t e 0 , so ψ ( s ) ψ ( e 0 ) = ψ ( t ) ψ ( e 0 ) . Then ψ ( s ) ∼ ψ ( t ) . Thus e ψ is a well defined mapping. Thus S 7→ S / ∼ , ψ 7→ e ψ , is a functor RGrp → Grp . Proposition 4.3. F or a right gr oup S, ther e is a one-to-one corr espondence between the set of the semigr oup morphisms that are right in verses of the canonical pr ojection π : S → S / ∼ and the set E ( S ) . If e 0 ∈ E ( S ) , the right in verse homomorphism of π corr esponding to e 0 is the semigr oup morphism r e 0 : S / ∼ → S induced by right multiplication r e 0 : S → S by e 0 . Pr oof. If ϕ : S / ∼ → S is any semigroup morphism such that π ϕ = id S / ∼ , then ϕ must map the identity E of the group S / ∼ to an idempotent element e 0 of S , that is, to an element e 0 ∈ E : = E ( S ) . W e ha ve the direct-product decomposition S = S e 0 × E and, correspondingly , the trivial direct-product decomposition S / ∼ = ( S / ∼ ) E × { E } . Let us sho w that the restriction π | Se 0 : Se 0 → S / ∼ = ( S / ∼ ) E of the canonical projection π : S → S / ∼ is a (semi)group isomorphism. The mapping π | Se 0 is defined by π | Se 0 ( se 0 ) = se 0 E = sE for ev ery se 0 ∈ S e 0 . It is injectiv e because if s , s ′ ∈ S and sE = s ′ E , then, multiplying by e 0 on the right we get that se 0 = s ′ e 0 . It is surjec- ti ve, because if sE ∈ S / ∼ , then, multiplying s by e 0 on the right, we see that π | Se 0 ( se 0 ) = sE . Therefore π | Se 0 is an isomorphism. In the notations of Proposition 4.1, the triplet ( ε : E ( S / ∼ ) → E ( S ) , E , ϕ | S / ∼ : S / ∼ → Se 0 ) corresponding to the semigroup morphism ϕ : S / ∼ → S is such that ε : E ( S / ∼ ) → E ( S ) maps the unique element E of E ( S / ∼ ) to e 0 . Also, π ϕ = id S / ∼ implies π | Se 0 ϕ | Se 0 = id S / ∼ . Since π | Se 0 : Se 0 → S / ∼ = ( S / ∼ ) E is an isomorphism, it follo ws that π | Se 0 : Se 0 → S / ∼ and ϕ | Se 0 : S / ∼ → Se 0 are mutually in verse group isomorphisms. Thus ϕ | Se 0 : S / ∼ → Se 0 maps any element sE of S / ∼ to se 0 . Now r e 0 : S → S induces an injectiv e homo- morphism r e 0 : S / ∼ → S , and it is easily seen that r e 0 also corresponds to the same triplet ( ε , E , ϕ | S / ∼ ) as ϕ . □ Proposition 4.4. (a) If ( S , · ) is a right gr oup and e , f ∈ E ( S ) , then ( S , · , e ) and ( S , · , f ) ar e isomorphic pointed right gr oups. (b) The for getful functor RGrp ∗ → RGrp that associates to each pointed right gr oup ( S , · , e ) the right gr oup ( S , · ) is a faithful, essentially surjective functor . A PRETORSION THEOR Y FOR RIGHT GR OUPS 15 Pr oof. (a) Let ε : E ( S ) → E ( S ) be any bijection that maps e to f . Let r f : S e → S f be the group isomorphism given by right multiplication by f . The triplet ( ε , e , r f ) corresponds to an isomorphism ( S , · , e ) → ( S , · , f ) . The proof of (b) is easy . □ Theorem 4.5. The cate gory of right gr oups is equivalent to the pr oduct cate gory of the cate gory Set  = / 0 of non-empty sets and the cate gory Grp of gr oups. Pr oof. The category equi valence is the functor F : RGrp → Set  = / 0 × Grp that associates to ev ery right group S the pair ( E ( S ) , S / ∼ ) , where ∼ is the kernel of an y right multiplication r e : S → S (Remark 2.7). The functor F associates to e very right group morphism f : S → S ′ the pair of morphisms ( f | E : E ( S ) → E ( S ′ ) , e f : S / ∼ → S ′ / ∼ ) , where f | E is the restriction of f to E ( S ) , and e f is the group morphism of S / ∼ into S ′ / ∼ induced by f . In order to prov e that F is full and faithful, we must prov e that the mapping (3) Hom RGrp ( S , S ′ ) → Hom Set ( E ( S ) , E ( S ′ )) × Hom Grp ( S / ∼ , S ′ ∼ ) , f 7→ ( f | E ( S ) , e f ) is a bijection for all right groups S , S ′ . Fix any element e 0 ∈ E ( S ) . Then right multiplication r e 0 : S → S e 0 is a surjecti ve semigroup morphism with kernel ∼ , hence it induces a (semi)group isomorphism r e 0 : S / ∼ → Se 0 . Since f is a semigroup morphism, we hav e the commutativ e diagram S r e 0 / / f   Se 0 f | Se 0   S ′ r f ( e 0 ) / / S ′ f ( e 0 ) , so that e f = ( r f ( e 0 ) ) − 1 f | Se 0 r e 0 . Hence, if f corresponds to the triplet ( f | E , e 0 , f | Se 0 ) in the sense of Proposition 4.1, then the mapping in (3) associates to f the pair ( f | E ( S ) , e f = ( r f ( e 0 ) ) − 1 f | Se 0 r e 0 ) . The proof that the mapping in (3) is bijecti ve follo ws therefore from Proposition 4.1. Finally , the functor F is essentially surjective. In order to see it, associate to any set X and an y group G , the direct product of G and the right zero semigroup ( X , π 2 ) . □ In vie w of Theorem 4.5, it is natural to define as kernel of a morphism ψ : S → S ′ between two right groups the pair ( ∼ ψ | E , K ) , where ∼ ψ | E is the 16 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O kernel of the mapping ψ | E : E → E ′ (it is a partition of the set E ) and K is the kernel of the group morphism e ψ : S / ∼ → S ′ / ∼ ) (it is a normal subgroup of the group S / ∼ ). The in verse image of an element s ′ ∈ S ′ via the semigroup morphism ψ : S → S ′ can be computed as follows. Giv en s ′ ∈ S ′ , we ha ve that s ′ E ′ ∈ S ′ / ∼ and π E ′ ( s ′ ) ∈ E ′ . Assume that s ′ E ′ ∈ Im ( e ψ ) and π E ′ ( s ′ ) ∈ Im ( ψ | E ) . Then there exist s 1 ∈ S such that ψ ( s 1 ) ∈ s ′ E and s 2 ∈ E ( S ) such that ψ ( s 2 ) = π E ′ ( s ′ ) . In this case, we ha ve ψ − 1 ( s ′ ) = ( e ψ ) − 1 ( s ′ E ′ ) · ( ψ | E ) − 1 ( π E ′ ( s ′ )) = = s 1 E · K · [ s 2 ] ∼ ψ | E = s 1 K · [ s 2 ] ∼ ψ | E . In the other case, that either s ′ E ′ / ∈ Im ( e ψ ) or π E ′ ( s ′ ) / ∈ Im ( ψ | E ) , we hav e that ψ − 1 ( s ′ ) = / 0. 5. P R E T O R S I O N T H E O RY F O R R I G H T G RO U P S W e now briefly recall the notions dev eloped in [5] and [7] about pre- torsion theories in arbitrary categories. Let C be a category and Z be a non-empty class of objects of C . For ev ery pair A , A ′ of objects of C , we indicate by T riv Z ( A , B ) the set of all morphisms in C that factor through an object of Z . W e call these morphisms Z -trivial . If f : A → A ′ is a morphism in C , a morphism ε : X → A in C is a Z - pr ekernel of f if: (1) f ε is a Z -tri vial morphism. (2) If λ : Y → A is any morphism in C for which f λ is Z -tri vial, then there exists a unique morphism λ ′ : Y → X in C such that λ = ε λ ′ . Dually , a Z -pr ecokernel of f is a morphism η : A ′ → X such that: (1) η f is a Z -trivial morphism. (2) If µ : A ′ → Y is any morphism in C for which µ f is Z -tri vial, then there exists a unique morphism µ ′ : X → Y with µ = µ ′ η . If f : A → B and g : B → C are morphisms in C , we say that A f / / B g / / C is a short Z -pr eexact sequence in C if f is a Z -prekernel of g and g is a Z -precokernel of f . Definition 5.1. Let C be a cate gory , and T , F be two replete (that is, closed under isomorphism) full subcategories of C . Set Z : = T ∩ F . The pair ( T , F ) is a pr etorsion theory in the category C if the following properties hold. (1) Hom C ( T , F ) = T riv Z ( T , F ) for ev ery object T ∈ T , F ∈ F . A PRETORSION THEOR Y FOR RIGHT GR OUPS 17 (2) For e very object B of C there is a short Z -pree xact sequence A f / / B g / / C with A ∈ T and C ∈ F . No w let C : = RGrp be the cate gory of right groups, let T : = Rzs (resp., F : = Grp ) be the full subcategories of RGrp consisting of right zero semi- groups (resp., groups), and Z : = T ∩ F . Clearly Z consists of all (semi)groups of order 1, i.e., the terminal objects of RGrp . Thus Z -tri vial morphisms S → S ′ are the semigroup morphisms whose image is a singleton, that is, they are e xactly the constant morphisms S → S ′ , that is, the mappings f : S → S ′ for which there exists an element e ′ 0 ∈ E ( S ′ ) for which f ( s ) = e ′ 0 for ev ery s ∈ S . Our first goal is to characterize morphisms in RGrp that admit a Z -pre- kernel. Lemma 5.2. Let f : S → S ′ be a morphism in RGrp and E : = E ( S ) . Then f has a Z -pr ekernel in RGrp if and only if f ( E ) is a singleton. Moreo ver , if f has a Z -prek ernel in RGrp and f ( E ) = { g 0 } , then K : = f − 1 ( { g 0 } ) is a right subgr oup of S , and the inclusion morphism i : K → S is a Z -pr ekernel of f . Pr oof. Assume that f ( E ) = { g 0 } , so that, in particular , g 0 must be an idem- potent of S ′ . Then K : = f − 1 ( { g 0 } ) is a subsemigroup of S . Now , let a , b ∈ K and let x ∈ S be the unique element such that ax = b (Theorem 2.6(a)). Then g 2 0 = g 0 = f ( b ) = f ( a ) f ( x ) = g 0 f ( x ) , and thus f ( x ) = g 0 , because S is left cancelati ve. This prov es that x ∈ K , and thus K is a right subgroup of S . Let i : K → S be the inclusion. By construction, the composition f i is Z -trivial. Consider now any morphism λ : Y → S of right groups such that f λ is Z -trivial, that is, there is an idem- potent g 1 ∈ S ′ such that f ( λ ( Y )) = { g 1 } . Since Y has an idempotent e 0 and λ ( e 0 ) ∈ E , it immediately follo ws that g 1 = g 0 , pro ving that λ ( Y ) ⊆ K . Hence the mapping λ ′ : Y → K , y 7→ λ ( y ) , is the unique morphism in RGrp such that λ = i λ ′ . This pro ves that i is a Z -prek ernel of f . Con versely , assume that f has a Z -prekernel j : L → S . In particular , f j is Z -tri vial, that is, f ( j ( L )) = { g 0 } for some idempotent g 0 ∈ S ′ . Since L has an idempotent l 0 and j ( l 0 ) ∈ E , it follows that g 0 ∈ f ( E ) . Consider no w any element e ∈ E . Then the inclusion ι : { e } → S is a morphism in RGrp and the composition f ι is a constant morphism of right groups, and thus it is Z -trivial. Since j is a Z -prekernel of f , there is a unique morphism ι 0 : { e } → L such that ι = j ι 0 . It follo ws that e ∈ j ( L ) and thus f ( j ( L )) = { g 0 } implies that f ( e ) = g 0 . This pro ves that f ( E ) = { g 0 } . □ 18 ALBER TO F ACCHINI AND CARMELO ANT ONIO FINOCCHIAR