Ordinal Sums of De Morgan Triples

In this paper we consider the ordinal sum of the summands of t-norms, t-conorms, and fuzzy negations and prove some new results about them. In particular, we provide some conditions in order to guarantee that when the summands are De Morgan triples their ordinal sums also are De Morgan triple. We also proved that De Morgan laws are preserved for the the action of automorphisms and N-duality.

Keywords:
t-norm; t-conorm; fuzzy negation; De Morgan triples; ordinal sum

Authorship

I. MEZZOMO

Centro de Ciências Exatas e Naturais - DCEN, Universidade Federal Rural do Semi-Árido - UFERSA, Mossoró, Rio Grande do Norte, Brazil. E-mail: imezzomo@ufersa.edu.brUniversidade Federal Rural do Semi-ÁridoBrazilMossoró, Rio Grande do Norte, BrazilCentro de Ciências Exatas e Naturais - DCEN, Universidade Federal Rural do Semi-Árido - UFERSA, Mossoró, Rio Grande do Norte, Brazil. E-mail: imezzomo@ufersa.edu.br

B. BEDREGAL

Departamento de Informática e Matemática Aplicada - DIMAp, Universidade Federal do Rio Grande do Norte - UFRN, Natal, Rio Grande do Norte, Brazil. E-mail: bedregal@dimap.ufrn.brUniversidade Federal do Rio Grande do NorteBrazilNatal, Rio Grande do Norte, BrazilDepartamento de Informática e Matemática Aplicada - DIMAp, Universidade Federal do Rio Grande do Norte - UFRN, Natal, Rio Grande do Norte, Brazil. E-mail: bedregal@dimap.ufrn.br

*Corresponding author: Ivan Mezzomo - E-mail: imezzomo@ufersa.edu.br

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (29)

(2.1)

(2.2)

(LC)

(LEM)

(2.3)

(2.4)

(2.5)

(2.6)

(4.1)

(4.2)

(4.3)

\begin{array}{l} T_{D} (x, y) = {\begin{matrix} 0 \begin{matrix} i f (x, y) \in [0,1 [^{2}; \end{matrix} \\ \min (x, y) \begin{matrix}  \end{matrix} o t h e r w i s e . \begin{matrix}  \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} □ \end{array}

\begin{array}{l} S_{D} (x, y) = {\begin{matrix} 1 \begin{matrix} i f (x, y) \in] 0,1]^{2}; \end{matrix} \\ \max (x, y) \begin{matrix}  \end{matrix} o t h e r w i s e . \begin{matrix}  \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} □ \end{array}

N_{⊥} (x) = {\begin{matrix} 0 \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} x > 0 \\ 1 \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} x = 0 \end{matrix}

N_{┬} (x) = {\begin{matrix} 0 \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} x = 1 \\ 1 \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} x < 1 \end{matrix}

(2.1)

N_{α} (x) = {\begin{matrix} 0 \begin{matrix}  \end{matrix} if \begin{matrix}  \end{matrix} x > α \\ 1 \begin{matrix}  \end{matrix} if \begin{matrix}  \end{matrix} x \leq α \end{matrix}

(2.2)

N^{α} (x) = {\begin{matrix} 0 \begin{matrix}  \end{matrix} if \begin{matrix}  \end{matrix} x \geq α \\ 1 \begin{matrix}  \end{matrix} if \begin{matrix}  \end{matrix} x < α \end{matrix}

(LC)

T (x, N (x)) = 0, \forall x \in [0,1]

(LEM)

S (x, N (x)) = 1, \forall x \in [0,1]

\begin{array}{l} T_{1} \land T_{2} (x, y) = \min {T_{1} (x, y), T_{2} (x, y)} \\ T_{1} \lor T_{2} (x, y) = \max {T_{1} (x, y), T_{2} (x, y)} . \end{array}

\begin{array}{l} T_{1} \land T_{2} (x, y) = \min {T_{1} (x, y), T_{2} (x, y)} \\ \begin{matrix}  \end{matrix} = \min {T_{1} (y, x), T_{2} (y, x)} \\ \begin{matrix}  \end{matrix} = T_{1} \land T_{2} (y, x) . \end{array}

