The Structure and Properties of (ξ, s) - Generated Implications | Wuhan University Journal of Natural Sciences

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Table 1

Basic properties of generator generated implications

I	$(U G)$ ¹	$(N P)$	$(E P)$	$I (x, y) = 1$	$C P (N)$	$N_{I}$
$I_{f}$	$c$	√	√	$\Leftrightarrow x = 0 \lor y = 1$	$\Leftrightarrow f (0) < \infty \land N = N_{f_{1}}$ is strong²	$N_{I} = N_{G}$ if $f (0) = \infty$
						$N_{I}$ is strict if $f (0) < \infty$
						$N_{I}$ is strong if $f (0) < \infty \land N_{f_{1}}$ is strong
$I_{g}$	$c$	√	√	$\Leftrightarrow g (1) < \infty, x g (1) \leq g (y)$	×	$N_{I} = N_{G}$
$I_{g}$	$c$	√	√	$\Leftrightarrow g (1) = \infty, x = 0 \lor y = 1$	×	$N_{I} = N_{G}$
$I_{h}$	1	√	√	$\Leftrightarrow h (1) = 0, x = 0 \lor y = 1$	$\Leftrightarrow h = h^{- 1} \land N$ $= N_{h}$	$N_{I}$ is strict if $h (1) = 0$
$I_{h}$	1	√	√	$\Leftrightarrow h (1) > 0, x h (y) \leq h (1)$	$\Leftrightarrow h = h^{- 1} \land N$ $= N_{h}$	$N_{I}$ is strong if $h = h^{- 1}$
$I_{k}$	1	√	√	$\Leftrightarrow x \leq k (y)$	×	$N_{I} = N_{G}$ if $k (0) = 0$
$I_{k}$	1	√	√	$\Leftrightarrow x \leq k (y)$	×	$N_{I}$ is a NCDN³if $k (0) > 0$
$I_{θ, t}$	See Ref. [25] Theorem 1	√	√ $(θ$ cont. on $[0,1)$ )	$\Leftrightarrow t (x) \geq θ (1^{-}) - θ (y)$	$\Leftrightarrow N = N_{θ, t}$ is strong	$N_{θ, t} = N_{⊤} \Leftrightarrow t (1^{-}) \geq θ (1^{-}) - θ (0)$
						$N_{θ, t}$ is continuous $\Leftrightarrow$ so is $t$
						$N_{θ, t}$ is a NCDN $\Leftrightarrow$ $θ$ is continuous on $[0,1)$ and $t$ is continuous on $[0,1]$
						$N_{θ, t}$ is strict $\Leftrightarrow$ $θ$ is continuous on $[0,1)$ and $t$ is continuous on $[0,1]$ with $t (0) = θ (1^{-}) - θ (0)$
						$N_{θ, t}$ is strong $\Leftrightarrow$ $θ$ is continuous on $[0,1)$ and $t$ is given by Eq. (5) in Ref. [25]⁴
$I_{ξ, s}$	See Prop. 1	√	√	$\Leftrightarrow s (x) + ξ (y) \leq s (1) + ξ (1)$	$\Leftrightarrow N = N_{ξ, s}$ is strong	$N_{ξ, s} = N_{⊤} \Leftrightarrow s (1) + ξ (1) \geq 1 + s (x)$
						$N_{ξ, s}$ is continuous $\Leftrightarrow$ so is $s$
						$N_{ξ, s}$ is a NCDN $\Leftrightarrow$ $ξ$ and $s$ are continuous on $[0,1]$
						$N_{ξ, s}$ is strict $\Leftrightarrow$ $ξ$ and $s$ are continuous on $[0,1]$ with $s (1) + ξ (1) = 1$
						$N_{ξ, s}$ is strong $\Leftrightarrow$ $ξ$ and $s$ are continuous on $[0,1]$ with $s (1) + ξ (x) = 1 + s (x)$

¹ This column (UG) denotes the uniqueness of generators, e.g., $f$ Mathematical equation -generators are unique up to a positive multiplicative constant $c$ .

² $f_{1}$ Mathematical equation means the normalization of $f$ when $f (0) < \infty$ , more precisely, $f_{1} (x) = \frac{f (x)}{f (0)}$ for all $x \in [0,1]$ .

³ NCDN means normal $C D$ Mathematical equation -negation. ⁴ t(x)=min(θ∘φ^－¹(1－φ(x)),θ(1^－))－θ(0).

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