O Lemma 5.3. Let µ : S → T be a morphism of right gr oups such that µ ( E ( S )) is a singleton. Then the kernel ∼ of the canonical pr ojection π G : S → S / ∼ is contained in the kernel of µ . In particular , ther e exists a unique morphism µ ′ : S / ∼ → T such that µ = µ ′ π G . Pr oof. Fix any idempotent element e 0 of S . As we saw in the proof of Proposition 4.1(a) (Identity 2), for ev ery morphism µ : S → T of right groups we hav e µ ( s ) = µ | Se 0 ( se 0 ) µ | E ( π E ( s )) for ev ery s ∈ S . Assume that µ ( E ( S )) is a singleton, and let t be the unique element of µ ( E ( S )) . In order to sho w that ∼ is contained in the kernel of µ , let s , s ′ be two elements of S such that s ∼ s ′ . Then se 0 = s ′ e 0 (Remark 2.7) and µ | E ( π E ( s )) = t = µ | E ( π E ( s ′ )) , so that µ ( s ) = µ ( s ′ ) . This proves that ∼ is contained in the kernel of µ . From this it follows that µ : S → T induces a unique semigroup morphism µ ′ : S / ∼ → T , i.e., there is a unique morphism µ ′ : S / ∼ → T such that µ = µ ′ π G . □ W e ha ve alreay remarked that the full subcate gory T = Rzs of RGrp con- sisting of right zero semigroups is isomorphic to the category Set . Theorem 5.4. The pair ( Rzs , Grp ) is a pr etorsion theory for RGrp . Pr oof. Set Z = Rzs ∩ Grp . Consider a right zero semigroup D , a group G with identity 1, and a morphism ϕ : D → G in RGrp . If D is empty then ϕ is obviously the empty mapping and it factors as the composition of the empty mapping D → { 1 } and the inclusion { 1 } → G , that is, ϕ is Z -trivial. No w suppose that D  = / 0. Since images of idempotents are idempotents, all elements of D are idempotents, and the only idempotent in the group G is 1, we immediately hav e ϕ ( D ) = 1. Thus ϕ is Z -tri vial. No w , let S be any right group, let E : = E ( S ) be the set of all idempotents of S , consider the group G : = S / ∼ and the canonical projection π G : S → G , defined by s 7→ sE , for ev ery s ∈ S . From Lemma 5.2 applied to the morphism π G : S → G , we know that π G has a Z -prekernel in RGrp , because π G ( E ) is the singleton { E } (it is the tri vial subgroup of the group G = S / ∼ ). Moreov er , E = π − 1 G ( { E } ) and the inclusion morphism i : E → S is a Z - prekernel of f (Lemma 5.2). It remains to sho w that π is a Z -precokernel of i . Let µ : S → T be any morphism in RGrp such that µ i is Z -trivial. This means that µ ( E ) is a singleton. Hence we can apply Lemma 5.3, getting that there exists a unique morphism µ ′ : S / ∼ → T such that µ = µ ′ π G . The mapping µ ′ is defined by sE 7→ µ ( s ) . This allo ws us to conclude. □ A PRETORSION THEOR Y FOR RIGHT GR OUPS 19 R E F E R E N C E S [1] S. Burris and H. P . 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(Alberto Facchini) D I PA RT I M E N T O D I M AT E M A T I C A “ T U L L I O L E V I - C I V I TA ” , U N I V E R S I T ` A D I P A D OV A , 3 5 1 2 1 P A D OV A , I TA LY Email addr ess : facchini@math.unipd.it (Carmelo Antonio Finocchiano) D I PA R T I M E N T O D I M A T E M A T I C A E I N F O R M AT I C A , U N I V E R S I T ` A D I C A TAN I A , C I T T ` A U N I V E R S I TA R I A , V I A L E A N D R E A D O R I A 6 , 9 5 1 2 5 C AT A N I A , I TA LY Email addr ess : cafinocchiaro@unict.it