\begin{array}{l} T_{1} \land T_{2} (x,1) = \min {T_{1} (x,1), T_{2} (x,1)} \\ \begin{matrix}  \end{matrix} = \min {x, x} \\ \begin{matrix}  \end{matrix} = x . \end{array}

\begin{array}{l} T_{N} (x, y) = N^{- 1} (T (N (x), N (y))) \\ \begin{matrix}  \end{matrix} = N^{- 1} (T (N (y), N (x))) \\ \begin{matrix}  \end{matrix} = T_{N} (y, x) . \end{array}

\begin{array}{l} T_{N} (x, T_{N} (y, z)) = N^{- 1} (T (N (x), N (N^{- 1} (T (N (y), N (z)))))) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} = N^{- 1} (T (N (x), T (N (y), N (z)))) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} = N^{- 1} (T (T (N (x), N (y)), N (z))) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} = N^{- 1} (T (N (N^{- 1} (T (N (x), N (y))), N (z)))) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} = T_{N} (T_{N} (x, y), z) \end{array}

\begin{array}{l} T_{N} (x,0) = N^{- 1} (T (N (x), N (0))) \\ \begin{matrix}  \end{matrix} = N^{- 1} (T (N (x),1)) \\ \begin{matrix}  \end{matrix} = N^{- 1} (N (x)) \\ \begin{matrix}  \end{matrix} = x . \end{array}

(2.3)

T^{ρ} (x, y) = ρ^{- 1} (T (ρ (x), ρ (y))),

(2.4)

S^{ρ} (x, y) = ρ^{- 1} (S (ρ (x), ρ (y))), \forall x, y \in [0,1] .

\begin{array}{l} T (x, y) = N (S (N (x), N (y))); \\ S (x, y) = N (T (N (x), N (y))), \end{array}

(2.5)

N (T (x, y)) = S (N (x), N (y));

(2.6)

N (S (x, y)) = T (N (x), N (y)) .

\begin{array}{l} N^{- 1} (T (x, y)) = N^{- 1} (T (N (N^{- 1} (x)), N (N^{- 1} (y)))) \\ \begin{matrix} \begin{matrix} = N^{- 1} (N (S (N^{- 1} (x), N^{1} (y)))) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = S (N^{- 1} (x), N^{- 1} (y)) \end{matrix} \end{matrix} . \end{array}

\begin{array}{l} N (S_{N} (x, y)) = N (N^{- 1} (S (N (x), N (y)))) \\ \begin{matrix} \begin{matrix} = S (N (x), N (y)) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = N (T (x, y)) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = N (T (N^{- 1} (N (x)), N^{- 1} (N (y)))) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = T_{N^{1}} (N (x), N (y)) . \end{matrix} \end{matrix} \end{array}

\begin{array}{l} N (T_{1} \land T_{2} (x, y)) = N (\min {T_{1} (x, y), T_{2} (x, y)}) \\ \begin{matrix} \begin{matrix} = \max {N (T_{1} (x, y)), N (T_{2} (x, y))} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = \max {S_{1} (N (x), N (y)), S_{2} (N (x), N (y))} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = S_{1} \lor S_{2} (N (x), N (y)) . \end{matrix} \end{matrix} \end{array}

\begin{array}{l} N^{ρ} (T^{ρ} (x, y)) = N^{ρ} (ρ^{- 1} (T (ρ (x), ρ (y)))) \\ \begin{matrix} \begin{matrix} = ρ^{- 1} (N (ρ (ρ^{- 1} (T (ρ (x), ρ (y)))))) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = ρ^{- 1} (N (T (ρ (x), ρ (y)))) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = ρ^{- 1} (S (N (ρ (x)), N (ρ (y)))) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = ρ^{- 1} (S (ρ (ρ^{- 1} (N (ρ (x)))), ρ (ρ^{- 1} (N (ρ (y)))))) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} = S^{ρ} \end{matrix} \end{matrix} (ρ^{- 1} (N (ρ (x))), ρ^{- 1} (N (ρ (y)))) \\ \begin{matrix} \begin{matrix} = S^{ρ} \end{matrix} \end{matrix} (N^{ρ} (x), N^{ρ} (y)) . \end{array}

(4.1)

T (x, y) = {\begin{matrix} a_{i} + (b_{i} - a_{i}) T_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}, \frac{y - a_{i}}{b_{i} - a_{i}}) \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} (x, y) \in {[a_{i}, b_{i}]}^{2}, \\ \min (x, y) \begin{matrix}  \end{matrix} o t h e r w i s e . \end{matrix}

