Monoid and Topological groupoid Mohammad Qasim Mann'a Department of Mathematics College of Science, Al-Qassim University P.O. Box: 6644-Buraidah: 51452, Saudi Arabia e-mail:[email protected] Abstract., Here we introduce some new results which is relative to the concept of topological monoid-groupoid and prove that the category of topological monoid coverings of X is equivalent to the category covering groupoids of the monoid- groupoid 1 X . Also, show that the monoid structure of monoid-groupoid lifts to a universal covering groupoid. Keywords. Fundamental groupoid, covering groupoids, topological groupoid, topological semigroup and monoid. Mathematical Subject Classification 2010: 22A05, 20L05, 22A22, 22A15, 06F05. 0.Introduction The groupoid is one of important subjects of category theory. In this work we introduced the concepts of topological monoid-groupoids and discuss some properties which relative to this concept. Also we find the fundamental groupoid of topological monoid whose underlying space has a universal covering is a topological monoid-groupoid. Moreover, we show that the category of topological monoid covering of X is equivalent to the category of covering groupoids of monoid-groupoids. Further, let G be monoid-groupoid ~ ~ and G be a groupoid and p : G G be a universal covering on the underlying ~ ~ groupoids such that G and G are transitive groupoids such that OP (e) e , then the monoid structure of G lifts. 1.Preliminaries A groupoid [1, 2, 7]G is small category consists of two sets G andOG , called respectively the set of elements (or arrows) and the set of objects (or vertices) , of the groupoid, together with, two maps : G OG , called respectively the source and target maps, the map :OG G, written as (x)= 1 x , where O 1x is called the identity element at x in G , and is called the object map and G G G the partial multiplication map; : , (g, h) g h, where G G (g, h)G G :(g) (h) . These terms must satisfy the followings axioms: ( g h ) = (h), ( g h ) = (g), 1 ( g) ( g h ) k=g (h k), (1x ) (1x ) x , g 1 ( g) =g and g=g. For all g, h, k 1 G and xOG . The map : G G , g g is called the inversion map and each element g has an inverse g 1 such that (g 1 ) (g) , (g 1 ) (g) , 1 1 g g 1 (g ) , g g 1 (g ). y Further, for we write G(x, y) Gx as the set of all morphisms g G such that st (x) 1 (x) G (g) x and (g) y . each x, y OG . We will write G x 1 y y and costG (y) (y) G and Gx G(x, y) stG (x) costG (y). The vertex group at x isG(x) G(x, x) stG (x) costG (x). We say G is transitive if for y each x, y OG , Gx is non-empty. A morphism of groupoids H and G is a functor, that is, it consists of a pair of functions f : H G and O f :OH OG preserving all structures. A subgroupoid of G is a subcategory N of G which is also a groupoid. We say that, N y y is full (wide) subgroupoid if for all x, y ON then,Gx N x (ON OG ).Moreover, N is called normal subgroupoid if N wide and a N(x) a 1 N(y) or a 1 N(y) a N(x) a G y for any x, y OG and x . A topological groupoid is a groupoid G together with topologies on G and OG such that all the structure maps of G are continuous, that is: the source map , the target map , the object map , the inversion and the partial multiplication map are all continuous. A morphism of topological groupoids is a pair of maps f :G H and O f :OG OH such that f and O f are continuous. Note that in this definition the partial multiplication map and the inversion map are continuous if and only if the map :G G G, (g h) g h 1 , called the difference map, is continuous, where G G (g, h)G G :(g) (h) has a subspace topology fromG G . Further, if the source, target and the inversion are continuous then the other one is continouos. Moreover, in topological groupoid G, y x y for each g Gx left-translation lg : G G ,lg (h) g h and the right-translation rg :Gy Gx ,rg (h) h g are homeomorphisms. ~ A morphism P : G G of groupoids is called a covering morphism [2, p.365-372] ~ ~ ~ and G a covering groupoid of G if each x OG then the restriction of mapping ~ ~ (G) ~ G ~ x OP (x ) of P is a bijective. Let P : G