We introduce new classes of functions related to metric-preserving functions, b-metrics, and ultrametrics. We investigate their properties and compare them to those of metric-preserving functions.

In this paper, we define new classes of functions related to metrics and w-distances. We also provide characterizations of functions in these classes. As a consequence, we obtain relations between all classes.

Let [Formula: see text] be fixed. For each [Formula: see text], we define the set [Formula: see text] of sequences of infinite complex matrices and the norm [Formula: see text] on [Formula: see text] analogously to the classical sequence space [Formula: see text] and the [Formula: see text]-norm as follows: [Formula: see text] and [Formula: see text] where [Formula: see text] is referred to as the operator norm on [Formula: see text]. We first study the completeness of sequence space [Formula: see text] equipped with the norm [Formula: see text]. Some duality results are also discussed. The aim of this paper is to show that for the case where [Formula: see text], the dual [Formula: see text] of [Formula: see text] can be decomposed as an [Formula: see text] direct-sum of its two closed subspaces. This is done by a way analogous to the theorem of Dixmier on decomposing the dual [Formula: see text] of [Formula: see text].

In this paper, we introduce the notion of a solvable triple of binary relations on a set. This notion generalizes the notion of a regular relation and all other notions that are variants of the notion of the regularity, defined previously by many people. We also give some characterizations of the solvability of a triple of relations and use this to study Green’s relations on the monoid of binary relations on a set.