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Fuzzy quantic nuclei and conuclei with applications to fuzzy semiquantales and (L, M)quasifuzzy topologies
Journal of the Egyptian Mathematical Society volume 27, Article number: 28 (2019)
Abstract
The aim of this paper is to introduce and study the notions of Mfuzzy quantic nuclei and conuclei on quantales. Firstly, the concept of an Mfuzzy quantic nuclei is introduced and some of its properties are discussed. Secondly, the concept of an Mfuzzy quantic conuclei is introduced. As an application of Mfuzzy quantic conuclei on quantales, a characterization of an (L, M)quasifuzzy interior operator on a nonempty set X is given and the relationship between it and an (L, M)quasifuzzy topology is discussed. Finally, the concept of an Mfuzzy left (resp., right) ideal conucleus is introduced and the relationship with the concept of Mfuzzy left (resp., right) ideal is introduced.
Introduction
Quantales were introduced by C. J. Mulvey in [1], with the purpose of studying the foundations of quantum mechanics and the spectrum of C^{∗}algebras. In 2007, Rodabaugh [2] introduced the notion of semiquantale as a generalization of quantale and used it as an appropriate latticetheoretic basis to formulate powerset, topological, and fuzzy topological theories. The notion of semiquantale provides a useful tool to gather various latticetheoretic notions, which have been extensively studied in noncommutative structures; it has a wide application, especially in studying the noncommutative latticevalued quasitopology [2–6].
In 2015, Demirici [7] introduced the notion of Mfuzzy semiquantales as a fuzzy version of notion of Rodabaugh’s semiquantales, providing a common framework for (L, M)fuzzy topological spaces of Kubiak and S̆ostak [8], Lquasifuzzy topological spaces of Rodabaugh [2], and Lfuzzy topological spaces of Höhle and S̆ostak [9].
As we all know, the quantic nuclei and the quantic conuclei play an important role in quantale theory. In this paper, we aim to introduce the notions of Mfuzzy quantic nuclei and conuclei on quantales and study some of their properties. Firstly, in “The (direct) product of two quantales” section, we will define and study the (direct) product of two quantales which will be used through this paper. In “Mfuzzy quantic nuclei” section, the concept of an Mfuzzy quantic (or quantale) nuclei is introduced and a relationship between it and the notion of Mfuzzy semiquantales is discussed. In “Mfuzzy quantic conuclei” section, the concept of an Mfuzzy quantic (or quantale) conuclei is introduced. As an application of Mfuzzy quantic conuclei on quantales, we characterize and study the notion of (L, M)quasifuzzy interior operator on a nonempty set X and discuss the relationship between it and an (L, M)quasifuzzy topology on X. Finally, in “Mfuzzy ideal conuclei on quantales” section, the concept of an Mfuzzy left (resp., right) ideal conucleus is introduced and the relationship with the concept of Mfuzzy left (resp., right) quantale ideals is introduced.
Preliminaries
A semiquantale L=(L,≤,⊗) [2, 10] is a complete lattice L=(L,≤) equipped with a binary operation ⊗:L×L→L (called a tensor product) with no additional assumptions. As convention, we denote the join, meet, top, and bottom in the complete lattice (L,≤) by \(\bigvee, \bigwedge, \top _{L}\), and ⊥_{L}, respectively. Semiquantales include various classes of ordered algebraic structures (e.g., complete residuated lattices, unit interval [0,1] equipped with tnorms or tconorms, quantales, frames, semiframes) playing a major role in fuzzy set theory and fuzzy logics [11, 12]. Now, we list only some of their definitions that will be needed in the following text.
Definition 1
A semiquantale L=(L,≤,⊗) is called:

A unital semiquantale [2] if the groupoid (L,⊗) has an identity element e∈L called the unit. If the unit e coincides with the top element ⊤_{L} of L, then a unital semiquantale is called a strictly twosided semiquantale.

A commutative semiquantate [2] if ⊗ is commutative, i.e., a⊗b=b⊗a for every a, b∈L.

A quantale [13] if the binary operation ⊗ is associative and satisfies
$$ a\otimes(\mathop {\bigvee}\limits_{i\in I}b_{i})=\mathop {\bigvee}\limits_{i\in I}(a\otimes b_{i})\ \text{and}\ (\mathop {\bigvee}\limits_{i\in I}b_{i})\otimes a=\mathop {\bigvee}\limits_{i\in I}(b_{i}\otimes a)\ \text{for all}\ a\in L, \{b\}_{i\in I}\subseteq L. $$ 
A coquantale [14] if the multiplication ⊗ is associative and satisfies
$$ a\otimes(\mathop {\bigwedge}\limits_{i\in I}b_{i})=\mathop {\bigwedge}\limits_{i\in I}(a\otimes b_{i})\ \text{and}\ (\mathop {\bigwedge}\limits_{i\in I}b_{i})\otimes a=\mathop {\bigwedge}\limits_{i\in I}(b_{i}\otimes a)\ \text{for all}\ a\in L, \{b\}_{i\in I}\subseteq L.$$
A semiquantale morphism [2] h from a semiquantale L=(L,≤,⊗) to an other semiquantale M=(M,≤,⊙) is a map h:L→M preserving the tensor product and the arbitrary joins. If a semiquantale morphism h:L→M additionally preserves the top (resp., unit) element, i.e., h(⊤_{L})=⊤_{M}(resp., h(e_{L})=e_{M}), then it is said to be strong (resp., unital). The category SQuant comprises all semiquantales together with semiquantale morphisms. The nonfull subcategory UnSQuant of SQuant comprises all unital semiquantales and all unital semiquantale morphisms [2]. Quant is the full subcategory of SQuant, which has as objects all quantales.
CoQuant is the full subcategory of SQuant, which has as objects all coquantales and as morphisms, all maps that preserve the tensor product and arbitrary meets.
Let X be a nonempty set and L∈SQuant. An Lfuzzy subset (or Lsubset) of X is a mapping A:X→L. The family of all Lfuzzy subsets on X will be denoted by L^{X}. The smallest element and the largest element in L^{X} are denoted by \(\underline {\bot }\) and \(\underline {\top }\), respectively. The algebraic and latticetheoretic structures can be extended from the semiquantale \((L,\leq,\bigvee,\otimes)\) to L^{X} pointwisely:

A≤B⇔A(x)≤B(x),

(A⊗B)(x)=A(x)⊗B(x),
for all x∈X.
Obviously, (L,≤,⊗) is again a semiquantale with respect to the multiplication ⊗ and the joins of a subset {A_{i}:i∈I} of L^{X} is given by
\((\mathop {\bigvee }\limits _{i\in I}A_{i})(x)=\mathop {\bigvee }\limits _{i\in I}A_{i}(x)\,\,\,\forall \,x\in X\).
In the case where L is unital with unit e, then L^{X} becomes a unital semiquantale with the unit \(\underline {e}\). For an ordinary mapping f:X→Y, one can define the mappings \(f^{\rightarrow }_{L}:L^{X}\longrightarrow L^{Y}\) and \(f^{\leftarrow }_{L}:L^{Y}\longrightarrow L^{X}\) by \(f^{\rightarrow }_{L}(A)(y)=\bigvee \{A(x):x\in X, f(x)=y\}\) for every A∈L^{X} and every y∈Y, \(f^{\leftarrow }_{L}(B)=B\circ f\) for every B∈L^{Y}, respectively. For more details, we refer to [2,15].
Every quantale L has two residuals, which are induced by its binary operation ⊗ and which are defined by \(a\searrow b=\bigvee \{c:a\otimes c\leq b\}\) and \( b\swarrow a =\bigvee \{c:c\otimes a\leq b\}\), respectively, providing a single residuum → in case of a commutative multiplication (resulting complete residuated lattices of Denniston et al. [16]). These operations have the standard properties of poset adjunctions [17] or (order preserving) Galois connections [18], for example,
For the convenience of the reader, the following proposition recalls some of their other properties, which will be heavily used throughout this paper.
Proposition 1
[13,19] For L∈Quant with a, b,c∈L and B⊆L, we have the following properties:

a⊗(a↘b)≤b and (b↙a)⊗a≤b,

b↘(a↘c)=(a⊗b)↘c and (c↙b)↙a=c↙(a⊗b),

a↘(c↙b)=(a↘c)↙b.

a≤b implies c↘a≤c↘b and b↘c≤a↘c.
Before going too much further, we recall that if L=(L,≤) is a poset, an order preserving function g:L→L is called a closure (resp., coclosure) operator on the poset L=(L,≤) [13,17] iff it satisfies the following conditions:

a≤g(a) (resp., g(a)≤a), for all a∈L,

g(g(a))=g(a), for all a∈L.
Definition 2
[4,13] Let (L,≤,⊗)∈SQuant. A quantic nucleus (resp., conucleus) on L is a closure (resp., coclosure) operator g:L→L such that g(a)⊗g(b)≤g(a⊗b)) for all a, b∈L.
Definition 3
[20] Let L be a quantale. A nonempty subset I⊆L is called a left (resp., right) ideal of L if it satisfies the following three conditions:

a∨b∈I for all a, b∈I,

For all a, b∈L, if a∈I and b≤a, then b∈I;

For all a∈L and x∈I, a⊗x∈I (resp., x⊗a∈I).

A subset I is an ideal if it is both a left ideal and a right ideal.
Definition 4
[7] Let (L,≤,⊗), (M,≤,⊙)∈SQuant.

An Mfuzzy semiquantale on L is a map μ:L→M satisfying the following conditions: For all a, b∈L and {a_{j}j∈J}⊆L,

μ(a)⊙μ(b)≤μ(a⊗b),

\(\bigwedge _{j\in J}\mu (a_{j})\leq \mu \left (\bigvee _{j\in J}a_{j}\right)\).


An Mfuzzy semiquantale μ is called strong if μ(⊤_{L})=⊤_{M}.

In case where (L,≤,⊗) is a unital semiquantale with the unit e_{L}, an Mfuzzy semiquantale μ is called unital if μ(e_{L})=⊤_{M}.
Definition 5
[7] Let (L,≤,⊗), (M,≤,⊙)∈SQuant, and X be a nonempty set.

A map τ:L^{X}→M is called an (L, M)quasifuzzy topology on X iff τ is an Mfuzzy semiquantale on L^{X}, i.e., the next conditions are satisfied for all A, B∈L^{X} and {A_{j}j∈J}⊆L^{X}:

τ(A)⊙τ(B)≤τ(A⊗B),

\(\bigwedge _{j\in J}\tau (A_{j})\leq \tau (\bigvee _{j\in J}A_{j})\).


An (L, M)quasifuzzy topology is strong iff \(\tau (\underline {\top })=\top _{M}\).

Let L be a unital semiquantale with unit e. An (L, M)quasifuzzy topology is then called an (L, M)fuzzy topology iff \(\tau (\underline {e}) = \top _{M}\).