(4.2)

S (x, y) = {\begin{matrix} a_{i} + (b_{i} - a_{i}) S_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}, \frac{y - a_{i}}{b_{i} - a_{i}}) \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} (x, y) \in {[a_{i}, b_{i}]}^{2}, \\ \max (x, y) \begin{matrix}  \end{matrix} o t h e r w i s e . \end{matrix}

(4.3)

N (x) = {\begin{matrix} (1 - b_{i}) + (b_{i} - a_{i}) N_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}) \begin{matrix}  \end{matrix} i f \begin{matrix}  \end{matrix} x \in [a_{i}, b_{i}], \\ N_{S} (x) \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} o t h e r w i s e . \end{matrix}

\begin{array}{l} \begin{matrix}  \end{matrix} N (x) \in [(1 - b_{i}) + (b_{i} - a_{i}) N_{i} (\frac{b_{i} - a_{i}}{b_{i} - a_{i}}), (1 - b_{i}) + (b_{i} - a_{i}) N_{i} (\frac{a_{i} - a_{i}}{b_{i} - a_{i}})] \\ = [(1 - b_{i}) + (b_{i} - a_{i}) N_{i} (1), (1 - b_{i}) + (b_{i} - a_{i}) N_{i} (0)] = [1 - b_{i},1 - a_{i}] . \end{array}

\begin{array}{l} N (T (x, y)) = (1 - b_{i}) + (b_{i} - a_{i}) N_{i} (\frac{T (x, y) - a_{i}}{b_{i} - a_{i}}) \\ \begin{matrix}  \end{matrix} = (1 - b_{i}) + (b_{i} - a_{i}) N_{i} (\frac{a_{i} + (b_{i} - a_{i}) T_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}, \frac{y - a_{i}}{b_{i} - a_{i}}) - a_{i}}{b_{i} - a_{i}}) \\ \begin{matrix}  \end{matrix} = (1 - b_{i}) + (b_{i} - a_{i}) N_{i} (T_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}, \frac{y - a_{i}}{b_{i} - a_{i}})) \\ \begin{matrix}  \end{matrix} = (1 - b_{i}) + (b_{i} - a_{i}) S_{i} (N_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}), N_{i} (\frac{y - a_{i}}{b_{i} - a_{i}})) \\ \begin{matrix}  \end{matrix} = (1 - b_{i}) + (b_{i} - a_{i}) S_{i} (\frac{N (x) - (1 - b_{i})}{b_{i} - a_{i}}, \frac{N (y) - (1 - b_{i})}{b_{i} - a_{i}}) \\ \begin{matrix}  \end{matrix} = S (N (x), N (y)) . \end{array}

\begin{array}{l} N_{S} (S (x, y)) = a_{i} + (b_{i} - a_{i}) N_{S} (\frac{S (x, y) - (1 - b_{i})}{b_{i} - a_{i}}) \\ \begin{matrix}  \end{matrix} = a_{i} + (b_{i} - a_{i}) N_{S} (\frac{(1 - b_{i}) + (b_{i} - a_{i}) S_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}, \frac{y - a_{i}}{b_{i} - a_{i}}) - (1 - b_{i})}{b_{i} - a_{i}}) \\ \begin{matrix}  \end{matrix} = a_{i} + (b_{i} - a_{i}) N_{S} (S_{i} (\frac{x - a_{i}}{b_{i} - a_{i}}, \frac{y - a_{i}}{b_{i} - a_{i}})) \\ \begin{matrix}  \end{matrix} = a_{i} + (b_{i} - a_{i}) T_{i} (N_{S} (\frac{x - a_{i}}{b_{i} - a_{i}}), N_{S} (\frac{y - a_{i}}{b_{i} - a_{i}})) \\ \begin{matrix}  \end{matrix} = a_{i} + (b_{i} - a_{i}) T_{i} (\frac{N_{S} (x) - (1 - b_{i})}{b_{i} - a_{i}}, \frac{N_{S} (y) - (1 - b_{i})}{b_{i} - a_{i}}) \\ \begin{matrix}  \end{matrix} = T (N_{S} (x), N_{S} (y)) . \end{array}

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