On a connection between fuzzy subgroups and F -inverse covers of inverse monoids
aa r X i v : . [ m a t h . G M ] N ov On a connection between fuzzy subgroups and F -inverse covers of inverse monoids Elton PaskuUniversiteti i Tiran¨esFakulteti i Shkencave NatyroreDepartamenti i Matematik¨esTiran¨e, [email protected]
Abstract
We define two categories, the category FG of fuzzy subgroups, and the category FC of F -inverse covers of inverse monoids, and prove that FG fully embeds into FC .This shows that, at least from a categorical viewpoint, fuzzy subgroups belong tothe standard mathematics as much as they do to the fuzzy one. Keywords
Fuzzy subgroup, inverse monoid, F -inverse cover, dual premorphism.AMS Mathematics Subject Classification (2010): 03E72, 20N25, 20M18. The theory of fuzzy sets originates with the article [10] of Zadeh and has aimed sincethan to help other branches of mathematics that study ambiguity or uncertainty. Alongwith fuzzy sets, fuzzy analogues have been developed, in particular the theory of fuzzygroups which started with the paper [7] of Rosenfeld. Given a set X , a fuzzy subset A of X is a function A : X → [0 , x ∈ X , the value A ( x ) represents the degree ofmembership of x in A . This is what makes A to look like an uncertain set. On the otherhand, the definition of fuzzy groups is a bit more complex and is given below. Let G bea group. A fuzzy subgroup of G is a map µ : G → [0 ,
1] such that:(i) for all x, y ∈ G , µ ( xy ) ≥ min { µ ( x ) , µ ( y ) } , and(ii) for all x ∈ G , µ ( x − ) ≥ µ ( x ).It turns out that for all x ∈ G , µ ( x ) = µ ( x − ), and µ ( e ) ≥ µ ( x ) where e is the unit of G .There is no restriction if we replace [0 ,
1] in this definition by µ ([0 , G is a triple ( G, µ, U ) where G is a group, U ⊆ [0 ,
1] and µ : G → U is a surjective map satisfying the properties:(1) for all x, y ∈ G , µ ( xy ) ≥ min { µ ( x ) , µ ( y ) } , and(2) for all x ∈ G , µ ( x − ) = µ ( x ). 1here is a similarity between the definition of a fuzzy subgroup as a triple, with thedefinition of a dual premorphism from a group to an inverse monoid as defined in [1],apart from the fact that here U is not given an inverse monoid structure. Luckily, wecan overcome this difficulty very easily. The fact that for all x ∈ G , µ ( x ) ≤ µ ( e ) saysexactly that sup( U ) = µ ( e ) and this belongs to U . Also U is clearly a poset, where theorder is the one inherited by the usual order in [0 , U has agreatest element. Each poset U ⊆ [0 ,
1] which has a greatest element α can be regardedas an inverse monoid with multiplication ∧ , that is u ∧ v = min { u, v } . The unit of ( U, ∧ ) is clearly the element α . Now it is obvious that a fuzzy subgroup( G, µ, U ) is a dual premorphism from the group G to the inverse monoid U . It is wellknown that such dual premorphisms are closely related with F -inverse covers of inversemonoids. Before we explain what are the F -inverse covers of inverse monoids, let us recalla few basic concepts from inverse monoids. An inverse monoid is a monoid M such thatfor every x ∈ M there is a unique x − ∈ M , called the inverse of x , such that xx − x = x and x − xx − = x − . Every inverse monoid M comes equipped with a natural partialorder ≤ defined by: x ≤ y if and only if there is an idempotent e ∈ M such that x = ye .Every inverse monoid M has a smallest group congruence which is denoted by σ and ischaracterized by ( x, y ) ∈ σ if and only if there is an idempotent e ∈ M such that xe = ye .An inverse monoid satisfying the property that each σ -class contains a greatest elementwith respect to the natural partial order is called an F -inverse monoid. An F -inversemonoid F is called an F -inverse cover over the group F/σ if there exists a surjectiveidempotent separating homomorphism F → M . There are a number of important resultsconcerning F -inverse monoids which are related with covers and expansions of inversemonoids. The reader can find useful material in papers [1], [3] and [9]. Regarding F -inverse covers of inverse monoids, as we mentioned before, they are closely related withdual premorphisms between inverse monoids. A dual premorphism ψ : M → N betweeninverse monoids is a map such that ψ ( x − ) = ( ψ ( x )) − and ψ ( xy ) ≥ ψ ( x ) ψ ( y ) for all x, y ∈ M . The following is Theorem VII.6.11 of [6] and gives a relationship between dualpremorphisms and F -inverse covers. Theorem 1.1
Let H be a group and M an inverse monoid. If ψ : H → M is a dualpremorphism such that for every u ∈ M , there is an h ∈ H with u ≤ ψ ( h ) , then F = { ( u, h ) ∈ M × H | u ≤ ψ ( h ) } , is an F -inverse cover of M over H . Conversely, every F -inverse cover of M over H canbe so constructed (up to isomorphism). Finally, for everything unexplained here on inverse semigroups we refer the reader to themonographs [2] and [6]. While for basics on fuzzy sets and fuzzy groups we refer thereader to [5], [7] and [10]. The book of Mac Lane [4] contains the necessary material oncategories and functors, 2
The definitions of FG and FC We define the category of fuzzy subgroups FG in the following way. The objects of FG are triples ( G, µ, U ) as defined above, and if (
G, µ , U ) and ( H, µ , V ) are two suchtriples, a morphism from ( G, µ , U ) to ( H, µ , V ) is a pair ( f, λ ) where f : G → H is agroup homomorphism, and λ : U → V is an order preserving map with the property that λ (sup( U )) = sup( V ), and, for all x ∈ Gµ f ( x ) = λµ ( x ) . The unit morphism on an object (
G, µ, U ) is defined to be the pair (1 G , U ). Now if( K, µ , W ) is another object from FG , and ( g, λ ′ ) : ( H, µ , V ) → ( K, µ , W ) is another amorphism, we define the composition ( g, λ ′ ) ◦ ( f, λ ) as the pair ( gf, λ ′ λ ). We will showthat this pair is indeed a morphism from ( G, µ , U ) to ( K, µ , W ). For every x ∈ G , µ gf ( x ) = λ ′ µ f ( x )= λ ′ λµ ( x ) . Also λ ′ λ is order preserving, and λ ′ λ (sup( U )) = λ ′ (sup( V )) = sup( W ). The propertiesthat FG should satisfy to be a category are straightforward. It looks like, we used in thedefinition of morphisms of FG mixed concepts. On the one hand we have homomorphismsof groups, and on the other hand, order preserving maps between posets. In fact, for anytwo posets U and V with respective greatest elements α and β , any order preserving map λ : U → V which sends α to β , is in fact a homomorphism between monoids ( U, ∧ ) and( V, ∧ ). Indeed, let u, v ∈ U such that u ≤ v , then λ ( u ∧ v ) = λ ( u ) (since u ≤ v )= λ ( u ) ∧ λ ( v ) (since λ ( u ) ≤ λ ( v )).In addition to that, the fact that λ ( α ) = β says that λ is a homomorphism of monoids.The converse is also true, that is, any homomorphism of monoids λ : U → V , is an orderpreserving map since for every u, v ∈ [0 , α ] such that u ≤ v , λ ( u ) = λ ( u ∧ v ) (since u ≤ v )= λ ( u ) ∧ λ ( v ) (since λ is a homomorphism),which implies that λ ( u ) ≤ λ ( v ). The condition that λ ( α ) = β follows from the fact that λ is a monoid homomorphism. Finally, we remark that λ maps the greatest element α ofthe single σ -class of U to the greatest element β of the single σ -class of V .The definition of the category FC of F -inverse covers of inverse monoids is an extensionof the definition of the category F of F -inverse semigroups made in [9]. The objects of FC are triples ( T, M, ϕ ) where T is an F -inverse monoid, M is an inverse monoid, and ϕ is ahomomorphism of monoids which is surjective and idempotent separating. We say that T is an F -inverse cover of M over T /σ . If now ( T ′ , M ′ , ϕ ′ ) is another triple as above, then amorphism from ( T, M, ϕ ) to ( T ′ , M ′ , ϕ ′ ) is a pair ( f ∗ , λ ) with f ∗ : T → T ′ and λ : M → M ′ monoid morphisms which map the greatest element of a σ -class onto the greatest elementof some σ -class, and that satisfy the commutativity condition ϕ ′ f ∗ = λϕ . The identitymorphism on the object ( T, M, ϕ ) is defined the pair (1 T , M ) which clearly satisfies the3bove commutativity condition. The composition of morphisms is defined in the followingfashion. If ( f ∗ , λ ) : ( T, M, ϕ ) → ( T ′ , M ′ , ϕ ′ ) and ( f ′∗ , λ ′ ) : ( T ′ , M ′ , ϕ ′ ) → ( T ′′ , M ′′ , ϕ ′′ ) aretwo morphisms, then their composition is defined to be the pair ( f ′∗ f ∗ , λ ′ λ ). This is indeeda morphism from ( T, M, ϕ ) to ( T ′′ , M ′′ , ϕ ′′ ) since ϕ ′′ ( f ′∗ f ∗ ) = ( ϕ ′′ f ′∗ ) f ∗ = ( λ ′ ϕ ′ ) f ∗ = λ ′ ( ϕ ′ f ∗ )= λ ′ ( λϕ ) = ( λ ′ λ ) ϕ, and that both compositions f ′∗ f ∗ and λ ′ λ map the greatest element of a σ -class onto thegreatest element of some σ -class since their respective components do so. Finally, it iseasy to see that FC is indeed a category. Looking back to the definition of an object (
G, µ, U ) from FG , but with U regarded nowas an inverse monoid, we see that the map µ : G → U is nothing but a dual premorphismbetween inverse monoids satisfying the property that for every u ∈ U , there exists x ∈ G such that u ≤ µ ( x ). It follows from Theorem 1.1, that there is an F -inverse cover of U over G which we write with the long notation C ( G, µ, U ). More explicitly, C ( G, µ, U ) = { ( u, x ) ∈ U × G | u ≤ µ ( x ) } is an inverse monoid whose idempotents turn out to be all the pairs ( u, G , in particular the unit element is ( µ (1) , u, x ) ≤ ( v, y ) if and only if y = x and u ≤ v . The σ -class of an element( u, x ) consists of all the elements ( v, x ) with v ≤ µ ( x ) and its greatest element withrespect to the natural order is ( µ ( x ) , x ). Finally note that the projection in the firstcoordinate ϕ : C ( G, µ, U ) → U , ( u, x ) u is surjective and idempotent separating. Wecall the triple ( C ( G, µ, U ) , ϕ, U ) the F -inverse cover associated with the fuzzy subgroup( G, µ, U ). The monoid C ( G, µ, U ) seems to be useful in connecting inverse semigroupswith fuzzy subgroups. An argument which goes in favor to this is that the H -classes of C ( G, µ, U ) correspond in a way that will be made precise below, to the so called levelsubsets of (
G, µ, U ). Level subsets are defined in [8] as follows. Given a fuzzy subgroup(
G, µ, U ) and u ∈ U , then the level subset µ u of the fuzzy subset µ is defined by µ u = { h ∈ G | µ ( h ) ≥ u } . It is proved in Theorem 2.1 of [8] that such subsets are in fact subgroups of G . Before wesee the connection they have with the H -classes of C ( G, µ, U ), we note that C ( G, µ, U ) isa Clifford monoid. Indeed, it is inverse and its idempotents are central. To see the latter,let ( u,
1) be an idempotent, and ( v, h ) an arbitrary element, then( u, v, h ) = ( u ∧ v, h ) = ( v ∧ u, h ) = ( v, h )( u, . To see what an H -class looks like, we recall first that the relations H and R coincide inClifford semigroups. Let now ( v, h ) ∈ C ( G, µ, U ) be such that ( v, h ) R ( u,
1) where ( u, w, a ) , ( w ′ , b ) ∈ C ( G, µ, U ) such that( v ∧ w, ha ) = ( v, h )( w, a ) = ( u, u ∧ w ′ , bh ) = ( u, w ′ , b ) = ( v, h ) , which both imply that u = v . Therefore, if ( v, h ) ∈ H ( u, , then necessarily v = u .Conversely, any ( u, h ) ∈ C ( G, µ, U ) is R (hence H )-equivalent with ( u, u, h )( u, h − ) = ( u,
1) and ( u, u, h ) = ( u, h ) . All we said means that for any fixed u ∈ U , H ( u, = { ( u, h ) | u ≤ µ ( h ) } and this forms asubgroup of C ( G, µ, U ). Now we show that each level subgroup µ u is in fact isomorphicto H ( u, . Indeed, the map φ : H ( u, → µ u such that ( u, h ) h, is clearly bijective and a homomorphism.Now we prove our main result. Theorem 3.1
There is a full and faithful embedding of the category FG of fuzzy sub-groups into the category FC of F -inverse covers of inverse monoids. Proof.