The ordered pair (X,τ) is called an (L, M)quasifuzzy (resp., strong (L, M)quasifuzzy, (L, M)fuzzy) topological space if τ is an (L, M)quasifuzzy (resp., strong (L, M)quasifuzzy, (L, M)fuzzy) topology on X.
The (direct) product of two quantales
It is known that the (direct) product of two ordered sets (P,≤) and (Q,≤) is the ordered set (P×Q,≤) [21], where the order relation on the product P×Q is defined as follows:
Also, the (direct) product of two semigroups (G,⊗) and (H,⊙) is a semigroup (G×H,∗) [22], where the binary operation ∗ on G×H is defined as follows:
Furthermore, the direct product of two complete lattices is a complete lattice [23]. So, we can conclude that the direct product of any two semiquantales is again a semiquantale with the tensor product ∗ denoted by Eq. (3).
Lemma 1
The direct product of any two quantales is a quantale.
Proof
Since the direct product of any two semiquantales is a semiquantale, then we only prove the distributively of \(\bigvee \) over the product ∗.
Let \(Q=\left (Q,\leq,\bigvee,\ast \right)=Q_{1}\times Q_{2}\) where \(Q_{1}=\left (Q_{1},\leq,\bigvee,\otimes \right)\) and \(Q_{2}=\left (Q_{2},\leq,\bigvee,\odot \right)\) are quantales.
For all \(a,\bigvee _{i}a_{i}\in Q_{1}\), \(b,\bigvee _{i}b_{i}\in Q_{2}\), we have
Similarly, we can prove that \(\bigvee _{i}(a_{i},b_{i})\ast (a,b)=\bigvee _{i}((a_{i}\otimes a),(b_{i}\odot b))\). □
Lemma 2
For \((Q_{1},\leq,\bigvee,\otimes)\), \((Q_{2},\leq,\bigvee,\odot)\in \mathbf {Quant}\), let a_{1},b_{1}∈Q_{1} and a_{2},b_{2}∈Q_{2}. Then

(a_{1},a_{2})↘(b_{1},b_{2})=(a_{1}↘b_{1},a_{2}↘b_{2}),

(b_{1},b_{2})↙(a_{1},a_{2})=(b_{1}↙a_{1},b_{2}↙a_{2}).
Proof
The item (2) can be proved similarly. □
Proposition 2
Let \((Q_{1},\leq,\bigvee,\otimes)\), \((Q_{2},\leq,\bigvee,\odot)\in \mathbf {Quant}\), a, b,c∈Q_{1} and a_{1},b_{1},c_{1}∈Q_{2}. Then

(a, a_{1})∗((a, a_{1})↘(b, b_{1}))≤(b, b_{1}),

((b, b_{1})↙(a, a_{1}))∗(a, a_{1})≤(b, b_{1}),

(b, b_{1})↘((a, a_{1})↘(c, c_{1}))=((a, a_{1})∗(b, b_{1}))↘(c, c_{1}),

((c, c_{1})↙(b, b_{1}))↙(a, a_{1})=(c, c_{1})↙((a, a_{1})∗(b, b_{1})),

(a, a_{1})↘((c, c_{1})↙(b, b_{1}))=((a, a_{1})↘(c, c_{1}))↙(b, b_{1}).
Proof

$$\begin{aligned} (a, a_{1})\ast((a, a_{1})\searrow &(b, b_{1}))=(a, a_{1})\ast(a\searrow b,a_{1}\searrow b_{1})\\ &=(a\otimes(a\searrow b),a_{1}\odot(a_{1}\searrow b_{1})) (\text{by Proposition~1(1)})\\ &\leq (b, b_{1}). \end{aligned} $$

$$\begin{aligned} ((b, b_{1})\swarrow(a, a_{1})) \ast (a, a_{1})&=(b\swarrow a, b_{1}\swarrow a_{1})\ast(a, a_{1})\\ &=((b\swarrow a)\otimes a,(b_{1}\swarrow a_{1})\odot a_{1}) (\text{by Proposition~1(1)})\\ &\leq (b, b_{1}). \end{aligned} $$

$$\begin{aligned} (b, b_{1})\searrow ((a, a_{1})\searrow (c, c_{1})) &= (b, b_{1})\searrow (a\searrow c,a_{1}\searrow c_{1})\\ &=(b\searrow (a\searrow c),b_{1}\searrow (a_{1}\searrow c_{1}))\\ &=((a\otimes b)\searrow c,(a_{1}\odot b_{1})\searrow c_{1}) (\text{by Proposition (2)})\\ &=(a\otimes b,a_{1}\odot b_{1})\searrow (c,c_{1}) (\text{by Lemma 2})\\ &=((a, a_{1})\ast (b, b_{1}))\searrow (c, c_{1}). \end{aligned} $$

$$\begin{aligned} ((c, c_{1})\swarrow (b, b_{1}))\swarrow(a, a_{1})&=(c\swarrow b,c_{1}\swarrow b_{1})\swarrow(a, a_{1})\\ &=((c\swarrow b)\swarrow a,(c_{1}\swarrow b_{1})\swarrow a_{1})\\ &=(c\swarrow(a\otimes b),c_{1}\swarrow(a_{1}\odot b_{1})) (\text{by Proposition 1(2)})\\ &=(c,c_{1})\swarrow(a\otimes b,a_{1}\odot b_{1}) (\text{by Lemma 2})\\ &=(c,c_{1})\swarrow((a, a_{1})\ast (b, b_{1})). \end{aligned} $$

$$\begin{aligned} (a, a_{1})\searrow ((c, c_{1})\swarrow(b, b_{1})) &=(a, a_{1})\searrow(c\swarrow b,c_{1}\swarrow b_{1})\\ &=(a\searrow (c\swarrow b),a_{1}\searrow (c_{1}\swarrow b_{1}))\\ &=((a\searrow c)\swarrow b,(a_{1}\searrow c_{1})\swarrow b_{1}) (\text{by Proposition~1(3)})\\ &=(a\searrow c,a_{1}\searrow c_{1})\swarrow(b, b_{1}) (\text{by Lemma 2})\\ &=((a, a_{1})\searrow (c, c_{1}))\swarrow(b, b_{1}). \end{aligned} $$
□
Mfuzzy quantic nuclei
In this section, we will introduce the concept of an Mfuzzy quantic nuclei as a fuzzy version of the wellknown quantic nuclei. Some properties of such Mfuzzy quantic nuclei will be studied, and the relationship between it and the notion of Mfuzzy semiquantales will be discussed.
Before we go further into this section, let us begin with introducing a fuzzy version of the known closure operator on a partially ordered set.
Definition 6
For a complete lattice (M,≤) and an ordered set (L,≤), a mapping C:L×M→L is called an Mfuzzy closure operator on L if it satisfies the following conditions: for all a, b∈L and α,β∈M,

