Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U, V)-group over X if (a) X is a subspace of G, (b) GE U, and (c) every continuous f:X+ H with H EV extends uniquely to a continuous homomorphism _?: G + H. For certain classes U and V, we consider the question of the existence of free (U, V)groups. Our principal result...

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its the Stone-Čech compactification βS provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y ...

Comfort, W.W. and J. van Mill, Some topological groups with, and some without, proper dense subgroups, Topology and its Applications 41 (1991) 3-15. Continuing earlier investigations into the question cf the existence of a proper dense subgroup of a given topological group, the authors obtain some positive and some negative results, as follows: (a) Every non-degenerate connected Abelian group c...

The paper contains results that characterize the Donkin–Koppinen filtration of coordinate superalgebra K[G] general linear supergroup G = GL(m|n) by its subsupermodules CΓ OΓ(K[G]). Here, supermodule is largest subsupermodule whose composition factors are irreducible supermodules highest weight ⋋, where ⋋ belongs to a finitely-generated ideal Γ poset X(T)+ dominant weights G. A decomposition as...

<p>For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on C ∗ (X, its subring bounded functions, and Σ(X, denote collection all rings that lie between C). We show there is natural correlation absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in P(X, their real-valued counterparts ∩ C(X). These corr...

Let C(X) be the ring of all continuous real-valued functions defined on a completely regular T1-space. Let CΨ (X) and CK(X) be the ideal of functions with pseudocompact support and compact support, respectively. Further equivalent conditions are given to characterize when an ideal of C(X) is a P -ideal, a concept which was originally defined and characterized by Rudd (1975). We used this new ch...