Abstract
The aim of this paper was to study the Belluce-semilattice associated with a monadic residuated lattice. Some characterizations of \(\cap \)-prime monadic filters and maximal monadic filters are derived, respectively. The \(\cap \)-prime monadic filter theorem is also established. The relationships among strong monadic filters, \(\cap \)-prime monadic filters, prime monadic filters and maximal monadic filters are discussed. It is proven that there are monadic residuated lattices having no prime monadic filters and that \(\cap \)-prime monadic filters and prime monadic filters coincide in strong monadic residuated lattices. We consider the Belluce-semilattice associated with a monadic residuated lattice and demonstrate that the Belluce-semilattice associated with a monadic residuated lattice (a strong monadic residuated lattice) is a bounded distributive meet-semilattice (lattice). The homeomorphism between \(\cap \)-prime monadic filter space of a monadic residuated lattice and \(\cap \)-prime filter space of its Belluce-semilattice is obtained. In addition, we derive that there is a meet-semilattice isomorphism between the Belluce-semilattices associated with a monadic residuated lattice and its underlying m-relatively complete subalgebra.
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Acknowledgements
The authors are very grateful to the anonymous referees for their constructive comments and suggestions.
Funding
This work was supported by the NSFC (Nos. 61273017, 11971417, 12271319) and by the Shanghai Key Laboratory of All Solid-State Laser and Applied Techniques open research programme (ADL_2020002).
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Liu, L., Zhang, X. The Belluce-semilattice associated with a monadic residuated lattice. Soft Comput 27, 6983–6998 (2023). https://doi.org/10.1007/s00500-023-08023-0
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DOI: https://doi.org/10.1007/s00500-023-08023-0