C(a,α)≥a;

C(a,α)≤C(b,β), if a≤b and α≤β;

C(a,α)=C(C(a,α),α).
Proposition 3
Let (L,≤,⊗), (M,≤,⊙)∈SQuant and μ:L→M be an Mfuzzy semiquantale. A mapping C_{μ}:L×M→L defined by the equality.
\({C}_{\mu }(a, \alpha)= \bigwedge \{ x\in L: x\geq a, \mu (x) \geq \alpha \} \forall a \in L, \alpha \in M \),
is an Mfuzzy closure operator on L.
Proof
Let μ:L→M be an Mfuzzy semiquantale. To prove that the map C_{μ}:L×M→L defined by
\( {C}_{\mu }(a, \alpha)= \bigwedge \{ x \in L: x \geq a, \mu (x) \geq \alpha \} \forall a \in L, \alpha \in M \),
is an Mfuzzy closure operator on L, we will prove that the conditions (C_{1}−C_{3}) of the above definition hold.

By definition of C_{μ}, we have \( {C}_{\mu }(a, \alpha) = \bigwedge \{ x \in L: x \geq a, \mu (x) \geq \alpha \} \geq a \).

So, C_{μ}(a,α)≥a.

If a≤b and α≤β, then

\( {C}_{\mu }(b, \beta) = \bigwedge \{x \in L: x \geq b, \mu (x) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigwedge \{x \in L: x\geq b\geq a, \mu (x)\geq \beta \geq \alpha \},\)
=C_{μ}(a,α).

Hence, C_{μ}(a,α)≤C_{μ}(b,β).

Since C_{μ}(a,α)∈L and
\( {C}_{\mu }( {C}_{\mu }(a, \alpha), \alpha)=\bigwedge \{ x \in L: x \geq {C}_{\mu }(a, \alpha), \mu (x) \geq \alpha \}\),
we have that μ(x)≥μ(C_{μ}(a,α))≥α.
Then, putting x=C_{μ}(a,α), we have
\(C_{\mu }(C_{\mu }(a, \alpha), \alpha)=\bigwedge C_{\mu }(a, \alpha)\) and this implies
C_{μ}(a,α)≥C_{μ}(C_{μ}(a,α),α).
Also, from (C_{1}), we have that
C_{μ}(C_{μ}(a,α),α)≥C_{μ}(a,α).
Then, the equality holds.
□
Definition 7
Let (L,≤,⊗), (M,≤,⊙)∈SQuant. A mapping C:L×M→L is called an Mfuzzy quantic nucleus operator on L if it is an Mfuzzy closure operator on L and satisfies the following condition: for all a, b∈L and α,β∈M,

C(a,α)⊗C(b,β)≤C(a⊗b,α⊙β).
Proposition 4
Let (L,≤,⊗)∈CoQuant,(M,≤,⊙)∈SQuant and μ:L→M be an Mfuzzy semiquantale. The mapping C_{μ}:L×M→L defined by the equality.
\( {C}_{\mu }(a, \alpha)= \bigwedge \{u \in L: u \geq a, \mu (u) \geq \alpha \}\), ∀a∈L,α∈M
is an Mfuzzy quantic nucleus on L.
Proof
We only prove the condition C_{4}. For a, b∈L and α,β∈M, we have: \( {C}_{\mu }(a, \alpha) \otimes {C}_{\mu }(b, \beta)= \bigwedge \{u: u \in L, u \geq a, \alpha \leq \mu (u) \} \otimes \bigwedge \{v: v \in L, v \geq b, \beta \leq \mu (v) \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \bigwedge \{u \otimes v: u, v \in L, u \geq a,v \geq b, \alpha \leq \mu (u), \beta \leq \mu (v) \},\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \bigwedge \{u \otimes v: u, v \in L, u \geq a,v \geq b, \alpha \odot \beta \leq \mu (u)\odot \mu (v) \},\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigwedge \{u \otimes v: u, v \in L, a\otimes b\leq u\otimes v, \alpha \odot \beta \leq \mu (u\otimes v) \},\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigwedge \{w: w \in L, w \geq a \otimes b, \alpha \odot \beta \leq \mu (w) \} = {C}_{\mu }(a \otimes b, \alpha \odot \beta)\).
Then C_{μ}(a,α)⊗C_{μ}(b,β)≤C_{μ}(a⊗b,α⊙β). □
Proposition 5
Let (L,≤,⊗), (M,≤,⊙)∈Quant and C:L×M→L be an Mfuzzy quantic nucleus on a quantale L, then

C(a,α)↘C(b,β)≤a↘C(b,β) (resp., C(b,β)↙C(a,α)≤C(b,β)↙a).
Proof
Since C(a,α)≥a, then by Proposition 1 (4), we have
C(a,α)↘C(b,β)≤a↘C(b,β).
The argument for the residuum ↙ proceeds similarly. □
Corollary 1
For (L,≤,⊗), (M,≤,⊙)∈Quant. If C:L×M→L be an Mfuzzy quantic nucleus on a quantale L, then

C(b↙a,β↙α)≤C(b,β)↙a,

C(a↘b,α↘β)≤a↘C(b,β),

C(b↙a,β↙α)≤C(b,β)↙C(a,α),

C(a↘b,α↘β)≤C(a,α)↘C(b,β),
for all a, b∈L, α,β∈M.
Proof

Since C(b↙a,β↙α)⊗C(a,α)≤C((b↙a)⊗a,(β↙α))⊙α)≤C(b,β) and C:L×M→L is an Mfuzzy quantic nucleus on L, then from Proposition 5, we have that
C(b↙a,β↙α)≤C(b,β)↙a.

Similarly, we can prove (2).

Since C(a,α)⊗C(b↙a,β↙α)=C(a,α)⊗C[(b,β)↙(a,α)]
≤C[(a,α)⊗(b,β)↙(a,α)]≤C(b,β). Thus
C(b↙a,β↙α)≤C(b,β)↙C(a,α).

Similarly, we can prove (4).
□
Lemma 3
Let L be a unital quantale. For all a, b∈L, α,β∈M, a mapping C:L×M→L with C(a,α)↘C(b,β)≤a↘C(b,β) and C(b,β)↙C(a,α)≤C(b,β)↙a is an Mfuzzy quantic nucleus on L.
Proof
For all a, b∈L, α,β∈M, suppose that
C(a,α)↘C(b,β)≤a↘C(b,β) and C(b,β)↙C(a,α)≤C(b,β)↙a.
By the unital assumption e_{L}, we have that:

e_{L}⊗C(a,α)≤C(a,α)⇔e_{L}≤C(a,α)↙C(a,α)≤C(a,α)↙a
⇔e_{L}⊗a≤C(a,α)
⇔a≤C(a,α).

If a≤b and α≤β, then
e_{L}⊗a≤b≤C(b,β)⇔e_{L}≤C(b,β)↙a
⇔e_{L}≤C(b,β)↙C(a,α)
⇔e_{L}⊗C(a,α)≤C(b,β)
⇔C(a,α)≤C(b,β).

Since a≤C(a,α), from (2), we have C(a,α)≤C(C(a,α),α).
On the other hand, e_{L}⊗C(a,α)≤C(a,α)⇔e_{L}≤C(a,α)↙C(a,α)
⇔e_{L}≤C(a,α)↙C(C(a,α),α)
⇔e_{L}⊗C(C(a,α),α)≤C(a,α)
⇔C(C(a,α),α)≤C(a,α),
⇔C(C(a,α),α)=C(a,α).

From the items (1)–(2), we have, for all a, b∈L,α,β∈M
a⊗b≤C(a⊗b,α⊙β)⇔a≤C(a⊗b,α⊙β)↙b
⇔a≤C(a⊗b,α⊙β)↙C(b,β)
⇔a⊗C(b,β)≤C(a⊗b,α⊙β)
⇔C(b,β)≤a↘C(a⊗b,α⊙β)
⇔C(b,β)≤C(a,α)↘C(a⊗b,α⊙β)
⇔C(a,α)⊗C(b,β)≤C(a⊗b,α⊙β).
Thus, C is a an Mfuzzy quantic nucleus. The right unital case follows similarly.
□
Mfuzzy quantic conuclei
In this section, we will introduce and study the concept of an Mfuzzy quantic conuclei on a quantale L=(L,≤,⊗). A relationship between Mfuzzy quantic conuclei and Mfuzzy semiquantales will be discussed. Also, we characterize and study the notion of (L, M)quasifuzzy interior operator, as an application of Mfuzzy quantic conuclei on quantales, and discuss the relationship between such an (L, M)quasifuzzy interior operator and an (L, M)quasifuzzy topology on a nonempty set X.
Definition 8
Let (L,≤) and (M,≤) be posets. A mapping κ:L×M→L is called an Mfuzzy coclosure operator on L if, for all a, b∈L and α,β∈M, it satisfies the following conditions:

κ(a,α)≤κ(b,β) whenever a≤b, β≤α.

κ(a,α)≤a.

κ(a,α)=κ(κ(a,α),α).
Definition 9
An Mfuzzy coclosure operator κ:L×M→L is said to be:

Strong if κ(⊤_{L},α)=⊤_{L}.

Unital if (L,≤,⊗) is unital and κ(e_{L},α)=⊤_{L}.
Proposition 6
Let (L,≤,⊗),(M,≤,⊙)∈SQuant and μ:L→M be an Mfuzzy (resp., a strong Mfuzzy) semiquantale. The mapping κ_{μ}:L×M→L defined by the equality.
\(\kappa _{\mu }(a, \alpha)= \bigvee \{x \in L: x \leq a, \mu (x) \geq \alpha \} \forall a \in L\) and α∈M;
is an Mfuzzy (resp., a strong Mfuzzy) coclosure operator on L.
Proof
Let μ:L→M be an Mfuzzy semiquantale. To prove that the map κ_{μ}:L×M→L defined by
\(\kappa _{\mu }(a, \alpha)= \bigvee \{x \in L: x \leq a \), μ(x)≥α}∀a∈L and α∈M
is an Mfuzzy coclosure operator on L, we will prove the conditions (κ_{1}−κ_{3}) of the above definition.

For a, b∈L and α,β∈M with a≤b,β≤α, we have

\(\kappa _{\mu }(a, \alpha)= \bigvee \{x \in L: x \leq a, \mu (x) \geq \alpha \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{x \in L: x \leq b, \mu (x) \geq \beta \}\),
=κ_{μ}(b,β).

