We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. We present some properties of the associated congruence subgroups. Namely, we show that a Hurwitz group defined by a congruence subgroup associated with an odd ideal, is necessarily defined by a principal congruence subgroup. All such Hurwitz gro...

Let S be topological semigroup, we consider an appropriate semigroup compacti cation b S of S. In this paper we study the connection between subgroups of a maximal group in a minimal left ideal of b S, which arise as equivalence classes of some closed left congruence, and the minimal ow characterized by the left congruence. A particular topology is de ned on a maximal group and it is shown that...

Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups...

A number of studies on political representation focus on the comparative assessments of citizen ideological congruence. But this literature has largely overlooked an important topic: the representation of social groups. While present studies of congruence investigate whether some countries perform better than others in terms of the levels of median citizen congruence, they cannot say much about...

The desymmetrized PSL(2,Z) group is considered. congruence subgroups of the are analysed according to hyperbolic reflections contained. Laplace-Beltrami problem and Hamiltonian associated with operator discussed boundary conditions for grouppal structures. New constructions leaky tori found; in particular, they obtained after generators group, from desymmetrized-triangle ’square-box’ Γ0 subgroup

Myhill-Nerode Theorem is regarded as a basic theorem in the theories of languages and automata and is used to prove the equivalence between automata and their languages. The significance of this theorem has stimulated researchers to develop that on different automata thus leading to optimizing computational models. In this article, we aim at developing the concept of congruence in general fuzzy...

Recently, the well-known Diie-Hellman key exchange protocol was extended to real quadratic congruence function elds in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals. This set does not possess a group structure, but instead exhibits a so-called infrastructure. The techniques are the same as in the protocol based on real quadratic number elds. A...