aa r X i v : . [ m a t h . G M ] O c t GENERALIZED GROUPS AND MODULE GROUPOIDS
P. G. ROMEO AND SNEHA K K
Abstract.
In this paper we discuss generalized group, provides some inter-esting examples. Further we introduce a generalized module as a module likestructure obtained from a generalized group and discuss some of its propertiesand we also describes generalized module groupoids. Introduction
The generalized groups introduced by Molaei is an interesting generalization ofgroups(cf.[3]). The identity element in a group is unique, but in a generalized groupthere exists an identity forall each elements. Clearly every group is a generalizedgroup. A groupid is another generalization of a group is a small category in whichevery morphism is invertible and was first defined by Brandt in the year 1926.Groupoids are studied by many Mathematicians with different objective. One of thedifferent approach is the structured groupoid which is obtained by adding anotherstructure in such a way that the added structure is compatible with the groupoidoperation.In this paper we introduce the module action on a generalized group and wecall the resulting structure as a generalized module. Then we discuss some in-teresting examples and properties of generalized modules. Further, analogoues togeneralized group groupoid we describe the generalized module groupoid over aring and obtained a relation between category of generalized module and categoryof generalized module groupoids.2.
Preliminaries
In this section we briefly recall all basic definitions and the elementary conceptsneeded in the sequel. In particular we recall the definitions of categories, groupoids,generalized groups with examples and discuss some intersetng properties of thesestructures.
Definition 1. (cf. [1] ) A Category C consists of the following data : (1) A class called the class of vertices or objects ν C . (2) A class of disjoint sets C ( a, b ) one for each pair ( a, b ) ∈ ν C × ν C . An element f ∈ C is called a morphism from a to b, written f : a → b ; a = dom f thedomain of f and b = cod f called the codomain of f. (3) For a, b, c, ∈ ν C , a map ◦ : C ( a, b ) × C ( b, c ) → C ( a, c ) given by ( f, g ) f ◦ g is the composition of morphisms in C . Mathematics Subject Classification.
Key words and phrases.
Category, Functors, Groupoid, Generalized group, Generalized groupgroupoid Generalized module, Generalized module groupoid. (4) for each a ∈ ν C , a unique a ∈ C ( a, a ) is the identity morphism on a. These must satisfy the following axioms: • (cat1) for f ∈ C ( a, b ) , g ∈ C ( b, c ) and h ∈ C ( c, d ) , then f ◦ ( g ◦ h ) = ( f ◦ g ) ◦ h • (cat 2) for each a ∈ ν C , f ∈ C ( a, b ) and g ∈ C ( c, a ) , then a ◦ f = f and g ◦ a = g . Clearly ν C can be identify as a subclass of C , and with this identification itis possible to regard categories in terms of morphisms alone. The category C issaid to be small if the class C is a set. A morphism f ∈ C ( a, b ) is said to bean isomorphism if there exists f − ∈ C ( b, a ) such that f f − = 1 a = e a , domainidentity and f − f = 1 b = f b , range identity. Example 1.
A group G can be regarded as category in the following way; definecategory C with just one object say ν C = G and morphisma C = { g : g ∈ G } withcomposition in C is the binary operation in G. Identity element in the group will bethe identity morphism on the vertex G. Definition 2.
A groupoid G = ( ν G , G ) is a small category such that for morphisms f, g ∈ G with cod f = dom g then f g ∈ G and every morphism is an isomorphism.A groupoid G is said to be connected if for all a ∈ ν G , G ( a, a ) = φ Example 2.
Every group can be regarded as a groupoid with only one object.
Example 3.
For a set X the cartesian product X × X is a groupoid over X withmorphisms are the elements in X × X with the composition ( x, y ) · ( u, v ) exists onlywhen y = u and is given by ( x, y )( u, v ) = ( x, v ) . In particular ( x, x ) is the uniqueleft identity and ( y, y ) is the unique right identity. Definition 3.
