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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

New topology in residuated lattices

L.C. Holdon
  • Corresponding author
  • Faculty of Exact Sciences, Department of Mathematics, University of Craiova, 13, Al.I.Cuza st., 200585, Craiova, Romania
  • International Theoretical High School of Informatics Bucharest, 648, Colentina st., 021187, Bucharest, Romania
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Published Online: 2018-10-19 | DOI: https://doi.org/10.1515/math-2018-0092

Abstract

In this paper, by using the notion of upsets in residuated lattices and defining the operator Da(X), for an upset X of a residuated lattice L we construct a new topology denoted by τa and (L, τa) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τa/F and τa. Finally, we study the uniform topology τΛ¯ and we obtain some conditions under which (L/J,τΛ¯) is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

Keywords: Residuated lattice; divisible residuated lattice; Semitopological algebra

MSC 2010: 03G10; 03G25; 06D05; 06D35; 08A72

1 Introduction

Residuation is a fundamental concept of ordered structures and categories. The origin of residuated lattices is in Mathematical Logic without contraction. The general definition of a residuated lattice was given by Galatos et al. (2007)[1]. They first developed the structural theory of this kind of algebra about residuated lattices.

Hajek (1998)[2] introduced the notion of BL-algebras and the concepts of filters and prime filters in BL-algebras in order to provide an algebraic proof of the completeness theorem of basic logic (BL, for short), arising from the continuous triangular norms, familiar in the fuzzy logic frame-work. Using prime filters in BL-algebras, he proved the completeness of Basic Logic. Soon after, Turunen (1999)[3] published a study on BL-algebras and their deductive systems.

A weaker logic than BL called Monoidal t-norm based logic (MTL, for short) was defined by Esteva and Godo (2001) [4] and proved by Jenei and Montagna (2002)[5] to be the logic of left continuous t-norms and their residua. The algebraic counterpart of this logic is MTL-algebra, also introduced by Esteva and Godo (2001)[4]. In Esteva and Godo (2001)[4] a residuated lattice L is called MTL-algebra if the prelinearity property holds in L.

Recently, Turunen and Mertanen (2008)[6] and D. Buşneag et al. (2013)[7] defined the notion of semidi-visible residuated lattice and investigated their properties. Also, D. Buşneag et al. (2015) [8] investigated the notion of Stonean residuated lattices and they discussed it from the view of ideal theory.

Semitopological and topological BL-algebras were defined by R. A. Borzooei et al. (2011)[9], and they establish the relationships between them. A. Borumand Saeid and S. Motamed (2009)[10] introduced the set of double complemented elements for any filter F in BL-algebras and studied their properties. In Borzooei and Zahiri (2014)[11] the definition of double complemented elements for any filter F in BL-algebras was generalized to the concept of Dy(F), for any upset F of the BL-algebra L. In fact, they show that any BL-algebra L with that topology is a semitopological BL- algebra.

In the present paper, we work on a special type of topology induced by a modal and closure operator denoted by Da(X), for an upset X of a residuated lattice L, where a is an element of L. We show that any divisible residuated lattice L with this topology is a semitopological algebra.

The paper is organized as follows:

In Section 2, we recall the basic definitions and we put in evidence rules of calculus in a residuated lattice which we need in the rest of the paper.

In Section 3, we consider the definition of upsets in the case of residuated lattices and we establish the relationship between upsets and filters. We define the operator Da(X), for an upset X of a residuated lattice L and we construct a new topology denoted by τa, where (L, τa) becomes a topological space. We discuss briefly the properties and applications of the operator Da in residuated lattices. Moreover, we obtain some of the topological aspects of these structures such as connectivity and compactness.

In Section 4, we investigate some properties of quotient topologies on residuated lattices as τa/F and τa. Finally, we study some properties of the direct product of residuated lattices. One of the most important findings is the characterization when (L/J,τΛ¯) becomes a Hausdorff space, a discrete space or a regular space relative to the uniform topology τΛ¯.

In Section 5, we apply our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

2 Preliminaries

In this section, we recall some basic notions relevant to residuated lattices which will need in the sequel.

Definition 2.1

([1]). A residuated lattice is an algebra (L, ∨, ∧, ⊙, →, 0,1) of type (2, 2, 2, 2, 0, 0) such that

(Lr1) (L, ∨, ∧, 0, 1) is a bounded lattice;

(Lr2) (L, ⊙, 1) is a commutative monoid;

(Lr3) ⊙ andform an adjoint pair, i.e., axb iff xab.

We call them simply residuated lattices. For examples of residuated lattices see [1217].

In what follows (unless otherwise speciied) we denote by L a residuated lattice. If L is totally ordered, then L is called a chain.

For xL and n ≥ 1 we define x* = x → 0, x** = (x**)**, x0 = 1 and xn = xn−1x.

We refer to [321] for detailed proofs of these well-known results:

Lemma 2.2

Let L be a residuated lattice. Then for every x, y, zL, we have:

(r1) xx = 1, x → 1 = 1, 1 → x = x;

(r2) xy iff xy = 1;

(r3) If xy = 1, then xy = xy;

(r4) If xy, then zxzy, zxzy, yzxz;

(r5) x → (yz) = (xy) → z = y → (xz);

(r6) x ⊙ (yz) = (xy) ∨ (xz), and x ⊙ (yz) ≤ (xy) ∧ (xz);

(r7) (xz) ∧ (yz) = (xy) → z;

(r8) (xz) ∨ (yz) ≤ (xy) → z;

(r9) x → (yz) = (xy) ∧ (xz);

(r10) (xy) ∨ (xz) ≤ x → (yz);

(r11) xy ≤ ((xy) → y) ∧ ((yx) → x);

(r12) (xy)* = x*y*, (xy)*x*y*;

(r13) (xy**)** = xy**;

(r14) x**y** = y*x* = xy** = (xy**)**;

(r15) xx* = 0, 1* = 0, 0* = 1, x*** = x*;

(r16) xx**, x**x*x, xyy*x*;

(r17) xy ≤ (xy)**x**y**;

(r18) x**y** ≤ (xy)**, so (x**)n ≤ (xn)**

for every natural number n;

(r19) x*y* ≤ (xy)*;

(r20) (xy)**x**y** ≤ (xy)**.

Following the above mentioned literature, we consider the identities:

(i1) xy = x ⊙ (xy) (divisibility);

(i2) (x*y*)* = [x* ⊙ (x*y*)]* (semi – divisibility);

(i3) (xy) ∨ (yx) = 1 (pre – linearity);

(i4) x*x** = 1;

(i5) x2 = x;

(i6) x = x**.

Then the residuated lattice L is called:

(i) Divisible if L verifies (i1);

(ii) Semi-divisible if L verifies (i2);

(iii) MTL-algebra if L verifies (i3);

(iv) BL-algebra if L verifies (i1) and (i3);

(v) Stonean if L verifies (i4);

(v) G-algebraif L verifies (i5);

(vi) Involutiveif L verifies (i6).

An MV-algebra is an algebra ℒ = (L, ⊕, *, 0) of type (2, 1, 0) satisfying the following equations:

(mv1) x ⊕ (yz) = (xy) ⊕ z;

(mv2) xy = yx;

(mv3) x ⊕ 0 = x;

(mv4) x** = x;

(mv5) x ⊕ 0* = 0*;

(mv6) (x*y)*y = (y*x)*x, for all x, y, zL.

Note that axioms (mv1) – (mv3) state that (L, ⊕, *, 0) is a commutative monoid.

Every BL-algebra L with x** = x for all xL, is an MV-algebra.

By Lemma 2.2, in the case of divisible residuated lattices we have:

Corollary 2.3

([7]). If L is a divisible residuated lattice, then for every x, yL we have:

(r21) (x**x)* = 0;

(r22) (xy)** = x**y**;

(r23) (xy)** = x**y**;

(r24) x ⊙ (yz) = (xy) ∧ (xz);

(r25) x ∧ (yz) = (xy) ∨ (xz).

On any residuated lattice L ([6, 8]) we may define an operator ⊕ by setting for all x, yL,

xy=(x*y*)*.(1)

By (r5), the identity (1) is equivalent with

xy=x*y**=y*x**, for allx,yL.(2)

Lemma 2.4

([8]). Let L be a residuated lattice and x, y, zL. Then:

(r26)x ⊕ 0 = x**, x ⊕ 1 = 1, xx* = 1;

(r27) xy = yx, x, yxy;

(r28) x ⊕ (yz) = (xy) ⊕ z;

(r29) If xy, then xzyz.

In what follows we will establish other necessary properties of operator ⊕ in a residuated lattice L.

Proposition 2.5

Let L be a residuated lattice and x, y, zL. Then:

(r30) (xy)** = xy = x**y**;

(r31) If xy = 1, then xy = 1;

(r32) x ⊕ (yz)* = (xy*) ∧ (xz*).

(r33) x ⊕ (y*z*)* = (xy) ∧ (xz).

Proof

(r30). We obtain successively (xy)**=(x*y*)***(r15)=(x*y*)*=xy(r15)=[(x**)*(y**)*]*=x**y**..

(r31). Since xx**, yy** and x**, y**xy implies xyxy. Since 1 = xyxy, then xy = 1.

(r32). By (r9), (r12), (r15) and (2) we obtain successively x(yz)*2=x*(yz)***(r15)=x*(yz)*(r12)=x*(y*z*)(r9)=(x*y*)(x*z*)(r14)=(y**x**)(z**x**)(2)=(xy*)(xz*).

(r33). By (r9), (r12), (r15) and (2) we obtain successively x(y*z*)*2=x*(y*z*)***(r15)=x*(y*z*)*(r12)=x*(y**z**)(r9)=(x*y**)(x*z**)(2)=(xy)(xz).

Corollary 2.6

Let L be a residuated lattice and x, yL. Then:

(r34) (x*y)*y = (y*x)*x iff (x**y**)**y** = (y** → x**)**x**.

Proof

(r34). Since (x*y)*y2=(x**y**)*y2=(x**y**)**y** and (y*x)*x2=(y**x**)*x2=(y**x**)**x**. Thus, our claim holds.

Deductive systems correspond to subsets closed with respect to Modus Ponens, their are also called filters, too. Apart from their logical interest, filters have important algebraic properties and they have been intensively studied from an algebraic point of view. Filter theory plays an important role in studying logical algebras.

We consider L as a residuated lattice.

Definition 2.7

([2]). An implicative filter (filter, for short) is a nonempty subset F of L such that

(F1) If xy and xF, then yF;

(F2) If x, yF, then xyF.

We notice that:

  1. F is an implicative filter ([13]) of L iff 1 ∈ F and x, x ͖ yF, then yF (that is, F is a deductive system ofL).

  2. Every filter is a latticeal filter in the lattice (L, ∧, ∨), but the converse does not hold ([3,14]).

Hence, if we denote by ℱ(L) (ℱi(L)) the set of all latticeal filters (implicative filters) of L, then ℱi(L) ⊆ ℱ (L).

We have ([13]) ℱi(L) = ℱ(L) iff xy = xy for every x, yL.

We recall ([3, 14]) that for a nonempty subset D of L we denote by 〈D〉 the filter generated by D, and 〈D〉 = {xL : d1 ⊙ ... ⊙ dnx, for some d1,..., dnD}. If aL, the filter generated by {a} will be denoted by 〈a〉, and (a) = {xL : anx for some n ≥ 1}. If F ∈ ℱi(L) and aL \ F, then 〈F ∪ {a}〉 will be denoted by F(a), and F(a) = {xL : anxF for some n ≥ 1}. If F, G ∈ ℱi(L), then FG = Fi.(L) G = 〈FG〉 = {xL : bcx for some bF, cG}.

We say that P ∈ ℱi(L), PL is a prime filter ([14]) if for x, yL and xyP, then xP or yP. We denote by Speci(L) the set of all prime filters of L.

We recall that a filter M of L is called maximal if ML and M is not strictly contained in any proper filter of L.

Every maximal filter M of L is obvious prime because, if there exist two proper filters N, P ∈ ℱi(L) such that M = NP, then MN and MP, by the maximality of M we deduce that M = N = P, that is, M is an inf-irreducible, so prime element in the lattice of filters (ℱi(L), ⊆) of L (by the distributivity of the lattice of filters (ℱi(L), ⊆) of L).

So, if we denote by Maxi(L) the set of all maximal filters of L, then Maxi(L) ⊆ Speci(L).

Proposition 2.8

([1, 3, 14]). For M ∈ ℱi(L), ML, the following are equivalent:

(i) M is maximal;

(ii) If xM, then there exists n ≥ 1 such that (xn)*M.

For Fi (L) we define a relation ≡F on L by xF y iff xy, yxF, for all x, yL iff (xy) ⊙ (yx) ∊ F.

Then ([3, 14]) ≡F is a congruence relation on L. For xL we denote by [x] = x/F the class of congruence of x modulo ≡F and L/F = {x/F: xL}.

Define the binary operations ∨, ⋀, ⊙ and → on L/F by (x/F) ∨ (y/F) = (xy)/F, (x/F)⋀(y/F) = (xy)/F, (x/F) ⊙ (y/F) = (xy)/F and (x/F) → (y/F) = (xy)/F for all x, yL.

Then (L/F, ∨, ⋀, ⊙, 0, 1) is a residuated lattice, which is called the quotient residuated lattice of L with respect to F, where 0 = 0/F and 1 = 1/F.

The relation of order on L/F is defined by (x/F) ≤ (y/F) iff xyF.

For a nonempty subset S of L we denote by S/F = {x/F : xS}. Clearly, for xL, x/F = 0 iff x* ∊ F and x/F = 1 iff xF.

We denote by π : LL/F the canonical epimorphism defined by π(x) = [x].

Definition 2.9

If L, L′ are residuated lattices, a map f : LL is called morphism of residuated lattices iff is a morphism of bounded lattices, f (xy) = f (x) ⊙ f (y) and f (xy) = f (x) → f (y) for every x, yL.

Iff is a bijective map (one-to-one and onto), then we say that L and L′ are isomorphic and we write LL′.

Remark 2.10

If Fi(L′), then f−1(F) ∊ i(L). Inparticular f−1 ({1}) = {xL : f (x) = 1} is denoted by Ker(f) and Ker(f) ∊ i (L).

Clearly, f is one-to-one iff Ker(f) = {1}. Also, f (L) is a subalgebra of L′ denoted by Im(f).

3 Construction of some topologies on residuated lattices

In general, the concept of topology represents the study of topological spaces. Important topological properties include connectedness and compactness.

A topology tells how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

Let X be a set and let τ be a family of subsets of X. We denote by 𝒫 (X) the family of all subsets of X. Then τ is called a topology on X if:

T1. Both the empty set ∅ and X are elements of τ;

T2. Any union of elements of τ is an element of τ;

T3. Any intersection of finitely many elements of τ is an element of τ.

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation XT is used to denote a set X endowed with the particular topology τ.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set ∅ and X itself are always both closed and open. An open set containing a point x is called a neighborhood of x.

A set with a topology is called a topological space. A topological space (X, τ) is called connected if {∅, X} is the set of all closed and open subsets of X.

A base (or basis) β for a topological space X with topology τ is a collection of open sets in τ such that every open set in τ can be written as a union of elements of β. We say that the base generates the topology τ.

Definition 3.1

([11]). Let (X, ≤) be an ordered set. Then we define ↑ : 𝒫(X) → 𝒫(X), byS = {xX|ax, for some aS}, for any subset S of X. A subset F of X is called an upset ifF = F. We denote by U(X) the set of all upsets of X. Anupset F is called finitely generated if there exists nN suchthat F =↑ {x1, x2, xn}, for some x1, x2, ... , xnX.

Remark 3.2

Clearly, if F is a filter of L, then F is an upset of L, but the converse does not hold. Indeed, we consider L = {0, n, a, b, c, d, e, f, m, 1} with 0 < n < a < c < m < 1, 0 < n < a < e < m < 1, 0 < n < b < c < m < 1, 0 < n < b < f < m < 1, 0 < n < d < e < m < 1 and elements {a, b}, {a, d}, {b, d}, {a, f}, {c, d}, {b , e} , {c, e} , {c, f} and {e, f} are pairwise incomparable.

Then ([12]) L becomes a residuated lattice relative to the following operations:

Thenn = {n, a, b, c, d, e, f, m, 1} is an upset, but nn = 0, that isn is nota filter.

In [11], O. Zahiri and R. A. Borzooei construct a topology on BL-algebras considering the notion of upsets, and inspired by their work we construct a topology in the case of residuated lattices and some of their results will become particular cases. We offer complete proofs for the results available in residuated lattices.

Definition 3.3

([11]). Let τ and τ be two topologies on a given set X. If τ′τ, then we say that τ is finer than τ. Let (X, τ) and (Y, τ ) be two topological spaces. A map f : XY is called continuous if the inverse image of each open set of Y is open in X. A homeomorphism is a continuous function, bijective and has a continuous inverse.

Lemma 3.4

([11]). Let β and β′ be two bases for topologies τ and τ′ , respectively on X. Then the following are equivalent:

(i) τ′ is iner than τ;

(ii) For any xX and each basis element Bβ containing x, there is a basis element B′β such that xB′B.

Definition 3.5

([11]). Let (A, *) be an algebra of type 2 and τ be a topology on A. Then (A, *, τ) is called:

(i) Right (left) topological algebra, if for all a s A the map * : AA defined by xa * x (xx * a) is continuous;

(ii) Semitopological algebra if A is a right and left topological algebra.

If (A, *) is a commutative algebra, then right and left topological algebras are equivalent.

Definition 3.6

([11]). Let A be a nonempty set, {*i}iI be a family of binary operations on A and τ be a topology on A. Then:

(i) (A, {*i}iI, τ) is a right (left) topological algebra, if for i s I, (A, *i, τ) is a right (left) topological algebra;

(ii) (A, {*i , τ) is a right (left) semitopological algebra, if for iI, (A, *i, τ) is a right (left) semitopological algebra.

On any residuated lattice L we define an operator ⊖ by setting for all x, yL,

xy=x*y.(3)

By (r5), the identity (3) is equivalent with

xy=x*y=x**y**=y*x*, for allx,yL.(4)

Definition 3.7

Consider L a residuated lattice and aL. For any nonempty upset X of L we define the set

Da(X)={xL|anxX,for all n},

where anx = a ⊖ (an–1x), for any n ∊ {2, 3, 4,...}.

Lemma 3.8

Let L be a residuated lattice and a, xL. Then:

(r35) anx = (a**)nx**;

(r36) (anx)** = anx, for any n ∊ ℕ.

Proof

(r35). Mathematical induction relative to n.

If n = 2, by (r5), (r3O) and identity (4) we obtain successively a2x=a(ax)=id.(4)a**(a*x)**=(r30)a**(a*x)=id.(4)a**(a**x**)=(r5)(a**a**)x**=(a**)2x**.

Suppose an–1x = (a**)n–1x**. By (r5) we obtain successively anx=a(an1x)=a**((a**)n1x**)=(r5)(a**(a**)n1)x**=(a**)nx**. Therefore, anx = (a**)nx**, for any n ∊ ℕ.

(r36). By (r30) we have (anx)**=[a(an1x)]**=id.(4)[a*(an1x)]**=(r30)a*(an1x)=id.(4)a(an1x)=anx, for any n ∊ ℕ.

A directly consequence of (r35) is the following result:

Proposition 3.9

Let L be a residuated lattice and aL. For any nonempty upset X of L theset Da (X) = {xL : (a**)nx** ∊ X, for some n ∊ ℕ}.

Remark 3.10

If a = 1, then Da (X) = D1(X) = {xL : x** ∊ X} is the set of double complemented elements of X.

Corollary 3.11

If L is a residuated lattice and a, xL, then:

(rM37) axaxaxa** ∨ x** ≤ ax;

(r38) x** ≤ axanx, for any n ∊ ℕ.

Proof

(r37). Indeed, axaxaxa** ∨ x**. By (r27) and identity (2) we obtain successively ax=id.(2)a*x**x** and ax=(r27)xa=id.(2)x*a**a**, hence a** ∨ xax.

(r38). Since anx=(r35)(a**)nx**(r4)a**x**=axx**, hence x** ≤ axanx, for any n ∊ ℕ.

The following properties hold for any residuated lattice:

(r39) zy ≤ (xz) → (xy) and zy ≤ (yx) → (zx);

(r40 (xy) ⊙ (yz) ≤ xz.

In the next result, we investigate some properties of the operator Da (X) for nonempty upsets. Clearly, Da (∅) = ∅.

In Proposition 3.3, [11], there are the properties of operator Da for BL-algebras, we offer complete proofs for them in the case of residuated lattices:

Theorem 3.12

If L is a residuated lattice and a, xL. Consider X, Y two nonempty upsets of L. Then:

(i) Da(X) is an upset of L;

(ii) 1 ∊ Da(X), aDa(X) and XDa(X);

(iii) am ⊖ (anx) = am+nx, for any m, n ∊ ℕ;

(iv) if XY, then Da (X) ⊆ Da (Y);

(v) Da (Da (X)) = Da (X);

(vi) if F is a filter of L, then Da(F) is a filter of L;

(vii) if as, then Ds (X) ⊆ Da (X);

(viii) if {Xα|αI} is a family of upsets of L, then Da(∪{Xα|αI}) = ∪(Da ({Xα|αI}));

(ix) if {X1, X2,..., Xn} is a finite set of upsets of L, then Da (∩{Xi| i = 1, 2,..., n}) = ∩{Da (Xi)| i = 1, 2,..., n};

(x) Da(Dx(X)) = Dx(Da(X));

(xi) if X1, X2, X3 are upsets of L, then Da(X1) ∩ [Da (X2) ∩ Da (X3)] = [Da (X1) ∩ Da (X2)] ∪ [Da (X1) ∩ Da (X3)] and Da(X1) ∪ [Da(X2) ∩ Da(X1)] = [Da(X1) ∪ Da(X2)] ∩ [Da(X1) ∪ Da(X3)].

Proof

(i). Clearly, Da(X) ⊆↑ Da(X). We consider s ∊↑ Da(X), then there exists xDa(X) such that xs. Since xDa(X) we have anxX, by (r35) we deduce that (a**)nx** ∊ X, for some n ∊ ℕ. By (r4) and (r16), since xs we obtain successively x** ≤ s**, (a**)nx** ≤ (a**)ns**. Since X is an upset of L and (a**)nx** ∊ X, then (a**)ns** ∊ X, hence sDa(X) and so ↑ Da(X) ⊆ Da(X). We deduce that Da (X) is an upset of L.

(ii). Since 1X, by (r35) we have an ⊖ 1 = (a**)n → 1** = (a**)n → 1 = 1 ∊ X, for any n ∊ ℕ. Hence 1 ∊ Da (X).

Since 1 ∊ X, by (r18 and (r35) we have ana = (a**)na** = 1 ∊ X, for any n ∊ ℕ. Hence aDa(X).

Now, we consider xX. Then by and (r38) we have xx** ≤ anxX, hence xDa(X). We deduce that XDa (X).

(iii). Consider m, n ∊ ℕ and a, xL. By (r5) and (r36) we obtain successively am(anx)=id.(4)(a**)m(anx)**=(r36)(a**)m(anx)=id.(4)(a**)m((a**)nx**)=(r5)[(a**)m(a**)n]x**=(a**)m+nx**=id.(4)am+nx.

(iv). Consider XY and xDa(X). Then there exists n ∊ ℕ such that anxXY and so xDa(Y). Hence Da(X) ⊆ Da(Y).

(v). Since XDa (X), by (iv) we have Da (X) ⊆ Da (Da (X)). Consider xDa (Da (X)). Then there exists m ∊ ℕ such that amxDa (X) and so an ⊖ (amx) ∊ X, for some n ∊ ℕ. By (iii) we have an+mxX and so xDa (X). Hence Da (X) = Da (Da (X)).

(iv). Consider XY and xDa(X). Then there exists n ∊ ℕ such that anxXY and so xDa(Y). Hence Da(X) ⊆ Da(Y).

(v). Since XDa (X), by (iv) we have Da (X) ⊆ Da (Da (X)). Consider xDa (Da (X)). Then there exists m ∊ ℕ such that amxDa (X) and so an ⊖ (amx) ∊ X, for some n ∊ ℕ. By (iii) we have an+mxX and so xDa (X). Hence Da (X) = Da (Da (X)).

(vi). