Representations into the hyperspace of a compact group

Embedding topological semigroups into the hyperspaces over topological groups

“…Theorem 2 allows us to construct many examples of algebraically regular topological semigroups non-embeddable into the hypersemigroups over topological groups. The first two items of this proposition imply the result of [3] that non-trivial rectangular semigroups and connected topological semilattices do not belong to the class H. The last two items imply that the class H does not contain neither Brandt nor bicyclic semigroups. A bicyclic semigroup is a semigroup generated by two elements p, q connected by the relation qp = 1.…”

“…In is clear that the class H contains all topological groups. On the other hand, the compact topological semigroup ([0, 1], min) does not belong to H, see [3]. In this paper we establish some inheritance properties of the class H and on this base detect compact Clifford semigroups belonging to H: those are precisely compact Clifford inverse semigroups with zero-dimensional idempotent semilattice.…”

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

“…This proposition allows one to construct many examples of topological regular semigroups non-embeddable into the hypersemigroups or convolution semigroups over a topological groups. The first two assertions of this proposition imply the result of [8] to the effect that non-trivial semigroups of left (or right) zeros as well as connected topological semilattices do not embed into the hypersemigroup exp(G) over a topological group G. The last two assertions imply that the semigroups exp(G) and P (G) do not contain Brandt semigroups and bicyclic semigroups. By a Brandt semigroup we understand a semigroup of the form B(H, I) = I × H × I ∪{0} where H is a group, I is a non-empty set, and the product (α, h, β) * (α ′ , h ′ , β ′ ) of two non-zero elements of B(H, I) is equal to (α, hh ′ , β ′ ) if β = α ′ and 0 otherwise.…”

“…This problem was resolved in [6] for the class of Clifford compact topological semigroups: such a semigroup S embeds into the hypersemigroup over a topological group if and only if the set E of idempotents of S is a zero-dimensional commutative subsemigroup of S. This characterization implies the result of [8] that the closed interval [0, 1] with the operation of the minimum does not embed into the hypersemigroup over a topological group.…”

Extending binary operations to funtor-spaces

“…Assume conversely that (xy)z = x(yz) for some points x, y, z ∈ X. Since X is Hausdorff, the points (xy This problem was addressed in [10], [11] for the monadic functor G of inclusion hyperspaces, in [2]- [5] for the functor of superextension λ, in [1], [12], [15] for the functor P of probability measures and in [6], [7], [8], [18] for the hyperspace functor exp.…”