Given two categories C and D , a functor F : C → D consists oftwo functions: the object function denoted by νF which assigns to each object a of C , an object νF ( a ) of the category D and a morphism function which assigns toeach morphism f : a → b of C , a morphism F ( f ) : F ( a ) → F ( b ) in D satisfies thefollowing F (1 c ) = 1 F ( c ) for every c ∈ ν C , and F ( f g ) = F ( f ) F ( g ) whenever f g is defined in C . We denote by
Gpd the category of groupoids in which objects are the groupoidsand morphisms are the functors.A semigroup is a pair ( S, · ) where S is a nonempty set and · is an associativebinary operation on S. A semigroup S is said to be regular if for each x ∈ S thereexists x ′ ∈ S such that xx ′ x = x. An element e of S is said to be idempotentif e = e. A band is a semigroup in which all elements are idempotents. Inversesemigroup is a regular semigroup in which for each x ∈ S there exists an unique y ∈ S such that x = xyx and y = yxy. ENERALIZED GROUPS AND MODULE GROUPOIDS 3
Generalized Groups and Generalized group groupoids.
In the following we recall the definitions of generalized groups and describe someof its properties.
Definition 4. (cf. [3] ) A generalized group G is a non-empty set together with abinary operation called multiplication subject to the set of rules given below:(1) ( ab ) c = a ( bc ) for all a, b, c ∈ G (2) for each a ∈ G there exists unique e ( a ) ∈ G with ae ( a ) = e ( a ) a = a (3) for each a ∈ G there exists a − ∈ G with aa − = a − a = e ( a )It is seen that for each element a in a generalized group the inverse is uniqueand both a and a − have the same identity. Every abelian generalized group is agroup. Definition 5. (cf. [3] ) A generalized group G is said to be normal generalized groupif e ( ab ) = e ( a ) e ( b ) for all elements a, b ∈ G Definition 6. (cf. [3] ) A non-empty subset H of a generalized group G is a gener-alized subgroup of G if and only if for all a, b ∈ H, ab − ∈ H. Theorem 1.
Rectangular band semigroup is a normal generalized group.Proof.
A band B is a regular semigroup in which every element is an idempotent.Let I, Λ be non empty sets and B = I × Λ with multipication defined by( i, λ )( j, µ ) = ( i, µ ) f or all ( i, λ ) , ( j, µ ) ∈ I × Λis a semigroup and in which all elements are idempotents. Then B is a rectangularband.For any ( i, λ ) , ( j, µ ) ∈ B ( i, λ )( j, µ ) = ( j, µ )( i, λ ) = ( i, λ ) gives ( j, µ ) = ( i, λ )Thus we have e ( i, λ ) = ( i, λ ) f or all ( i, λ ) ∈ B and ( i, λ ) − = ( i, λ ) . Moreover e (( i, λ )( j, µ )) = e ( i, λ ) e ( j, λ )hence B is a normal generalized group. (cid:3) Example 4.
Let G = ( V ( G ) , E ( G )) be a complete digraph without multiple edges,then each edge can be uniquely identified with the starting and ending vertices. Let d ( g ) stands for the domain and r ( g ) stands for the range of the edge g. For any f, g ∈ E ( G ) define the composition of f and g as the unique edge starting from thedomain of f and ending at the range of g.ie ; f ◦ g = h r ( g ) d ( f ) with d ( h r ( g ) d ( f ) ) = d ( f ) and r ( h r ( g ) d ( f ) ) = r ( g ) . The composition ◦ is an associative binary operation. For, f, g, h ∈ E ( G ) with d ( f ) = v r ( f ) = v , d ( g ) = u , r ( g ) = u , d ( h ) = w and r ( h ) = w ( f ◦ g ) ◦ h = ( h u v ) ◦ h = h w v f ◦ ( g ◦ h ) = f ◦ ( h w u ) = h w v ie., ( f ◦ g ) ◦ h = f ◦ ( g ◦ h ) P. G. ROMEO AND SNEHA K K also for each edge f, f ◦ f = f hence e ( f ) = f − = f . Thus the complete digraph is a generalized group. Example 5. G = ( A = (cid:18) a b (cid:19) : a = 0 , a, b ∈ R ) Then, G is a generalized group and for all A ∈ G , e ( A ) = (cid:18) b/a (cid:19) and A − = (cid:18) /a b/a (cid:19) where e ( A ) and A − are the identity and the inverse of matrix A respectively. Also e ( AB ) = e ( B ) for all A, B ∈ G Definition 7. (cf. [3] ) Let G and H be two generalized groups. A generalized grouphomomorphism from G to H is a map f : G → H such that f ( ab ) = f ( a ) f ( b ) Theorem 2. (cf. [3] ) Let f : G → H be a homomorphism of generalized groups G and H. Then(1) f ( e ( a )) = e ( f ( a )) is an identity element in H for all a ∈ G (2) f ( a − ) = f ( a ) − (3) if K is a generalized subgroup of G then f ( K ) is a generalized subgroup of H. Generalized groups and their homomorphisms form a category and is denotedby GG Definition 8. (cf. [2] ) A generalized group groupoid G is a groupoid ( ν G , G ) endowedwith the structure of generalized group such that the following maps(1) + : G × G → G, ( f, g ) → f + g, (2) u : G → G, f → − f, (3) e : G → G, f → e ( f ) , are functorial. Since + is a functorial we have( f ◦ g ) + ( h ◦ k ) = +( f ◦ g, h ◦ k )= +[( f, h ) ◦ ( g, k )]= +( f, h ) ◦ +( g, k )= ( f + h ) ◦ ( g + k )( f ◦ g ) + ( h ◦ k ) = ( f + h ) ◦ ( g + k )thus the interchange law( f ◦ g ) + ( h ◦ k ) = ( f + h ) ◦ ( g + k )exists between groupoid composition and generalized group operation.In other words, a generalized group groupoid is a groupoid endowed with a struc-ture of generalized group such that the structure maps of groupoid are generalizedgroup homomorphisms. ENERALIZED GROUPS AND MODULE GROUPOIDS 5
Example 6.
Let G be a generalized group. Then G × G is a generalized groupgroupoid with object set G . For each morphism ( x, y ) ∈ G × G the identity arrow of ( x, y ) is ( e ( x ) , e ( y )) and the inverse is ( − x, − y ) and the interchange law also holds.For, [( f, g ) ◦ ( g, h )] + [( f ′ , g ′ ) ◦ ( g ′ , h ′ )] = ( f, h ) + ( f ′ , h ′ )= ( f + f ′ , h + h ′ )[( f, g ) + ( f ′ , g ′ )] ◦ [( g, h ) + ( g ′ , h ′ )] = ( f + f ′ , g + g ′ ) ◦ ( g + g ′ , h + h ′ )= ( f + f ′ , h + h ′ )[( f, g ) ◦ ( g, h )] + [( f ′ , g ′ ) ◦ ( g ′ , h ′ )] = [( f, g ) + ( f ′ , g ′ )] ◦ [( g, h ) + ( g ′ , h ′ )]2.2. Generalized Groups from a connected groupoid.
We proceed to describe the generalized group obtained from a connected groupoid.Let G be a connected groupoid with G ( a, b ) contains exactly one morphism for all a, b ∈ G . Define a binary operation which is deoted by + on G . For, let f ∈ G ( a, b )and g ∈ G ( c, d ) f + g = f ◦ k bc ◦ g where ◦ is the composition in the groupoid and k bc is the unique morphism in G ( b, c ). Then G is a generalized group, since(1) The operation ” + ” is associative.for f ∈ G ( a, b ) g ∈ G ( c, d ) h ∈ G ( i, j )( f + g ) + h = ( f ◦ k bc ◦ g ) + h = ( f ◦ k bc ◦ g ) ◦ ( k di ◦ h )= f k bc gk di hf + ( g + h ) = f + ( g ◦ k di ◦ h )= ( f ◦ ( k bc ) ◦ ( g ◦ k di ◦ h )= f k bc gk di hie ; ( f + g ) + h = f + ( g + h ) and + is an associative binary opration.(2) for each f ∈ G f + f = f ◦ f − ◦ f = f hence e ( f ) = − f = f Thus the groupoid G together with the operation + , forms a generalizedgroup and is denoted by G ∗ . More over this generalized group G ∗ is normal,since e ( f + g ) = f + g = e ( f ) + e ( g )Now we extend this construction of a generalized group in to an arbitrary con-nected groupoid. We start with a connected groupid G with partial composition ◦ . Choose a morphism from each homset G ( a, b ) and is denote by h ab in such a waythat h ba = h ab − and h a = 1 a . Define an addition + on G for f ∈ G ( a, b ) , g ∈ G ( c, d )as