On Modal Logics of Full Products of Neighborhood Frames

Rajab AghamovRajab Aghamov, Andrey KudinovAndrey Kudinov, Maik NguyenMaik Nguyen, Jakob PiribauerJakob Piribauer

Rajab Aghamov, Andrey Kudinov, Maik Nguyen, Jakob Piribauer
On Modal Logics of Full Products of Neighborhood Frames
Advances in Modal Logic, volume 16, 2026. College Publications

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KurzfassungAbstract
On the product of two neighborhood frames, three natural neighborhood functions can be defined: the horizontal one assigning to a point $(x,y)$ the set of all supersets of sets $U\times\{y\}$, where $U$ is a neighborhood of $x$; the vertical analog; and the product neighborhood function assigning as neighborhoods all supersets of sets $U\times V$, for neighborhoods $U$ of $x$ and $V$ of $y$.
We define the tri-modal logics $\mathsf{T}\times_n^+ \mathsf{T}$ and $\mathsf{D}\times_n^+ \mathsf{D}$ of classes of full products equipped with all three neighborhood functions of neighborhood frames validating the logic $\mathsf{T}$ or $\mathsf{D}$; thereby extending known product results for $\mathsf{S4}$ and $\mathsf{D4}$ to weaker systems. Two interaction principles arise: $(\mathsf{sub})=\Box p \rightarrow \Box_1 p \land \Box_2 p$ and $(\mathsf{mix})=\Box p \rightarrow \Box_1\Box_2 p \land \Box_2\Box_1 p$, where $\Box$ for the product neighborhood function and $\Box_1,\Box_2$ the horizontal and vertical ones. Namely, we show that $\mathsf{T}\times_n^+ \mathsf{T} = \mathsf{T}\otimes \mathsf{T}\otimes \mathsf{T} + (\mathsf{mix})$ and
$\mathsf{D}\times_n^+ \mathsf{D} = \mathsf{D}\otimes \mathsf{D}\otimes \mathsf{D} + (\mathsf{mix})$, where $\otimes$ denotes fusion. Notably, $(\mathsf{sub})$ and $(\mathsf{mix})$ are equivalent over $\mathsf{S4}\otimes\mathsf{S4}\otimes\mathsf{S4}$ and thus $\mathsf{S4}\otimes\mathsf{S4}\otimes\mathsf{S4}+(\mathsf{mix})$ axiomatizes the logic of full products of topological spaces.
Bemerkung: Note: To appear
Projekt:Project: CPEC, SECAI
Forschungsgruppe:Research Group: Algebraische und logische Grundlagen der InformatikAlgebraic and Logical Foundations of Computer Science

@inproceedings{AKNP2026,
  author    = {Rajab Aghamov and Andrey Kudinov and Maik Nguyen and Jakob
               Piribauer},
  title     = {On Modal Logics of Full Products of Neighborhood Frames},
  booktitle = {Advances in Modal Logic},
  volume    = {16},
  publisher = {College Publications},
  year      = {2026}
}