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Note on “construction of uninorms on bounded lattices”. (English) Zbl 07396272

The existence of uninorms on bounded lattice \((L,0,1,\vee,\wedge)\) was proved by F. Karaçal and R. Mesiar [Fuzzy Sets Syst. 261, 33–43 (2015; Zbl 1366.03229)]. Then many papers provided particular constructions of such uninorms with neutral element \(e\neq 0,1\). Authors of this note observed that in the paper by G. D. Çaylı and F. Karaçal [Kybernetika 53, No. 3, 394–417 (2017; Zbl 1424.03060)] some assumptions about lattice \(L\) are false (e.g. assumption “for every \(x,y\in L\), if \(x,y\| e\), then \(x\vee y>e\)” is false, if such \(x\in L\) exists, because then \(x\vee x=x\| e\) ). Such assumption is formally true, if all lattice elements are comparable with \(e\), i.e. \(L=[0,e]\cup[e,1]\) with lattice intervals, what reduces many considerations and results of the mentioned paper. Another correction concerns direct descriptions in \(L^2\) of constructed uninorms, where domains of diverse formulas should be disjoint (except for \([0,e]^2\cap[e,1]^2=\{(e,e)\}\)).
This note brings a generalized construction of uninorms, new assumptions and new proof. The construction uses closure and interior operators (cf. Y. Ouyang and H.-P. Zhang [Fuzzy Sets Syst. 395, 93–106 (2020; Zbl 1452.03120)]).
Reviewer’s remark: Authors use notation \(I_e\) without explanation (it is copied from reference papers and denotes the set of all elements of lattice \(L\) non-comparable with \(e\)).

MSC:

03G10 Logical aspects of lattices and related structures
03B52 Fuzzy logic; logic of vagueness
06B05 Structure theory of lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B20 Varieties of lattices
03E72 Theory of fuzzy sets, etc.
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