follows f + g = f ◦ h bc ◦ g. Clerarly + is an associative binary operation on G and for each f ∈ G ( a, b ) P. G. ROMEO AND SNEHA K K f + h ab = f ◦ h ba ◦ h ab = f ◦ h ab − ◦ h ab = f ◦ b = fh ab + f = h ab ◦ h ba ◦ f = h ab ◦ h ab − ◦ f = 1 a ◦ f = ff + h ab = h ab + f = f and e ( f ) = h ab ∀ f ∈ G f + ( h ab f − h ab ) = f ◦ b ◦ f − ◦ h ab = h ab ( h ab f − h ab ) + f = h ab ◦ f − ◦ a ◦ f = h ab f + ( h ab f − h ab ) = ( h ab f − h ab ) + f = h ab = e ( f ) − f = h ab f − h ab Thus the connected groupoid G is a generalized group with respect to the additionwe defined above.Next we recall the generalized rings which was introduced by Molaei. Definition 9. (cf. [4] ) A generalized ring R is a nonempty set R with two differentoperations adition and multiplication denoted by ‘ + ‘ and ‘ × ‘ respectively in which ( R, +) is a generalized group and satisfies the following conditions.(1) multiplication is an associative binary operation.(2) for all x, y, z ∈ R x ( y + z ) = xy + xz and ( x + y ) z = xz + yz Note that in a generalized ring
R, e ( ab ) = e ( a ) e ( b ) for all a, b ∈ R. Example 7. R with the operations ( a, b )+ ( c, d ) = ( a, d ) and ( a, b )( c, d ) = ( ac, bd ) is a generalized ring. Generalized modules
Let M be a generalized group and R be a ring, in the following we proceed todefine a generalized module using the generalized group M . Definition 10.
Let R be a ring with unity. A generalized group M is said to be(left) generalized R module if for each element r in R and each m in M we have aproduct rm in M such that for r, s ∈ R and m, n ∈ M (1) (r+s)m = rm+sm(2) r(m+n) = rm+ rn(3) r(sm) = (rs)m(4) re ( m ) = e ( m ) for all r ∈ R and m ∈ M (5) 1.m = m ENERALIZED GROUPS AND MODULE GROUPOIDS 7
Proposition 1.
Let M be an R − module then M × M is a generalized R modulewith the operations ( x, y ) + ( m, n ) = ( x, y + n ) r ( x, y ) = ( x, ry ) and for each x ∈ M the subset M x = { ( x, y ) : y ∈ M } is an R module.Proof. It can be seen that ( M, +) is a generalized group in which for all ( m, n ) ∈ M, e ( m, n ) = ( m,
0) and ( m, n ) − = ( m, − n ) . To show that M is a generalized R module consider r, s ∈ R and ( x, y ) , ( m, n ) ∈ M × M, (1) ( r + s )( x, y ) = ( x, ( r + s ) y )= ( x, ry + rs ) r ( x, y ) + s ( x, y ) = ( x, ry ) + ( x, rs )= ( x, ry + rs ) ie ; ( r + s )( x, y ) = r ( x, y ) + s ( x, y ) axiom(1)is satisfied(2) r [( x, y ) + ( m, n )] = r [( x, y + n )]= ( x, r ( y + n ))= ( x, ry + rn ) r ( x, y ) + r ( m, n ) = ( x, ry ) + ( m, rn )= ( x, ry + rn )hence r [( x, y ) + ( m, n )] = r ( x, y ) + r ( m, n ) axiom(2)is satisfied(3) rs ( x, y ) = ( x, rsy )= ( x, r ( sy ))= r ( x, sy )= r ( s ( x, y )) rs ( x, y ) = r ( s ( x, y )) axiom(3)is satisfied(4) re ( x, y ) = r ( x, x, r x, e ( x, y ) re ( x, y ) = e ( x, y ) axiom(4)is satisfied(5) 1 · ( x, y ) = ( x, y ) ∀ ( x, y ) ∈ M To show that for each x ∈ M the subset M x = { ( x, y ) : y ∈ M } is an R moduleit is enough to show that M x is an abelian group and the scalar multipication isclosed. Let m + n = ( x, y ) and n = ( x, z ) be elements in M x then m + n = ( x, y ) + ( x, z ) = ( x, y + z ) P. G. ROMEO AND SNEHA K K and n + m = ( x, z ) + ( x, y ) = ( x, z + y ) = ( x, y + z )Therefore m + n = n + m and + is a commutative binary operation on M x . Forevery m ∈ M x ( x, y ) + ( x,
0) =( x,
0) + ( x, y ) = ( x, y ) and ( x, y ) − = ( x, − y ) . Thus M x is an abelian subgroup of M and for any r ∈ R r ( x, y ) = ( x, ry ) ∈ M x . Hence M x is a R module. (cid:3) Thus we have the following example for a generalized R module. Example 8.