Define Ω : FG → FC on objects by sending each fuzzy subgroup ( G, µ, U )to its corresponding F -inverse cover ( C ( G, µ, U ) , ϕ, U ). Further, for each morphism( f, λ ) : ( G, µ , U ) → ( H, µ , V ) in FG , if ( C ( G, µ, U ) , ϕ, U ) and ( C ( H, µ ′ , V ) , ϕ ′ , V ) arethe corresponding F -inverse covers, we define f ∗ : C ( G, µ, U ) → C ( H, µ ′ , V )by setting f ∗ ( u, x ) = ( λ ( u ) , f ( x )) . This map is correct since λ ( u ) ≤ µ ( f ( x )). Indeed, from the definition of the morphism( f, λ ), we see that µ f ( x ) = λµ ( x ) ≥ λ ( u ) (since µ ( x ) ≥ u ) . Also f ∗ is a monoid homomorphism since if ( u, x ) , ( v, y ) ∈ C ( G, µ, U ) such that u ≤ v ,then f ∗ (( u, x )( v, y )) = f ∗ ( u ∧ v, xy )= f ∗ ( u, xy )= ( λ ( u ) , f ( xy ))= ( λ ( u ) ∧ λ ( v ) , f ( x ) f ( y )) (since λ ( u ) ≤ λ ( v ))= ( λ ( u ) , f ( x ))( λ ( v ) , f ( y ))= f ∗ ( u, x ) f ∗ ( v, y ) . Also if α, β are the respective units of
U, V , and e , e the units of G, H respectively, then f ∗ ( α, e ) = ( λ ( α ) , f ( e )) = ( β, e ) ,
5o that f ∗ preserves the unit element. Further, letting ϕ and ϕ be the correspondingmaps of the above covers, we see that for every ( u, x ) ∈ C ( G, µ, U ) ,ϕ f ∗ ( u, x ) = ϕ ( λ ( u ) , f ( x ))= λ ( u )= λϕ ( u, x ) . Lastly, if ( µ ( x ) , x ) is the greatest element of its σ -class, then f ∗ ( µ ( x ) , x ) = ( λµ ( x ) , f ( x )) = ( µ ′ f ( x ) , f ( x ))where ( µ ′ f ( x ) , f ( x )) is the greatest element of its σ -class. Since in addition to what wesaid, λ is a homomorphism of inverse monoids that maps the greatest element α of theonly σ -class of U to the greatest element β of the only σ -class of V , then it follows thatthe pair Ω( f, λ ) = ( f ∗ , λ ) is a morphism from ( C ( G, µ, U ) , ϕ, U ) to ( C ( H, µ ′ , V ) , ϕ ′ , V ).Next we show that Ω is functorial. It is obvious that when ( f, λ ) = (1 G , U ) is the identityon ( G, µ, U ), then Ω(1 G , U ) = id ( C ( G,µ,U ) ,ϕ,U ) . Let now( f, λ ) : ( G, µ , U ) → ( H, µ , V )and ( f ′ , λ ′ ) : ( H, µ , V ) → ( K, µ , W )be two morphisms in FG , and( f ′ f, λ ′ λ ) : ( G, µ , U ) → ( K, µ , W )their composition, and want to prove thatΩ( f ′ f, λ ′ λ ) = Ω( f ′ , λ ′ )Ω( f, λ ) , or equivalently that (( f ′ f ) ∗ , λ ′ λ ) = ( f ′∗ , λ ′ )( f ∗ , λ ) . This is the same as to prove that ( f ′ f ) ∗ = f ′∗ f ∗ . The latter is true since for every( u, x ) ∈ C ( G, µ , U ) we have that( f ′ f ) ∗ ( u, x ) = (( λ ′ λ )( u ) , ( f ′ f )( x ))= ( λ ′ ( λ ( u )) , f ′ ( f ( x )))= f ′∗ ( λ ( u ) , f ( x ))= f ′∗ ( f ( u, x )) . Next we prove that Ω is faithful. Let (
