In this section, we define M-fuzzy left (resp., right) ideals of quantales and discuss some of their properties. Also, the concept of an M-fuzzy left (resp., right) ideal conucleus is introduced and the relationship with the concept of M-fuzzy left (resp., right) ideals on a quantale is introduced.
Definition 12
Let (L,≤,⊗),(M,≤,⊙)∈|SQuant|. An M-fuzzy 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 M-fuzzy left and right ideal, is called an M-fuzzy 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 {μj}j∈J be a family of M-fuzzy left (resp., right) ideals of a quantale L. Then, \(\bigwedge _{j\in J}\mu _{j}\) is also an M-fuzzy left (resp., right) ideal of L.
Proof
Suppose that {μj}j∈J be a family of M-fuzzy left (resp., right) ideals of L. Statement (I1) is clear. To prove (I2) notice that since every μj is an M-fuzzy 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 (I3) 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 M-fuzzy left (resp., right) ideal of L. □
Proposition 13
An onto quantale homomorphic preimage of an M-fuzzy left (resp., right) ideal is an M-fuzzy left (resp., right) ideal.
Proof
Let f:L1→L2 be an onto homomorphism. Let ρ be an M-fuzzy left ideal and let μ be the preimage of ρ under f, i.e., \(\mu =f^{\leftarrow }_{M}(\rho)\). Property (I1) is clear. For any a, b∈L1,
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 M-fuzzy left ideal of L1. The other case is similar. □
Now, we are in a position to introduce and study the notion of M-fuzzy left (resp., right) ideal conucleus on quantales, and study the relationship with M-fuzzy left (resp., right) ideals.
Definition 13
For (L,≤,⊗),(M,≤,⊙)∈|Quant| and all a, b∈L,α,β∈M, an M-fuzzy coclosure operator κ:L×M→L is said to be :
-
An M-fuzzy left ideal conucleus if a⊗κ(b,β)≤κ(a⊗b,β),
-
An M-fuzzy right ideal conucleus if κ(a,α)⊗b≤κ(a⊗b,α).
Proposition 14
Let (L,≤,⊗),(M,≤,⊙)∈|Quant|. If μ:L→M is an M-fuzzy left (resp., right) ideal on L, the mapping κμ: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 M-fuzzy left (resp., right) ideal conucleus on L.
Proof
Let μ:L→M be an M-fuzzy left ideal of L and let κμ: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 κμ is an M-fuzzy left conucleus.
By Proposition 6, we have that the mapping κμ:L×M→L is an M-fuzzy coclosure on L. Now, we prove only the condition (Lκ4). To this end, for a, b∈L, β∈M and since μ∈ML is an M-fuzzy left ideal, then μ(a⊗x)≥μ(x), and therefore,
\(a \otimes \kappa _{\mu }(b, \beta)= a \otimes \bigvee \{x \in L: x \leq b, \mu (x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{a \otimes x \in L: a \leq b, x \leq b, \mu (x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{a \otimes x \in L: a \otimes x \leq a \otimes b, \mu (a \otimes x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{y \in L: y \leq a \otimes b, \mu (y) \geq \beta \}\),
=κμ(a⊗b,β).
Then, a⊗κμ(b,β)≤κμ(a⊗b,β). The right case follows similarly. □
Corollary 3
For (L,≤,⊗),(M,≤,⊙)∈|Quant| and an M-fuzzy ideal μ∈ML on L, the mapping κμ: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 M-fuzzy 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 M-fuzzy 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 M-fuzzy left (resp., right) ideal conucleus κ:L×M→L, the mapping μκ:L→M defined by \(\mu _{\kappa }(a)=\bigvee \{\alpha \in M, \kappa (a, \alpha) \geq a, a \in L \}\) is an M-fuzzy left (resp., right) ideal of L.
Proof
Let κ:L×M→L be an M-fuzzy left (resp., right) ideal conucleus on L. For a, b∈L and α,β∈M with a≤b and α≥β, we have
-
\(\mu _{\kappa }(a)= \bigvee \{\alpha \in M:\kappa (a, \alpha) \geq a \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigvee \{\beta \in M:\kappa (b, \beta) \geq b \}\),
=μκ(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},
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \wedge \beta : \kappa (a, \alpha)\vee \kappa (b, \beta)\geq a\vee b\}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \wedge \beta : \kappa (a\vee b,\alpha \wedge \beta)\geq \kappa (a,\alpha)\vee \kappa (b,\beta)\geq a\vee b\}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{\alpha \wedge \beta : \kappa (a\vee b,\alpha \wedge \beta) \geq a\vee b\}\),
=μκ(a∨b).
-
\(\mu _{\kappa }(a \otimes b)=\bigvee \{\beta \in M: \kappa (a \otimes b, \beta) \geq a \otimes b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\beta \in M: a \otimes \kappa (b, \beta) \geq a \otimes b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigvee \{\beta \in M: \kappa (b, \beta) \geq b \}\),
=μκ(b).
Similarly, μκ(a⊗b)≥μκ(a).
□
Corollary 4
For (L,≤,⊗),(M,≤,⊙)∈|Quant| and an M-fuzzy ideal conucleus κ:L×M→L, the mapping μκ:L→M defined by
\(\mu _{\kappa }(a)=\bigvee \{\alpha \in M, \kappa (a, \alpha) \geq a, a \in L \}\) is an M-fuzzy ideal of L.
The following lemma provides an important description of M-fuzzy 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 M-fuzzy 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 M-fuzzy 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 M-fuzzy 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 eL be unit of L.
-
eL⊗κ(a,α)≤κ(a,α)⇒eL≤κ(a,α)↙κ(a,α)=a↙κ(a,α)
⇒eL⊗κ(a,α)≤a
⇒κ(a,α)≤a.
-
If a≤b, and β≤α, then
eL⊗κ(a,α)≤a≤b⇒eL≤b↙κ(a,α)=κ(b,β)↙κ(a,α)
⇒eL⊗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
eL⊗κ(a,α)≤κ(a,α)⇒eL≤κ(κ(a,α),α)↙κ(a,α),
⇒eL⊗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 M-fuzzy 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 M-fuzzy 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 M-fuzzy 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.