## 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.

### Citations:

Zbl 1366.03229; Zbl 1424.03060; Zbl 1452.03120
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