Consider M = R × R with the following operations ( x, y ) + ( m, n ) = ( x, y + n ) r ( x, y ) = ( x, ry ) is a generalized R module. Theorem 3. If M is a generalized R module then(1) e ( rm ) = re ( m ) for all r ∈ R, m ∈ M (2) ( rm ) − = r ( m − ) Proof.
Let r ∈ R and m ∈ M ,(1) rm + re ( m ) = r ( m + e ( m )) = rmre ( m ) + rm = r ( e ( m ) + m ) = rm Hence e ( rm ) = re ( m )(2) rm + rm − = r ( m + m − ) = re ( m ) rm − + rm = r ( m − + m ) = re ( m ) (cid:3) Definition 11.
Let M and N be two generalized R − modules. A function f : M → N is called generalized module homomorphism if f ( m + n ) = f ( m ) + f ( n ) f or all m, n ∈ Mf ( rm ) = rf ( m ) f or all r ∈ R, m ∈ M Definition 12.
Let M be a generalized R module and N ⊂ M is said to begeneralized submodule of M if N is a generalized subgroup of G and for each r ∈ R and m ∈ N rm ∈ N Proposition 2.
Let M be a generalized R -module which is also a normal gener-alized group. Then the set of identity elements in M is a submodule of M and wecall it the zero submodule or trivial submodule. ENERALIZED GROUPS AND MODULE GROUPOIDS 9
Proof.
We denote the set of identity elements in M by e ( M ) = { e ( x ) : x ∈ M } First we show that e ( M ) is a generalized subgroup of M. For m, n ∈ e ( M ) thereexists x, y ∈ M such that m = e ( x ) and n = e ( y ) .mn − = e ( x ) e ( y ) − = e ( x ) e ( y ) = e ( xy )hence xy − ∈ e ( M ) and e ( M ) is a generalized subgroup of M. Ler r be an element in the ring R and rm = re ( x ) = e ( x )hence rm ∈ M for all r ∈ R and m ∈ M. Thus set of identitt elements of M forma generalized submodule of M. (cid:3) Every nonzero generalized module M contains at least one submodule M itself. Theorem 4.
Let R is a ring and M and N are normal generalized R modules. If f : M → N is a generalized R − module homomorphism, then kerf = { x : x ∈ M, f ( x ) ∈ e ( N ) } where e ( N ) denotes the set of identities elements of N, is a submodule of M and Imf = { f ( x ) : x ∈ M } is a submodule of N. Proof.
Let M and N be a generalized modules over a ring R and f is a homomor-phism between M and N. To show that kerf is a submodule of M it is suffices toprove that it is a generalized subgroup of M and is closed under scalar multipica-tion.Let x, y ∈ kerf then f ( x ) = e ( n ) and f ( y ) = e ( n ′ ) for some n, n ′ ∈ N Nowconsider; f ( xy − ) = f ( x ) f ( y − )= f ( x )( f ( y )) − = e ( n ) e ( n ′ ) − = e ( n ) e ( n ′ )= e ( nn ′ )hence xy − ∈ kerf and kerf is a generalized subgroup of M. for any r ∈ R f ( rx ) = rf ( x )= re ( n )= e ( n ) rx ∈ kerf f or each r ∈ R. Therefore kerf is a submodule of M for every m ∈ M. Similarly we can prove that
Imf is a submodule of N. For; let x, y ∈ Imf then there exists m, n, ∈ M such that f ( m ) = x and f ( n ) = y. Now consider, xy − = f ( m ) f ( n ) − = f ( m ) f ( n − )= f ( mn − ) and mn − ∈ M hence xy − ∈ Imf.
