On Topological AG-groupoids and Paramedial Quasigroups with Multiple Identities (original) (raw)

On topological groupoids and multiple identities

2009

This paper studies some properties of (n, m)-homogeneous isotopies of medial topological groupoids. It also examines the relationship between paramediality and associativity. We extended some affirmations of the theory of topological groups on the class of topological (n, m)-homogeneous primitive goupoids with divisions. Mathematics subject classification: 20N15.

A note on topological semigroup-groupoid

2013

In this paper we prove that the set of homotopy classes of paths in topological semigroup is a semigroup-groupoid. Further, we define the category TSGCov/X of topological semigroup coverings of X and prove that its equivalent to the category SGpGpdCov/ of covering groupoids of the semigroup-groupoid . We also prove that the topological semigroup structure of a topological semigroup-groupoid lifts to a universal topological covering groupoid.

ν-AG-QUASIGROUPS

In this paper, we define and study ν-AG-guasigroups. We discuss some of its structural properties and its relation with other classes of AG-groupoids. At the end, we show that under some given conditions an ν-AG-guasigroup becomes an abelian group.

Topological Group-Groupoids and Equivalent Categories

Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi

The concept of groupoid was offered by Brandt (1926). The structure of the topological groupoid was given by Ehresmann (1958). A groupoid action is a significant appliance in algebraic topology offered by Ehresmann. Another algebraic notion is a covering given by Brown (1988). The topological group-groupoids (Γ-groupoid) were first put forward by Icen & Ozcan (2001). The definition of coverings of topological Γ groupoid and actions of topological Γ-groupoid were also presented by Icen et al. (2005). In this paper, we are going to create a category TΓGpdCov(Γ) of covering morphisms of TΓ-groupoid and a category TΓGpdOp(Γ) of actions of TΓ-groupoid. We will then prove that these categories are equivalent.

On weaker forms for concepts in theory of topological groupoids

Journal of the Egyptian Mathematical Society, 2013

In this paper, we investigate the topologically weak concepts of topological groupoids by giving the concepts of a-topological groupoid and a-topological subgroupoid. Furthermore, we show the role of the density condition to allow a-topological subgroupoid inherited properties from a-topological groupoid and the irresoluteness property for the structure maps in a-topological groupoid is studied. We also give some results about the fibers of a-topological groupoids.

ACTA UNIVERSITATIS APULENSIS No 15/2008 G−N-QUASIGROUPS AND FUNCTIONAL EQUATIONS ON QUASIGROUPS

2008

In this paper we present criteria for an n-quasigroup to be isotopic to an n-group. We call a such n-quasigroup G−n-quasigroup. Applications to functional equations on quasigroups are presented in a subsequent paper. 2000 Mathematics Subject Classification: 20N15. Some important n-quasigroup classes are the following. An n-quasigroup (A, α) of the form α(x1 ) = n ∑ i=1 fi(xi)+a, where (A, +) is a group, f1, . . . , fn are some automorphisms of (A, +), a is some fixed element of A is called linear nquasigroup (over group (A, +)). A linear quasigroup over an abelian group is called T − n-quasigroup. An n-quasigroup with identity α(α(x 11 ), . . . , α(x nn n1 )) = α(α(x n1 11 ), . . . , α(x nn 1n ) is called medial n-quasigroup. All these quasigroups are isotopic to n-groups. This motivates the purpose of our work to find criteria for an n-quasigroup to be isotopic to an n-group.

A group of continuous self-maps on a topological groupoid

Semigroup Forum, 2017

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f (x)) is a composable pair for every x ∈ G, is denoted by S G. We show that S G by a natural binary operation is a monoid. S G (α), the group of units in S G precisely consists of those f ∈ S G such that the map x → x f (x) is a bijection on G. Similar to the group of bisections, S G (α) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that S G (α) with the compact-open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of G 2 is isomorphic to the group S G (α) and the group of transitive bisections of G, Bis T (G), is embedded in S G (α), where G 2 is the groupoid of all composable pairs.

On CM-groupoids with multiple identities and medial topological left loops

Acta et commentationes: Ştiinţe Exacte şi ale Naturii, 2022

This paper studies some properties of CM-groupoids with multiple identities and medial topological left loops. The conditions for a CM-groupoid to become a CM-quasigroup were found. A new method of constructing non-associative medial topological quasigroups with left identity is given. Various examples of quasigroups with multiple identities have been constructed

Local Group-Groupoids

2010

It is known that if X is a topological group, then the fundamental groupoid π1(X) is a group-groupoid, i,e, a group object in the category of groupoids. The group structure of a group-groupoid lifts to a covering groupoid. Further if G is a group-groupoid, then the category GpGdAct(G) of groupgroupoid operations and the category GpGdCov/G of group-groupoid coverings of G are equivalent. In this paper we prove the corresponding results for local topological groups and local group objects in the category of groupoids.