Ultra Approach Groups: Characterizations, Examples, and Invariant Pseudo-Metrizability

The motivation of this paper is to discuss some aspects of approach groups, recall and reformulate a characterization theorem in a self-contained form, and provide various natural examples in a new setting; in so doing, we show that every bi-invariant Menger probabilistic metric structure on a group with respect to a triangular norm Tm gives rise to an ultra approach group. Furthermore, the ultra setting is natural here because the constructions considered below are governed by maximum-type estimates arising from the triangular norm Tm=min and from invariant ultra ∞p-metric structures. We discuss convergence approach groups and their associated approach uniform structure together with some of their relationships with approach groups. In this respect, we observe that given a convergence approach group, its induced approach uniform convergence structure is translation-invariant both on the left and on the right. Finally, we focus on the invariant pseudo-metrizability of approach groups and present several related results. Here and among others, we present an example of an invariant ∞p-metric on a group leading to an approach group.