Hence
Imf is a generalized subgroup of N and for any r ∈ R,rx = rf ( m )= f ( rm ) , rm ∈ M Hence rx ∈ Imf for any r ∈ R and x ∈ M. Therefore
Imf is a generalizedsubmodule of N. (cid:3) Theorem 5. If M is a generalized R -module and if there exists x ∈ M such that M = Rx then M is a module over R Proof.
First we show that M is an abelian group with respect to the generalizedgroup operation. For any m, n ∈ M there exists r, s ∈ R such that m = rx and n = sx Consider rx + sx = ( r + s ) x = ( s + r ) x = sx + rxie ; m + n = n + m f or all m, n ∈ M. Hence M is an abelian generalized group hence is an abelian group.Therefore M is a R − module. (cid:3) Note 6.
The generalized modules and their homomorphisms form a category inwhich objects are the generalized modules and morphisms are their homomorphismsdenoted by GM . Result 7. If M and N be two generalized modules over a ring R then their cartesianproduct M × N defined by M × N = { ( m, n ) : m ∈ M, n ∈ N } is a generalized module over R with respect to the component wise operations. ie ; for any ( m, n ) , ( x, y ) ∈ M × N and r ∈ R the addition and scalar multipication isgiven by ( m, n ) + ( x, y ) = ( m + x, n + y ) r ( m, n ) = ( rm, rn ) Definition 13.
A groupoid G is a generalized module groupoid over R if it has ageneralized module structure over R and it satisfies the following conditions.(1) G is a generalized group groupoid.(2) For each r ∈ R the mapping η r : G → G defined by η r ( g ) = rg is a functor on G . ie ; for any composable morphisms in G and any r ∈ R we should have r ( g ◦ h ) = rg ◦ rh. ENERALIZED GROUPS AND MODULE GROUPOIDS 11
Example 9.
Let M be a generalized module over a ring R. Then M × M is ageneralized module groupoid over R with object set M. It follows from the example(6)that M × M with operation ( x, y ) + ( m, n ) = ( x + m, y + n ) is a generalized groupgroupoid. The cartesian product of two generalized modules are again a generalizedmodule. To show that M × M is a generalized module groupoid over R it is enough toprove that for any r ∈ R the map η r : M × M → M × M defined by η r ( x, y ) = ( rx, ry ) is a functor. For, let x be any object in the category M × M then η r (1 x ) = η r ( x, x ) = ( rx, rx ) = 1 η r ( x ) . Consider two composable morphisms ( x, y ) , ( y, z ) in M × M then η r [( x, y ) ◦ ( y, z )] = η r [( x, z )]= ( rx, rz ) η r ( x, y ) ◦ η r ( y, z ) = ( rx, ry ) ◦ ( ry, rz )= ( rx, rz ) η r [( x, y ) ◦ ( y, z )] = η r ( x, y ) ◦ η r ( y, z ) hence M × M is a generalized module groupoid over R. Definition 14.
Let M and N be two generalized module groupoids over a ring R. A homomorphism f : M → N of generalized module groupoids is a functor ofunderlying groupoids preserving generalized module structure. Note That the generalized module groupoids and their homomorphisms form acategory denoted by
GMG
Proposition 3.
There is a functor from the category GM of generalized modulesto the category GMG of generalized module groupoidProof.
