![](data:image/png;base64,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as Dishkant implications. This hyperclass is known
as generalized Dishkant implications (or GD- impli-
cations for short). The downside of GD- implica-
tions is that they do not always satisfy (16). As a re-
sult, we refer to GD- operations rather than implica-
tions in general. The characterization of quadruples
(⊥,⊤,¬1,¬2), such that Σ⊥,⊤,¬1,¬2satisfies (16) re-
mains unsolved. The same problem holds for the
characterization of triples (⊤,⊥,¬), such that Σ⊤,⊥,¬
satisfies (16) (see page 108 in [2]).
On the other hand, it has been demonstrated that
the set of D- operations is a subset of the set of GD-
operations, and the findings are depicted in Figure 1.
It has been demonstrated that a necessary but not suf-
ficient condition for a GD- operation to be a fuzzy
implication exists (see Corollary 1 and Remark 9).
Theorem 2 excludes quadruples (⊥,⊤,¬1,¬2)that
do not generate GD- implications, and Theorem 3 in-
vestigates the relationship of F−conjugation in GD-
operations.
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A Method of Generating Fuzzy Implications
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Dimitrios S. Grammatikopoulos: Writing- original
draft.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.78
Dimitrios S. Grammatikopoulos, Basil Papadopoulos