# An Algorithm for Producing Fuzzy Implications via Conical Sections
## 3 Main Results  
We present an algorithm to produce fuzzy implications of a special form based only on a strong fuzzy negation. The specific strong fuzzy negation employed here is based on the properties of conical sections and was introduced by [PERSON] in [PERSON] and [PERSON] in [PERSON] and Papadopoulos (2018). To this end, first recall that, as demonstrated in Theorem 1, a t-conorm can be generated via a function \(g\) as  
\[S(u,v)=g^{-1}(\min\{g(u)+g(v),g(1)\}),\quad u,v\in[0,1]\]  
and note that \(g:[0,1]\to[0,\infty]\) is a continuous function, which is strictly increasing in its domain. Let \(g(u)=N_{cs}(1-u)\), where \(N_{cs}\) denotes the strong negations based on conical sections introduced in [PERSON] and [PERSON] (2018). As shown by the authors, this can take one of two forms. Specifically, in the first case, we have  
\[N_{cs}(u)=\frac{1-u}{1+mu},\quad m>-1, \tag{2}\]  
which leads to the so-called Sugeno family of negations. In the second case, we have  
\[N_{cs}(u)=\sqrt{(m^{2}-1)u^{2}+1}+mu,\quad m\leq 0. \tag{3}\]  
Note that, using the previous transformation, \(g\) is indeed a strictly increasing function, with the additional boundary condition \(g(0)=0\). Finally, recall that \(N_{cs}\) was proven to be strong; i.e., \(N_{cs}^{-1}=N_{cs}\). Now, note that we have  
\[g^{-1}(u)=(1-u)^{-1}\circ N_{cs}^{-1}(u)=1-N_{cs}(u). \tag{4}\]  
This leads to  
\[S(u,v)=1-N_{cs}(\min\{N_{cs}(1-u)+N_{cs}(1-v),1\}) \tag{5}\]  
and finally  
\[J_{cs}(u,v) =S(N(u),v)\] \[=1-N_{cs}(\min\{N_{cs}(1-N_{cs}(u))+N_{cs}(1-v),1\}). \tag{6}\]  
Thus, the family of strong negations presented in Equations (2) and (3) is utilized to parametrically produce infinitely many implications. Due to the fact that these negations are strong, the produced family of implications possesses several desirable properties, as shown in the sequel. To this end, for the above implication \(J_{cs}(u,v)\), the properties presented in the proposition below are true:  
**Proposition 2**. : _Let \(N_{cs}\) be a fuzzy negation as defined in (2) or (3). Then, for the fuzzy implication provided by (6), we have_  
1. \(J_{(S,N)}\in\mathcal{F}\mathcal{J}\)_; i.e., it satisfies properties (I-I-J5) and is indeed a fuzzy implication;_
2. _condition (NePr) and condition (ExPr) are valid;_
3. _condition (CP) as well as conditions (L-CP)-(R-CP) are valid;_
4. _condition (ldPr) and condition (OrdPr) are valid if_ \(-1<m\leq 0\)Proof of Proposition 2.: The proof of _(i)_ and _(ii)_ is a direct application of the corresponding points in Proposition 1. The proof of _(iii)_ is a direct application of points _(iii)_ and _(v)_ in Proposition 1 and the fact that every strong fuzzy negation is also strict [10] (p. 15). For point _(iv)_, note that, since \(N_{cs}\) was proven to be strong in [9], from _(vi)_ and _(vii)_ in Proposition 1, we only need to prove that, if \(-1<m\leq 0\), then the disjunction-negation pair \((S,N_{cs})\), with \(S\) defined as in (5) and \(N_{cs}\) defined as in Equations (2) and (3), satisfies Equation (LEM). To this end, first note that, from [9], we have  
\[-1<m\leq 0\Rightarrow N_{cs}(u)\geq 1-u. \tag{7}\]  
A straightforward substitution \(u\to 1-u\) can be performed in (7), which yields the following inequality:  
\[N_{cs}(1-u)\geq u. \tag{8}\]  
Similarly, an additional substitution \(u\to 1-N_{cs}(u)\) can also be performed in (7). In turn, this gives rise to the following series of inequalities:  
\[N_{cs}(1-N_{cs}(u))\geq 1-(1-N_{cs}(u))\Rightarrow N_{cs}(1-N_{cs}(u))\geq N _{cs}(u) \tag{9}\]  
The inequalities presented in (8) and (9) can be combined, which results in the following relation:  
\[N_{cs}(u)=\frac{1-u}{1+mu},\quad m>-1, \tag{2}\]0  
where the last inequality holds due to (7).