So, κ_{μ}(a,α)≤κ_{μ}(b,β).

By definition of κ_{μ}, we have

\(\kappa _{\mu }(a, \alpha) = \bigvee \{x \in L: x \leq a, \mu (x) \geq \alpha \} \leq a \).

Then, κ_{μ}(a,α)≤a.

Since κ_{μ}(a,α)∈L and \(\kappa _{\mu }(\kappa _{\mu }(a, \alpha), \alpha)=\bigvee \{x\in L: x\leq \kappa _{\mu }(a,\alpha),\mu (x) \geq \alpha \}\), we have that μ(κ_{μ}(a,α))≥μ(x)≥α.
Then, putting x=κ_{μ}(a,α), we have \(\kappa _{\mu }(\kappa _{\mu }(a, \alpha), \alpha)=\bigvee \kappa _{\mu }(a, \alpha)\) and this implies
κ_{μ}(a,α)≤κ_{μ}(κ_{μ}(a,α),α).
Also, from (κ_{2}), we have that
κ_{μ}(κ_{μ}(a,α),α)≤κ_{μ}(a,α).
Then, the equality holds.
If μ:L→M be a strong Mfuzzy semiquantale, then it is clear that
\(\kappa _{\mu }(\top _{L}, \alpha)=\bigvee \{x \in L, x= \top _{L}, \mu (x) \geq \alpha \}=\top _{L} \),
which means that κ_{μ} is a strong Mfuzzy coclosure operator on L. □
Definition 10
Let (L,≤,⊗),(M,≤,⊙)∈SQuant. A mapping κ:L×M→L is called an Mfuzzy quantic conucleus on L if it is an Mfuzzy coclosure operator on L and satisfies the following conditions: for all a, b∈L and α,β∈M,

κ(a,α)⊗κ(b,β)≤κ(a⊗b,α⊙β).
Proposition 7
Let (L,≤,⊗)∈Quant, (M,≤,⊙)∈SQuant and μ:L→M be an Mfuzzy semiquantale 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 quantic conucleus on L.
Proof
We only prove the condition κ_{4}:
\(\kappa _{\mu }(a,\alpha)\otimes \kappa _{\mu }(b, \beta)=\bigvee \{x: x\in L, x\leq a, \alpha \leq \mu (x)\}\otimes \bigvee \{y: y\in L, y\leq b, \beta \leq \mu (y)\}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \bigvee \{x \otimes y: x, y \in L, x \leq a,y \leq b, \alpha \leq \mu (x), \beta \leq \mu (y) \},\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{z: z \in L, z \leq a \otimes b, \alpha \odot \beta \leq \mu (z) \} = \kappa _{\mu }(a \otimes b, \alpha \odot \beta)\). Then, κ_{μ}(a,α)⊗κ_{μ}(b,β)≤κ_{μ}(a⊗b,α⊙β). □
Remark 1
If L∈UnSQuant and \(\kappa _{\mu }(e_{L}, \alpha)=\bigvee \{x \in L, x= e_{L}, \mu (x) \geq \alpha \}=\top _{L}\), then κ_{μ} is a unital Mfuzzy quantic conucleus on L.
Proposition 8
For (L,≤,⊗),(M,≤,⊙)∈Quant and given an Mfuzzy (resp., a strong Mfuzzy) quantic conucleus κ:L×M→L, then an Mfuzzy set μ_{κ}:L→M, which defined by
\(\mu _{\kappa }(a)=\bigvee \{\alpha \in M:\kappa (a, \alpha) \geq a, a \in L \}\),
is an Mfuzzy (resp., a strong Mfuzzy) semiquantale on L.
Proof
Let κ:L×M→L be an Mfuzzy quantic conucleus on L. We need to show that μ_{κ} is an Mfuzzy semiquantale. To this end

For a family of {a_{i}:i∈I}⊆L, we have
\(\mu _{\kappa }(\bigvee _{i} a_{i})=\bigvee \{\alpha \in M:\kappa (\bigvee _{i\in I} a_{i}, \alpha) \geq \bigvee _{i\in I} a_{i} \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \in M:\bigvee _{i\in I}\kappa (a_{i}, \alpha) \geq \bigvee _{i\in I} a_{i} \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigwedge _{i\in I}\bigvee \{\alpha \in M:\kappa (a_{i}, \alpha) \geq a_{i} \}=\bigwedge _{i\in I}\mu _{\kappa }(a_{i})\).

Then, \(\mu _{\kappa }(\bigvee _{i} a_{i}) \geq \bigwedge _{i\in I}\mu _{\kappa }(a_{i})\).

For a, b∈L and α,β∈M.
\(\mu _{\kappa }(a)\odot \mu _{\kappa }(b)=\bigvee \{\alpha \in M:\kappa (a, \alpha) \geq a \} \odot \bigvee \{\beta \in M:\kappa (b, \beta) \geq b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \odot \beta : \kappa (a, \alpha) \geq a, \kappa (b, \beta) \geq b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \odot \beta :\kappa (a,\alpha)\otimes \kappa (b,\beta) \geq a\otimes b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \odot \beta : \kappa (a\otimes b,\alpha \odot \beta)\geq \kappa (a,\alpha)\otimes \kappa (b,\beta) \geq a\otimes b \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{\gamma : \kappa (a \otimes b, \gamma) \geq a \otimes b \}\),
=μ_{κ}(a⊗b).

