In this section, we define Mfuzzy left (resp., right) ideals of quantales and discuss some of their properties. Also, the concept of an Mfuzzy left (resp., right) ideal conucleus is introduced and the relationship with the concept of Mfuzzy left (resp., right) ideals on a quantale is introduced.
Definition 12
Let (L,≤,⊗),(M,≤,⊙)∈SQuant. An Mfuzzy left (resp, right) ideal on a quantale L is a map μ:L→M satisfying the following conditions: for all a, b∈X

If a≤b, then μ(a)≥μ(b).

μ(a∨b)≥μ(a)∧μ(b).

μ(a⊗b)≥μ(b)(resp.,μ(a)).
A map μ:L→M, which is both Mfuzzy left and right ideal, is called an Mfuzzy ideal.
Example 1
Let L={⊥,a, b,c, d,⊤} be a set ordered by ⊥≤c≤b≤⊤,⊥≤d≤a≤⊤, and d≤b and equipped with associative binary operations:
Then, we can easily see that (L,≤,⊗) is a quantale. A mapping μ:L→{0,1} defined by
$$ \mu(x)=\left\{ \begin{array} [c]{c} 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, x\leq b\\ 0\,\,\,\,\,\,\,\,\,\,otherwise \end{array} \right. $$
is an M fuzzy left(resp., right) ideal of the quantale (L,≤,⊗).
Proposition 12
Let {μ_{j}}_{j∈J} be a family of Mfuzzy left (resp., right) ideals of a quantale L. Then, \(\bigwedge _{j\in J}\mu _{j}\) is also an Mfuzzy left (resp., right) ideal of L.
Proof
Suppose that {μ_{j}}_{j∈J} be a family of Mfuzzy left (resp., right) ideals of L. Statement (I_{1}) is clear. To prove (I_{2}) notice that since every μ_{j} is an Mfuzzy left (resp., right) ideal of L, we have
\((\bigwedge _{j\in J}\mu _{j})(a\vee b)=\bigwedge _{j\in J}\mu _{j}(a\vee b)\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\geq \bigwedge _{j\in J}(\mu _{j}(a)\wedge \mu _{j}(b))\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\geq \bigwedge _{j\in J}(\mu _{j}(a))\wedge \bigwedge _{j\in J}(\mu _{j}(b))\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,=(\bigwedge _{j\in J}\mu _{j})(a)\wedge (\bigwedge _{j\in J}\mu _{j})(b)\).
We prove property (I_{3}) as follows:
\((\bigwedge _{j\in J}\mu _{j})(a\otimes b)= \bigwedge _{j\in J}\mu _{j} (a\otimes b)\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigwedge _{j\in J}(\mu _{j}(b)) (resp., \bigwedge _{j\in J}(\mu _{j}(a)))\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq (\bigwedge _{j\in J}\mu _{j})(b)(resp., (\bigwedge _{j\in J}\mu _{j})(a))\).
Therefore, \(\bigwedge _{j\in J}\mu _{j}\) is an Mfuzzy left (resp., right) ideal of L. □
Proposition 13
An onto quantale homomorphic preimage of an Mfuzzy left (resp., right) ideal is an Mfuzzy left (resp., right) ideal.
Proof
Let f:L_{1}→L_{2} be an onto homomorphism. Let ρ be an Mfuzzy left ideal and let μ be the preimage of ρ under f, i.e., \(\mu =f^{\leftarrow }_{M}(\rho)\). Property (I_{1}) is clear. For any a, b∈L_{1},
μ(a∨b)=ρ(f(a∨b)),
=ρ(f(a)∨f(b)),
≥ρ(f(a))∧ρ(f(b)),
=μ(a)∧μ(b).
and μ(a⊗b)=ρ(f(a⊗b)),
=ρ(f(a)⊗f(b)),
≥ρ(f(b))=μ(b).
This shows that μ is an Mfuzzy left ideal of L_{1}. The other case is similar. □
Now, we are in a position to introduce and study the notion of Mfuzzy left (resp., right) ideal conucleus on quantales, and study the relationship with Mfuzzy left (resp., right) ideals.
Definition 13
For (L,≤,⊗),(M,≤,⊙)∈Quant and all a, b∈L,α,β∈M, an Mfuzzy coclosure operator κ:L×M→L is said to be :

