Abstract Let $\textbf{Lat}_{\lor }$ denote the category of lattices with morphisms preserving supremum of non-empty finite subsets. We prove the following: (1) There exists a distributive law of the ordered-ideal monad over the ordered-filter monad on $\textbf{Lat}_{\lor }$. (2) The category ${\textbf{Cnt}}^{\mathbb{co}}$ of cocontinuous lattices with morphisms preserving arbitrary non-empty sups and filtered infs is strictly monadic over $\textbf{Lat}_{\lor }$. (3) The category ${\textbf{MCnt}}^{\mathbb{co}}$ of meet-continuous and cocontinuous lattices with morphisms preserving arbitrary non-empty sups and filtered infs is isomorphic to a full subcategory of ${\textbf{Lat}_{\lor }}^{\mathbb{FilId}}$, consisting of $\mathbb{FilId}$-algebras and $\mathbb{FilId}$-homomorphisms on ${\textbf{Lat}_{\lor }}$. For bounded-complete posets, analogous Eilenberg–Moore algebras provide a categorical representation. Finally, using the concept of bounded down-sets, we derive an algebraic characterization of ${{\textbf{Cnt}}}^{\mathrm{co}}$.

Leveraging its exceptionally wide bandwidth, Terahertz (THz) communication offers ultra-high-speed data transmission and remarkably precise sensing, establishing itself as a cornerstone technology for integrated sensing and communication (ISAC) systems. This paper investigates a multi-base-station (multi-BS) cooperative extremely-large-scale multiple-input multiple-output (XL-MIMO) orthogonal frequency division multiplexing (OFDM) near-field THz-ISAC system designed to guarantee secure downlink communication for multiple users, while simultaneously enhancing multi-target localization accuracy. The core challenge lies in optimizing the secrecy rate subject to the Cramer-Rao Bound (CRB) constraint to strike an´ effective balance between communication security and sensing precision. To this end, we propose an unsupervised learning-based complex-valued deep neural network (CVNN) that jointly optimizes hybrid beamforming and radar sensing signals in a data-driven manner. Simulation results unveil that the proposed approach outperforms the conventional alternating optimization-based hybrid beamforming benchmark in terms of secrecy rate and computation time. These results validate the practicality of data-driven joint optimization in THz-ISAC systems by demonstrating its effectiveness in achieving an efficient tradeoff between communication security and sensing accuracy, while also providing actionable design guidelines for scalable, low-latency system in extremely-large-scale deployment.

Within the framework of fuzzy algebras with fuzzy equality and a complete lattice for membership values, this paper introduces a novel fuzzified version of classical Green's relations on E -fuzzy semigroups, referred to as E -fuzzy Green's relations. We reveal several properties of E -fuzzy semigroups, including the properties of fuzzy unit elements within E -fuzzy cancellative semigroups. Additionally, we derive equivalent forms of these newly defined fuzzy relations and examine the properties of their associated cut-quotient structures in certain E -fuzzy semigroups. The E -fuzzy Green's relations on E -fuzzy semigroups are found to be compatible fuzzy equivalence relations (fuzzy equalities) on some E -fuzzy bands. Furthermore, we present two significant decomposition theorems related to the cut-quotient structures over fuzzy Green's relations, demonstrating that these structures have additional algebraic properties compared with the quotient structures over ordinary fuzzy equality E μ .