Abstract
In this paper, the notions of \(\mathcal {L}\)-ideal and \(\mathcal {L}\)-filter, where \(\mathcal {L}\) is a complete lattice, are introduced in hoop algebras, and their fundamental properties are investigated. Also, (weak) prime \(\mathcal {L}\)-ideals, minimal prime \(\mathcal {L}\)-ideals, and maximal \(\mathcal {L}\)-ideals are introduced, and in addition to basic properties, the relationships among them are investigated. Particularly, it is proved that the lattices of \(\mathcal {L}\)-ideals and \(\mathcal {L}\)-filters are order isomorphic. Next, the Zariski topology induced by weak prime \(\mathcal {L}\)-ideals is studied; it is proved that it is a compact \(T_0\)-space that is connected, under suitable conditions. Moreover, the (minimal) prime \(\mathcal {L}\)-ideal space and maximal \(\mathcal {L}\)-ideal space are studied. Furthermore, a homeomorphism between prime ideal spaces and certain subclass of prime \(\mathcal {L}\)-ideal spaces is obtained.
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Notes
In some contexts, this is called ‘Double Negation Property’ (DNP for short) see Aaly and Borzooei (2020)
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Bakhshi, M. Prime \(\mathcal {L}\)-ideal spaces in hoop algebras. Soft Comput 27, 629–644 (2023). https://doi.org/10.1007/s00500-022-07599-3
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DOI: https://doi.org/10.1007/s00500-022-07599-3