An Mfuzzy left ideal conucleus if a⊗κ(b,β)≤κ(a⊗b,β),

An Mfuzzy right ideal conucleus if κ(a,α)⊗b≤κ(a⊗b,α).
Proposition 14
Let (L,≤,⊗),(M,≤,⊙)∈Quant. If μ:L→M is an Mfuzzy left (resp., right) ideal on L, the mapping κ_{μ}:L×M→L defined by the equality
\(\kappa _{\mu }(a, \alpha)=\bigvee \{x \in L: x \leq a, \mu (x) \geq \alpha \}, \forall a \in L, \alpha \in M,\)
is an Mfuzzy left (resp., right) ideal conucleus on L.
Proof
Let μ:L→M be an Mfuzzy left ideal of L and let κ_{μ}:L×M→L be a mapping defined by
\(\kappa _{\mu }(a, \alpha)=\bigvee \{x \in L: x \leq a, \mu (x) \geq \alpha \} \forall a \in L, \alpha \in M.\)
We need to show that the operator κ_{μ} is an Mfuzzy left conucleus.
By Proposition 6, we have that the mapping κ_{μ}:L×M→L is an Mfuzzy coclosure on L. Now, we prove only the condition (Lκ_{4}). To this end, for a, b∈L, β∈M and since μ∈M^{L} is an Mfuzzy left ideal, then μ(a⊗x)≥μ(x), and therefore,
\(a \otimes \kappa _{\mu }(b, \beta)= a \otimes \bigvee \{x \in L: x \leq b, \mu (x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{a \otimes x \in L: a \leq b, x \leq b, \mu (x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{a \otimes x \in L: a \otimes x \leq a \otimes b, \mu (a \otimes x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{y \in L: y \leq a \otimes b, \mu (y) \geq \beta \}\),
=κ_{μ}(a⊗b,β).
Then, a⊗κ_{μ}(b,β)≤κ_{μ}(a⊗b,β). The right case follows similarly. □
Corollary 3
For (L,≤,⊗),(M,≤,⊙)∈Quant and an Mfuzzy ideal μ∈M^{L} on L, the mapping κ_{μ}:L×M→L defined by the equality,
\(\kappa _{\mu } (a, \alpha)=\bigvee \{ x \in L: x \leq a, \mu (x) \geq \alpha \} \forall a \in L, \alpha \in M\),
is an Mfuzzy ideal conucleus on L.
Remark 3
For a, b∈L and α,β∈M, we have a, b≤a∨b and α∧β≤α,β, so for an Mfuzzy coclosure κ:L×M→L, we have that κ(a∨b,α∧β)≥κ(a,α),κ(b,β), which implies that κ(a∨b,α∧β)≥κ(a,α)∨κ(b,β).
Proposition 15
For (L,≤,⊗),(M,≤,⊙)∈Quant and an Mfuzzy left (resp., right) ideal conucleus κ:L×M→L, the mapping μ_{κ}:L→M defined by \(\mu _{\kappa }(a)=\bigvee \{\alpha \in M, \kappa (a, \alpha) \geq a, a \in L \}\) is an Mfuzzy left (resp., right) ideal of L.
Proof
Let κ:L×M→L be an Mfuzzy left (resp., right) ideal conucleus on L. For a, b∈L and α,β∈M with a≤b and α≥β, we have

\(\mu _{\kappa }(a)= \bigvee \{\alpha \in M:\kappa (a, \alpha) \geq a \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigvee \{\beta \in M:\kappa (b, \beta) \geq b \}\),
=μ_{κ}(b).

\(\mu _{\kappa }(a)\wedge \mu _{\kappa }(b)= \bigvee \{\alpha \in M:\kappa (a, \alpha) \geq a \}\wedge \bigvee \{\beta \in M:\kappa (b, \beta) \geq b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \wedge \beta :\kappa (a, \alpha) \geq a\) and κ(b,β)≥b},
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \wedge \beta : \kappa (a, \alpha)\vee \kappa (b, \beta)\geq a\vee b\}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \wedge \beta : \kappa (a\vee b,\alpha \wedge \beta)\geq \kappa (a,\alpha)\vee \kappa (b,\beta)\geq a\vee b\}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{\alpha \wedge \beta : \kappa (a\vee b,\alpha \wedge \beta) \geq a\vee b\}\),
=μ_{κ}(a∨b).

