Representation theorems for four classes of uninorms on bounded trellises

In this paper, we establish necessary and sufficient representation theorems for the classes ${\tilde{\mathcal{U}}}_{\mathrm{t}\mathrm{o}\mathrm{p}}$, ${\tilde{\mathcal{U}}}_{\mathrm{b}\mathrm{o}\mathrm{t}}$, ${\tilde{\mathcal{U}}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^r$, and ${\tilde{\mathcal{U}}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^r$ 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 ${\tilde{\mathcal{U}}}_{\mathrm{t}\mathrm{o}\mathrm{p}}$ (respectively, ${\tilde{\mathcal{U}}}_{\mathrm{b}\mathrm{o}\mathrm{t}}$) 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 ${\tilde{\mathcal{U}}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\star }$ and ${\tilde{\mathcal{U}}}_{\mathrm{b}\mathrm{o}\mathrm{t}}^{\star }$. For ${\tilde{\mathcal{U}}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^r$ (respectively, ${\tilde{\mathcal{U}}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^r$), we identify the pairs in $I_e^3\times I_e^3$ whose values lie in $[0,e[\cup I_e^1$ (respectively, $]e,1]\cup I_e^2$) 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.