G be a morphism of groupoids . Then ~ ~ ~ ~ ~ for an object x OG the subgroup P(G(x))of G(OP (x)) ) is called the characteristic ~ ~ ~ group of P at x . So if P is the covering morphism then P maps G(x) isomorphically ~ ~ ~ to P(G(x)) . We say that a covering morphism P : G G of transitive groupoids is a ~ universal covering morphism if G is covers every cover of G, i.e., if for every ~ covering q : A G there is a unique morphism of groupoids q : G A such that qq P (and hence q is also a covering morphism). This is equivalent to saying that ~ ~ ~ ~ ~y ~ ~ ~ for x, y OG the set G(x, y) G~x has one element at most. We recall from Howie [5] that a semigroup G is said to be monoid if there is a unique e (called the identity) such that g = eg ge for every g in G. From 6 A monoid-groupoid G is a groupoid endowed with monoid structure such that the following maps: 1. m : GG G , defined by (g, h) gh, a semigroup multiplication, 2. e : G , where is singleton. If the identityeOG , then 1e G is the of G, are morphisms of groupoids. A morphism f :G H of monoid-groupoids is a morphism of underlying groupoids preserving the monoid structure i.e. f (gh) f (g) f (h), and f (1 ) 1 ,for eG eH all g, hG. In a monoid-groupoid. The source, target and object maps are morphisms (homomorphisms) of monoid and the inversion map is an isomorphism of monoid. Also, the interchange law holds, i.e. (b a)(g h) (bg) (ah) . Let X be a topological space then 1 (X ) is called a fundamental groupoid of X which is the set of all relative to end points homotopy classes of paths in topological space[2, p.213]. Let X be a connected and locally path connected X has a universal covering if and only if X semi-locally 1-connected [8, Corollary 10.37]. Assume [1] that a topological space X are locally path connected and semi-locally 1- connected, so that each path component of X admits a simply connected cover. Recall ~ that a covering map p : X X of connected spaces is called universal if it covers ~ every cover of X in the sense that if q :Y X is another cover of X then there exists a ~ ~ map r : X Y such that p = qr (hence r becomes a cover). A covering map ~ ~ p : X X is called simply connected if X is simply connected. So a simply connected cover is a universal cover. A subset U of the space X is called liftable (see section 2, in 1) if it is open, path connected and U lifts to each cover of X, that is, if ~ ~ p : X X is a covering map, i :U X is the inclusion map, and ~x X satisfies ~ ~ ~ p(x) x U, then there exists a map (necessarily unique) i :U X such that ~ ~ pi i and i(x) x. It is easy to see that U is liftable if and only if it is open, path connected and for each x U, the fundamental group1(U, x) , is mapped to a singleton by the morphism induced by the inclusion map i :U X . Note that if a connected and locally path connected space X has a universal covering then each point x X has a liftable neighborhood. 2. Topological Monoid-groupoid The concept of the following definition from taken from [6]. Definition2.1 A topological monoid-groupoid G is a toplogical groupoid endowed with a topological monoid structure such that the following monoid structure maps are morphism of topological groupoids: Tmg1. m :GG G , the monoid multiplications, Tmg2. e : G , where is singleton. A morphism of topological monoid-groupoids f : G H is a morphism of the underlying topological groupoids preserving the topological monoid structures. Example 2.2 Let G be a topological monoid . Then G G is a topological monoid- groupoid with the object set G and the morphims are the pair(a,b) . The source, target, object and the inversion maps are defined respectively by (a,b) b and (a,b) a , (a) (a,a) and (a,b) (b, a).The composition of groupoid is defined by (a,b) (b,c) (a,c) , the monoid multiplication is defined by (a,b)(d,c) (ad,bc) and the identity (e,e) of G G where e is the identity of G. Observe that all of the structure maps are continuous. Proposition 2.3 Let G be a topological monoid-groupoid. Then the set of arrows of G and the set of objects OG are topological monoid.