Let M be a generalized module over a ring R. Then it can be seen thatcartesian product M × M is a generalized module groupoid over R. If f : M → M is a homomorphism of generalized modules then define F : GM → GMG by F ( M ) = M × M and F ( f ) : M × M → M × M F ( f )( m, n ) = ( f ( m ) , f ( n ))and it can be seen that F ( f ) is a functor between M × M and M × M so that F ( f ) is a morphism in the category of generalized module groupoid. Now we provethat F is a functor between the category of generalized modules and the generalizedmodule groupoids. For; let for any vertex M ∈ GM we have, F (1 M )( m, n ) = ( m, n ) F (1 M ) = 1 F ( M )2 P. G. ROMEO AND SNEHA K K let f : M → N and g : N → P be two composable homomorphisms in GM then F ( f g )( m, n ) = ( f g ( m ) , f g ( n ))= (( g ( f ( m )) , g ( f ( n ))) F ( f ) F ( g )( m, n ) = F ( g )( F ( f )( m, n ))= F ( g )( f ( m ) , f ( n ))= ( g ( f ( m )) , g ( f ( n ))) F ( f g )( m, n ) = F ( f ) F ( g )( m, n ) ∀ ( m, n ) ∈ M × MHence F ( f g ) = F ( f ) F ( g ) (cid:3) Proposition 4.
Let { M i i ∈ I } be a family of generalized R module groupoids.Then M = ( νM, M ) where νM = Q νM i and M = Q M i is a generalized R modulegroupoid.Proof. Q νM i has elements ( m i ) i ∈ I where m i ∈ νM i and morphisms ( f i ) i ∈ I from dom f i to cod f i in M i . The composition of morphisms is( f i ) i ∈ I · ( g i ) i ∈ I = ( f i · g i ) i ∈ I whenever f i and g i re composable morphisms in M i . Since each M i is a groupoideach morphism f i admits an inverse f − i , thus the product M = ( Q νM i , Q M i ) , i ∈ I is a groupoid. Moreover the product M = Q M i has a structure of generalizedmodule with respect to component wise operations( f i ) i ∈ I + ( g i ) i ∈ I = ( f i + g i ) i ∈ I and r ( f i ) i ∈ I = ( rf i ) i ∈ I thus M is a generalized group groupoid. It remains to show that the map η r : M → M by η r ( m i ) i ∈ I = ( rm i ) i ∈ I is a functor on M. For each ( x i ) i ∈ I ∈ νM consider η r (1 ( m i ) ) i ∈ I = ( r ( m i ) ) i ∈ I = (1 ( rm i ) ) i ∈ I ( each M i is a generalized module groupid )= 1 η r ( m i ) i ∈ I let ( m i ) i ∈ I , ( n i ) i ∈ I are two composable morpisms in M, then η r (( m i ) i ∈ I ◦ ( n i ) i ∈ I ) = r (( m i ) i ∈ I ◦ ( n i ) i ∈ I )= r ( m i ◦ n i ) i ∈ I = ( r ( m i ◦ n i ) i ∈ I )= ( rm i ◦ rn i ) i ∈ I = ( rm i ) i ∈ I ◦ ( rn i ) i ∈ I η r ( m i ) i ∈ I ◦ η r ( n i ) i ∈ I = ( rm i ) i ∈ I ◦ ( rn i ) i ∈ I η r (( m i ) i ∈ I ◦ ( n i ) i ∈ I ) = η r ( m i ) i ∈ I ◦ η r ( n i ) i ∈ I (cid:3) ENERALIZED GROUPS AND MODULE GROUPOIDS 13
References [1] K.S.S. Nambooripad, Theory of Regular Semigroups, Sayahna Foundation Trivandrum, 2018.[2] M.H. Gürsoy, H. Aslan, I. Icen, Generalized crossed modules and group-groupoids, DOI:10.3906/mat-1602-63, Turkish J. Math., 2017.[3] M. R. Molaei, Generalized groups, Bul. Inst. Politeh. Iasi. Sect. I. Mat. Mec. Teor. Fiz. 49(1999), 21–24.[4] M. R. Molaei, Generalized rings, Ital. J. Pure Appl. Math. 12 (2003), 105–111.[5] Mustafa Habil Gursoy: Generalized ring groupoids Annals of the University of Craiova,Mathematics and Computer Science Series .[6] Saunders Mac Lane: Categories for the Working Mathematician, Second edition, 0-387-9803-8, Springer-Verlag New york, Berlin Heidelberg Inc., 1998.
Dept. of Mathematics, Cochin University of Science and Technology, Kochi, Kerala,INDIA.
Email address : romeo − parackal @ yahoo.com, snehamuraleedharan007@