Affiliations: [a]
School of Science, Linyi University, Linyi Shandong, P.R. China | [b]
School of Informatics, Linyi University, Linyi Shandong, P.R. China
Correspondence:
[*]
Corresponding author. Zhen Ming Ma, School of Science, Linyi University, Linyi Shandong 276005, P.R. China. Tel.: +86 13754710423; E-mail: [email protected]
Abstract: Filter theory plays a vital role in studying fuzzy logical systems and their algebraic structures. Various types of filters have been extensively investigated in the literature. In this paper, several types of filters related to the Stonean axiom in residuated lattices are introduced such as Stonean filters, subimplicative filters, De Morgan filters, RDP-filters and (weak) subimplicative-Stonean filters, the relationships between them and some existing types of filters are obtained. Particularly, it is shown that a filter is a weak subimplicative-Stonean filter if and only if subimplicative filter coincides with Stonean filter. By these filters, the Stonean axiom and its alternative definitions in an arbitrary residuated lattice are investigated. In particular, it is proved that Stonean axiom and subimplicative axiom coincide when the residuated lattice is subimplicative-Stonean, specially, De Morgan or RDP.
Keywords: Non-classical logics, residuated lattice, Stonean filter, subimplicative filter, De Morgan filter, RDP-filter, (weak) subimplicative-Stonean filter