G, µ , U ) and ( H, µ , V ) be two objects in FG and( f, λ ) , ( f ′ , λ ′ ) : ( G, µ , U ) → ( H, µ , V )be two parallel morphisms, and assume that Ω( f, λ ) = Ω( f ′ , λ ′ ). Then, from the definitionof Ω, ( f ∗ , λ ) = ( f ′∗ , λ ′ ), consequently λ = λ ′ and f ∗ = f ′∗ . The second equality impliesthat for every ( u, x ) ∈ C ( G, µ , U ) we have that f ∗ ( u, x ) = f ′∗ ( u, x ). It follows that( λ ( u ) , f ( x )) = ( λ ′ ( u ) , f ′ ( x )), consequently f ( x ) = f ′ ( x ).6inally we prove that Ω is full. Let again ( G, µ , U ) and ( H, µ , V ) be two objects in FG and ( g, λ ) : ( C ( G, µ , U ) , ϕ , U ) → ( C ( H, µ , V ) , ϕ , V )be a morphism from Ω( G, µ , U ) to Ω( H, µ , V ). We show that g induces a homomorphism f : G → H such that g = f ∗ and µ f = λµ . This would prove that ( f, λ ) : ( G, µ , U ) → ( H, µ , V ) is a morphism in FG such that ( g, λ ) = Ω( f, λ ). For every x ∈ G , let ( µ ( x ) , x )be the greatest element of its σ -class in C ( G, µ , U ), and let ( µ ( x ′ ) , x ′ ) = g ( µ ( x ) , x )which is, from the assumption on g , the greatest element of its σ -class in C ( H, µ , V ). Itfollows that µ ( x ′ ) = ϕ ( µ ( x ′ ) , x ′ ) = ϕ g ( µ ( x ) , x ) = λϕ ( µ ( x ) , x ) = λµ ( x ) . (1)Since g preserves σ -classes, there is an induced homomorphism˜ f : C ( G, µ , U ) /σ → C ( H, µ , V ) /σ which maps the σ -class [( µ ( x ) , x )] to the σ -class [( µ ( x ′ ) , x ′ )]. Considering now theismorphisms γ : G → C ( G, µ , U ) /σ such that x [( µ ( x ) , x )] , and κ : H → C ( H, µ , V ) /σ such that y [( µ ( y ) , y )] , we obtain a homomorphism f = κ − ˜ f γ : G → H such that f ( x ) = x ′ where x ′ is determined as above. Using (1) we see that µ f ( x ) = λµ ( x ), hence ( f, λ ) is a morphism in FG from ( G, µ , U ) to ( H, µ , V ). Now we provethat ( g, λ ) = Ω( f, λ ) = ( f ∗ , λ ) which amounts to saying that g = f ∗ . Before we provethis, we observe that for every ( u, x ) ∈ C ( G, µ , U ), the second coordinate of g ( u, x ) is x ′ = f ( x ) as determined above, since g preserves σ -classes, while the first coordinate is ϕ g ( u, x ) = λϕ ( u, x ) = λ ( u ) . So g ( u, x ) = ( λ ( u ) , f ( x )) = f ∗ ( u, x ) , consequently, g = f ∗ as claimed. This completes the proof. References [1] K. Auinger, M.B. Szendrei, On F -inverse covers of inverse monoids , J. Pure Appl.Algebra, 204 (2006), 493-506[2] M.V. Lawson, Inverse Semigroup. The Theory of Partial Symmetries , World Scien-tific, 1998[3] M.V. Lawson, S. W. Margolis, B. Steinberg,
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