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 Ia3× Ia3, where Ia3 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 Ia3. 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.
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BibTeX
@misc{xiu2026complete,
title = {{A complete representation theorem for nullnorms on bounded trellises}},
author = {Zhenyu Xiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14509},
eprint = {2609.14509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14509},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A complete representation theorem for nullnorms on bounded trellises
AU - Zhenyu Xiu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14509
UR - https://arxiv.org/abs/2609.14509
AB - 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× 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.
ER -