\(\mu _{\kappa }(a \otimes b)=\bigvee \{\beta \in M: \kappa (a \otimes b, \beta) \geq a \otimes b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\beta \in M: a \otimes \kappa (b, \beta) \geq a \otimes b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigvee \{\beta \in M: \kappa (b, \beta) \geq b \}\),
=μ_{κ}(b).
Similarly, μ_{κ}(a⊗b)≥μ_{κ}(a).
□
Corollary 4
For (L,≤,⊗),(M,≤,⊙)∈Quant and an Mfuzzy ideal conucleus κ:L×M→L, the mapping μ_{κ}:L→M defined by
\(\mu _{\kappa }(a)=\bigvee \{\alpha \in M, \kappa (a, \alpha) \geq a, a \in L \}\) is an Mfuzzy ideal of L.
The following lemma provides an important description of Mfuzzy left (resp., right) ideal conuclei for a unital quantale in terms of the residuum ↘(resp., ↙).
Lemma 4
Let (L,≤,⊗),(M,≤,⊙)∈Quant and κ:L×M→L be an Mfuzzy left (resp., right) ideal conucleus on a quantale L. Then,
κ(a,α)↘κ(b,β)=κ(a,α)↘b (resp., κ(b,β)↙κ(a,α)=b↙κ(a,α)),
for all a, b∈L, α,β∈M.
Proof
By (Proposition 1(4)), a≤b⇒c↘a≤c↘b,
⇒b↘c≤a↘c.
If κ is an Mfuzzy left ideal conucleus on L, then since κ(b,β)≤b, we have
κ(a,α)↘κ(b,β)≤κ(a,α)↘b, and b↘κ(a,α)≤κ(b,β)↘κ(a,α).
Thus producing
$$\kappa(a,\alpha)\searrow \kappa(b,\beta) = \kappa(a,\alpha)\searrow b.$$
The argument for ↙ proceeds similarly. □
Lemma 5
Let L be a unital quantale. A mapping κ:L×M→L is an Mfuzzy left (resp., right) ideal conucleus on L if
κ(a,α)↘κ(b,β)=κ(a,α)↘b (resp., κ(b,β)↙κ(a,α)=b↙κ(a,α)),
for all a, b∈L, α,β∈M.
Proof
Suppose that κ(b,β)↙κ(a,α)=b↙κ(a,α) for all a, b∈L, α,β∈M, and e_{L} be unit of L.

e_{L}⊗κ(a,α)≤κ(a,α)⇒e_{L}≤κ(a,α)↙κ(a,α)=a↙κ(a,α)
⇒e_{L}⊗κ(a,α)≤a
⇒κ(a,α)≤a.

If a≤b, and β≤α, then
e_{L}⊗κ(a,α)≤a≤b⇒e_{L}≤b↙κ(a,α)=κ(b,β)↙κ(a,α)
⇒e_{L}⊗κ(a,α)≤κ(b,β)
⇒κ(a,α)≤κ(b,β).
That is, κ is order preserving.

Since κ(a,α)↙κ(a,α)≤κ(κ(a,α),α)↙κ(a,α), then
e_{L}⊗κ(a,α)≤κ(a,α)⇒e_{L}≤κ(κ(a,α),α)↙κ(a,α),
⇒e_{L}⊗κ(a,α)≤κ(κ(a,α),α),
⇒κ(a,α)≤κ(κ(a,α),α).
⇒κ(a,α)=κ(κ(a,α),α).
That is, κ is idempotent.
By (i), (ii), and (iii), we have that κ is an Mfuzzy coclosure, and therefore, we have
$$\begin{aligned} a \otimes \kappa(b, \beta)\leq \kappa(a \otimes \kappa(b, \beta))&\Rightarrow a \leq\kappa(a\otimes\kappa(b, \beta))\swarrow\kappa(b,\beta)\\ &\Rightarrow a \leq\kappa(a\otimes b, \beta)\swarrow\kappa(b, \beta)\\ &\Rightarrow a \otimes \kappa(b, \beta)\leq\vspace*{2pt} \kappa(a\otimes b, \beta). \end{aligned} $$
Thus, a⊗κ(b,β)≤κ(a⊗b,β). So κ is an Mfuzzy left ideal conucleus. The right case follows similarly.
□
As a consequence of the above lemmas, we have the following proposition:
Proposition 16
Let L be a unital quantale. A map κ:L×M→L is an Mfuzzy ideal conucleus if and only if
κ(a,α)↘κ(b,β)=κ(a,α)↘b and κ(b,β)↙κ(a,α)=b↙κ(a,α),
for all a, b∈L, α,β∈M.