Strictly Increasing Additive Generators of the Second Kind of Associative Binary Operations

The class of strictly increasing additive generators of the second kind is defined and analyzed. Necessary and sufficient conditions for a binary operation generated by a strictly increasing additive generator of the second kind to be associative are introduced. The relation between the class of strictly increasing additive generators of the second kind of associative binary operations and the class of discrete upper additive generators of associative binary operations is revealed.

An Extension of Semiuninorms: Weak-Neutral Semiuninorms

By weakening the neutral element condition of semiuninorms, we introduce a new concept called weak-neutral semiuninorms (shortly, wn-semiuninorms). After analyzing their structure, several classes of wn-semiuninorms are presented and discussed. Particularly, based on a kind of monotone unary functions which are not necessarily continuous and strictly monotone, we introduce representable wn-semiuninorms and discuss some of their properties in detail. We show that there is no idempotent proper wn-semiuninorm. Each representable wn-semiuninorm is Archimidean but not strictly monotone, and its additive generator is unique up to a positive multiplicative constant under some conditions. In the discussion about the representable wn-semiuninorms, we also characterize the solutions to a class of Cauchy functional equations on a restricted domain.