A complete representation theorem for nullnorms on bounded trellises

Abstract

We establish necessary and sufficient conditions under which a binary operation on a bounded trellis is a proper nullnorm. The representation combines a t-conorm on the lower interval, a t-norm on the upper interval, two order-preserving maps, and a commutative, increasing function on $I_a^3\times I_a^3$, where $I_a^3$ consists of the elements incomparable with the absorbing element $a$ that neither reach $a$ nor are reachable from $a$. Unlike earlier range-restricted constructions, this function may take values anywhere in the trellis. To preserve associativity for such unrestricted values, we introduce a mixed interaction function that evaluates every pair with at least one component in $I_a^3$. We also derive the specializations in which this region is empty or consists of a single element, including an exact description of the admissible value in the singleton case. A five-element lattice example shows that the mixed associativity condition is independent of the remaining hypotheses, while a fourteen-element nontransitive trellis example demonstrates the necessity of allowing the unrestricted range. Finally, the principal range-restricted subclasses and the bounded-lattice case are recovered as specializations of the general representation.