in this paper we introduce and study an algebraic structure, namely grouplike. a grouplike is something between semigroup and group and its axioms are generalizations of the four group axioms. every grouplike is a semigroup containing the minimum ideal that is also a maximal subgroup (but the converse is not valid). the first idea of grouplikes comes from b-parts and $b$-addition of real number...

We prove that the delooping, i. e., the classifying space, of a grouplike monoid is an H-space if and only if its multiplication is a homotopy homomorphism, extending and clarifying a result of Sugawara. Furthermore it is shown that the Moore loop space functor and the construction of the classifying space induce an adjunction of the according homotopy categories. 2000 Mathematics Subject Class...

in this paper we give a characterization for all semigroups whose square is a group. moreover, we axiomatize such semigroups and study some relations between the class of these semigroups and grouplikes,introduced by the author. also, we observe that this paper characterizes and axiomatizes a class of homogroups (semigroups containing an ideal subgroup).  finally, several equivalent conditions ...

In this paper we introduce and study an algebraic structure, namely Grouplike. A grouplike is something between semigroup and group and its axioms are generalizations of the four group axioms. Every grouplike is a semigroup containing the minimum ideal that is also a maximal subgroup (but the converse is not valid). The first idea of grouplikes comes from b-parts and $b$-addition of real number...

We extend the Larson–Sweedler theorem [10] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplik...

We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...

We show how to construct central and grouplike quantum determinants for FRT algebras A(R). As an application of the general construction we give a quantum determinant for the q-Lorentz group.