Then, μ_{κ}(a)⊙μ_{κ}(b)≤μ_{κ}(a⊗b).
In the case of a strong Mfuzzy quantic conucleus, i.e., κ(⊤_{L},α)=⊤_{L}, we have that μ_{κ}(⊤_{L})=⊤_{M} and this completes the proof. □
The following proposition lists some of the basic properties of the residuals ↘ and ↙ on an Mfuzzy quantic conucleus κ:L×M→L.
Proposition 9
For (L,≤,⊗),(M,≤,⊙)∈Quant, let κ:L×M→L be an Mfuzzy quantic conucleus on L. Then for all a, b,c∈L,α,β∈M, the following hold:

κ(a,α)⊗κ(a↘b,α↘β)≤κ(b,β),

κ(b↙a,β↙α)⊗κ(a,α)≤κ(b,β).
Proof

By Proposition 1 (1), we have
κ(a,α)⊗κ(a↘b,α↘β)=κ(a,α)⊗κ[(a,α)↘(b,β)]
≤κ[(a,α)⊗(a,α)↘(b,β)] (by Proposition 2(1))
≤κ(b,β).

By Proposition 1 (1), we have
κ(b↙a,β↙α)⊗κ(a,α)=κ[(b,β)↙(a,α)]⊗κ(a,α)
≤κ[(b,β)↙(a,α)⊗(a,α)] (by Proposition 2(2));
≤κ(b,β).
□
We conclude this section by given the notion of (L, M)quasifuzzy interior operator as an example of an Mfuzzy quantic conucleus on the power set quantale L^{X} and as a generalized form of an Linterior operator of [9]. Also, we study the relationship between (L, M)quasifuzzy interior operator and (L, M)quasifuzzy topology on a nonempty set X.
Definition 11
For (L,≤,⊗),(M,≤,⊙)∈SQuant and a nonempty set X, the mapping \(\mathcal {I}: L^{X}\times M\longrightarrow L^{X}\) is called:

An (L, M)quasifuzzy interior operator on X iff \(\mathcal {I}\) satisfies the following conditions: For all A, B∈L^{X},α,β∈M;

\(\mathcal {I}(A,\alpha) \leq \mathcal {I}(B,\beta)\) whenever A≤B, β≤α.

\(\mathcal {I}(A,\alpha) \leq A\).

\(\mathcal {I}(A,\alpha)\leq \mathcal {I}(\mathcal {I}(A,\alpha),\alpha)\).

\(\mathcal {I}(A, \alpha) \otimes \mathcal {I}(B, \beta) \leq \mathcal {I}(A\otimes B, \alpha \odot \beta)\).


A strong (L, M)quasifuzzy interior operator if it satisfies the following condition:

\(\mathcal {I}(\underline {\top },\alpha)=\underline {\top }\).


An (L, M)fuzzy interior operator if L∈UnSQuant with unit e and the following condition is satisfied:

\(\mathcal {I}(\underline {e}, \alpha)=\underline {\top }\).
Proposition 10
For (L,≤,⊗),(M,≤,⊙)∈Quant, a nonempty set X, and an (L, M)quasifuzzy topology τ:L^{X}→M, the mapping \(\mathcal {I}_{\tau }: L^{X}\times M \longrightarrow L^{X}\) defined by the equality.
\(\mathcal {I}_{\tau }(A, \alpha)= \bigvee \{u \in L^{X}: u \leq A, \tau (u) \geq \alpha \}, \forall A \in L^{X}, \alpha \in M,\)
is an (L, M)quasifuzzy interior operator on X.
Proof
Let τ:L^{X}→M be an (L, M)quasifuzzy topology on X. To prove that the map \(\mathcal {I}_{\tau }: L^{X}\times M\longrightarrow L^{X}\) defined by
\(\mathcal {I}_{\tau }(A, \alpha)= \bigvee \{u \in L^{X}: u \leq A, \tau (u) \geq \alpha \}, \forall A \in L^{X}, \alpha \in M,\)
is an (L, M)quasifuzzy interior operator on X, we will prove that the conditions \((\mathcal {I}_{1}\mathcal {I}_{4})\) of the above definition hold.

For A, B∈L^{X} and α,β∈M with A≤B,β≤α, we have

\(\mathcal {I}_{\tau }(A, \alpha)= \bigvee \{u \in L^{X}: u\leq A, \tau (u) \geq \alpha \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{u \in L^{X}: u \leq B, \tau (u) \geq \beta \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathcal {I}_{\tau }(B, \beta)\).

By definition of \(\mathcal {I}_{\tau }\), we have \(\mathcal {I}_{\tau }(A, \alpha) = \bigvee \{u \in L^{X}: u \leq A, \tau (u) \geq \alpha \} \leq A \).

Since \(\mathcal {I}_{\tau }(A, \alpha)\in L^{X}\) and \(\mathcal {I}_{\tau }(\mathcal {I}_{\tau }(A, \alpha), \alpha)=\bigvee \{u\in L^{X}: u\leq \mathcal {I}_{\tau }(A, \alpha),\tau (u) \geq \alpha \}\), then we have that \(\tau (\mathcal {I}_{\tau }(A, \alpha)) \geq \tau (u)\geq \alpha \).
Putting \(u=\mathcal {I}_{\tau }(A, \alpha)\), we have \(\mathcal {I}_{\tau }(\mathcal {I}_{\tau }(A, \alpha), \alpha)=\bigvee \mathcal {I}_{\tau }(A, \alpha)\) and this implies
\(\mathcal {I}_{\tau }(A, \alpha)\leq \mathcal {I}_{\tau }(\mathcal {I}_{\tau }(A, \alpha), \alpha)\).
Also, from (2), we have that
\(\mathcal {I}_{\tau }(\mathcal {I}_{\tau }(A, \alpha), \alpha)\leq \mathcal {I}_{\tau }(A, \alpha)\).
Then, the equality holds.

\(\mathcal {I}_{\tau }(A, \alpha) \otimes \mathcal {I}_{\tau }(B, \beta)= \bigvee \{u: u \in L^{X}, u \leq A, \alpha \leq \tau (u) \} \otimes \bigvee \{v: v \in L^{X}, v \leq B, \beta \leq \tau (v) \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \bigvee \{u \otimes v: u, v \in L^{X}, u \leq A,v \leq B, \alpha \leq \tau (u), \beta \leq \tau (v) \}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{w: w \in L^{X}, w \leq A \otimes B, \alpha \odot \beta \leq \tau (w) \} = \mathcal {I}_{\tau }(A \otimes B, \alpha \odot \beta)\).
That is, \(\mathcal {I}_{\tau }(A, \alpha) \otimes \mathcal {I}_{\tau }(B, \beta)\leq \mathcal {I}_{\tau }(A \otimes B, \alpha \odot \beta)\).
□
As consequences of the above proposition, we have the following result:
Corollary 2
Let (L,≤,⊗),(M,≤,⊙)∈Quant and X be a nonempty set.

For a strong (L, M)quasifuzzy topology τ:L^{X}→M, the mapping \(\mathcal {I}_{\tau }: L^{X}\times M\longrightarrow L^{X}\) is a strong (L, M)quasifuzzy interior operator on X.

For L∈UnQuant and an (L, M)fuzzy topology τ:L^{X}→M, the mapping \(\mathcal {I}_{\tau }: L^{X}\times M\longrightarrow L^{X}\) is an (L, M)fuzzy interior operator on X.
Proposition 11
For (L,≤,⊗),(M,≤,⊙)∈Quant and an (L, M)quasifuzzy (resp., strong (L, M)quasifuzzy, (L, M)fuzzy) interior operator \(\mathcal {I}: L^{X}\times M\longrightarrow L^{X}\), the mapping \(\tau _{_{\mathcal {I}}}:L^{X}\longrightarrow M\) defined by
\(\tau _{_{\mathcal {I}}}(A)=\bigvee \{\alpha \in M:\mathcal {I}(A, \alpha) \geq A\}\), A∈L^{X},
is an (L, M)quasifuzzy (resp., strong (L, M)quasifuzzy, (L, M)fuzzy) topology on X.
Proof
We prove only the case of (L, M)quasifuzzy interior operator, and the other cases can be proved similarly. Let \(\mathcal {I}: L^{X}\times M\longrightarrow L^{X}\) be an (L, M)quasifuzzy interior operator on X. Define the mapping \(\tau _{_{\mathcal {I}}}: L^{X}\longrightarrow M\) by
\(\tau _{_{\mathcal {I}}}(A)=\bigvee \{\alpha \in M:\mathcal {I}(A, \alpha) \geq A\}\), A∈L^{X}.
We need to show that \(\tau _{_{\mathcal {I}}}\) is an (L, M)quasifuzzy topology on X. To this end

For a family of {A_{i}:i∈I}⊆L^{X}, we have
\(\tau _{_{\mathcal {I}}}(\bigvee _{i} A_{i})=\bigvee \{\alpha \in M:\mathcal {I}(\bigvee _{i\in I} A_{i}, \alpha) \geq \bigvee _{i\in I} A_{i} \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \in M:\bigvee _{i\in I}\mathcal {I}(A_{i}, \alpha) \geq \bigvee _{i\in I} A_{i} \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq \bigwedge _{i\in I}\bigvee \{\alpha \in M:\mathcal {I}(A_{i}, \alpha) \geq A_{i} \}=\bigwedge _{i\in I}\tau _{_{\mathcal {I}}}(A_{i})\).

Then, \(\tau _{_{\mathcal {I}}}(\bigvee _{i} A_{i}) \geq \bigwedge _{i\in I}\tau _{_{\mathcal {I}}}(A_{i})\).

For A, B∈L^{X} and α,β∈M,
\(\tau _{_{\mathcal {I}}}(A)\odot \tau _{_{\mathcal {I}}}(B)=\bigvee \{\alpha \in M:\mathcal {I}(A, \alpha) \geq A \} \odot \bigvee \{\beta \in M:\mathcal {I}(B, \beta) \geq B \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \odot \beta : \mathcal {I}(A, \alpha) \geq A, \mathcal {I}(B, \beta) \geq B \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\bigvee \{\alpha \odot \beta : \mathcal {I}(A \otimes B, \alpha \odot \beta) \geq A \otimes B \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\leq \bigvee \{\gamma : \mathcal {I}(A \otimes B, \gamma) \geq A \otimes B \}\),
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\tau _{_{\mathcal {I}}}(A \otimes B)\).

Then, \(\tau _{_{\mathcal {I}}}(A)\odot \tau _{_{\mathcal {I}}}(B)\leq \tau _{_{\mathcal {I}}}(A \otimes B)\).
□
Remark 2
The correspondences \(\tau \longmapsto \mathcal {I}_{\tau }\) and \(\mathcal {I} \longmapsto \tau _{_{\mathcal {I}}}\) obtained in Propositions 10 and 11 are the generalizations of the correspondences between Lfuzzy interior operators and Lfuzzy topological spaces in [9].
Mfuzzy ideal conuclei on quantales
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
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
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.
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ElSaady, K., Sharqawy, S. & Temraz, A.A. Fuzzy quantic nuclei and conuclei with applications to fuzzy semiquantales and (L, M)quasifuzzy topologies. J Egypt Math Soc 27, 28 (2019). https://doi.org/10.1186/s427870190031z
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DOI: https://doi.org/10.1186/s427870190031z
Keywords
 Quantale
 Semiquantale
 Fuzzy semiquantale
 Quantic nucleus and conucleus
AMS subject classification
 06A06
 54A10
 54A40