We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C *-algebra C * (T) associated with it which is a subalgebra of the uniform Roe algebra C * u (X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently group-like, as determined by our invarian...

The concept of topological gyrogroups is a generalization group. In this work, ones prove that gyrogroup G metrizable iff has an {\omega}{\omega}-base and Frechet-Urysohn. Moreover, in gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn having are all weakly three-space properties with H closed L-subgyrogroup

An algebra A is uniform if for each θ ∈ ConA, every two classes of θ have the same cardinality. It was shown by W. Taylor that coherent varieties need not be uniform (and vice versa). We show that every coherent variety having transferable congruences is uniform.

Let A = (A;F ) be an algebra and ConA its congruence lattice. Recall that A is congruence uniform (see e.g. [1], [5]) if for each Θ ∈ ConA and every a, b ∈ A, card[a]Θ = card[b]Θ. Examples of congruence uniform algebras are e.g. groups, rings or Boolean algebras. A variety V is congruence uniform if each A ∈ V has this property. It was proved by W. Taylor [5] that every congruence uniform varie...

We introduce an algebra $\mathcal{K}_n$ which has a structure of left comodule over the quantum toroidal type $A_{n-1}$. Algebra is higher rank generalization $\mathcal{K}_1$, provides uniform description deformed $W$ algebras associated with Lie (super)algebras types BCD. show that possesses family commutative subalgebras.

We are to discuss how to view notions of computability for discontinuous functions. We con ne ourselves to real-valued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. I...