This paper describes the distributivity, the modularity, the semimodularity and the lower semimodularity of compactly gener- ated lattices from a view of cut sets of L-valued sets, respectively. Applying terms of cut sets of L-valued sets, it gives some sufﬁcient and necessary conditions which can be used to determine whether a compactly generated lattice is distributive, modular, semimodular and lower semimodular, respectively. Keywords L-valued set · Cut set · Compactly generated lattice · Distributive lattice · Modular lattice · Semimodular lattice · Lower semimodular lattice 1 Introduction possible to present an ordered structure by cut sets of a suit- able fuzzy set μ and consequently by the mapping μ itself Lattice-valued mathematics had undergone a signiﬁcant as pointed out by Šešelja and Tepavce ˇ vic( ´ 2003a, b). This development from Goguen’s (1967) ﬁrst paper. In this way, topic has been discussed by many scholars (see, e.g., Jiménez bridges had been created between fuzzy mathematics and et al. 2010, 2011a, b; Gorjanac-Ranitovic´ and Petojevic´ other branches like automata and tree series (Borchardt 2014; Gorjanac-Ranitovic´ and Tepavce ˇ vic´ 2018; Šešelja et al. et al. 2006), theoretical computer science (Chechik et al. 2008). On the other hand, the distributivity, the
Soft Computing – Springer Journals
Published: Jun 2, 2018
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