Representation theorems for four classes of uninorms on bounded trellises

Abstract

In this paper, we establish necessary and sufficient representation theorems for the classes 𝒰̃_top, 𝒰̃_bot, 𝒰̃_max^r, and 𝒰̃_min^r of uninorms on bounded trellises, extending the corresponding bounded-lattice results. For a non-extremal neutral element e belonging to no non-trivial cycle, we use a common decomposition of the elements incomparable with e; every neutral element of a uninorm is shown to be middle-transitive. Each uninorm in 𝒰̃_top (respectively, 𝒰̃_bot) is represented by an interior operator (respectively, a closure operator), a component uninorm, and an increasing, associative, and commutative operation, with all components uniquely determined. Two further conditions characterize the subclasses 𝒰̃_top^⋆ and 𝒰̃_bot^⋆. For 𝒰̃_max^r (respectively, 𝒰̃_min^r), we identify the pairs in I_e³× I_e³ whose values lie in [0,e[∪ I_e¹ (respectively, ]e,1]∪ I_e²) or equal e. The pairs producing e satisfy a symmetric unique-partner condition, while the remaining values are encoded by a partial operation. Regional order compatibility and explicit associativity conditions then yield necessary and sufficient representations. Finally, we compare the four classes, relate them to the corresponding bounded-psoset classes, prove their bounded-lattice specialization under transitivity, and provide finite examples on proper trellises illustrating the additional compatibility conditions.

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BibTeX RIS

BibTeX

@misc{xiu2026representation,
  title = {{Representation theorems for four classes of uninorms on bounded trellises}},
  author = {Zhenyu Xiu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08191},
  eprint = {2610.08191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08191},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Representation theorems for four classes of uninorms on bounded trellises
AU  - Zhenyu Xiu
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08191
UR  - https://arxiv.org/abs/2610.08191
AB  - In this paper, we establish necessary and sufficient representation theorems for the classes 𝒰̃_top, 𝒰̃_bot, 𝒰̃_max^r, and 𝒰̃_min^r of uninorms on bounded trellises, extending the corresponding bounded-lattice results. For a non-extremal neutral element e belonging to no non-trivial cycle, we use a common decomposition of the elements incomparable with e; every neutral element of a uninorm is shown to be middle-transitive. Each uninorm in 𝒰̃_top (respectively, 𝒰̃_bot) is represented by an interior operator (respectively, a closure operator), a component uninorm, and an increasing, associative, and commutative operation, with all components uniquely determined. Two further conditions characterize the subclasses 𝒰̃_top^⋆ and 𝒰̃_bot^⋆. For 𝒰̃_max^r (respectively, 𝒰̃_min^r), we identify the pairs in I_e^3× I_e^3 whose values lie in [0,e[∪ I_e^1 (respectively, ]e,1]∪ I_e^2) or equal e. The pairs producing e satisfy a symmetric unique-partner condition, while the remaining values are encoded by a partial operation. Regional order compatibility and explicit associativity conditions then yield necessary and sufficient representations. Finally, we compare the four classes, relate them to the corresponding bounded-psoset classes, prove their bounded-lattice specialization under transitivity, and provide finite examples on proper trellises illustrating the additional compatibility conditions.
ER  -

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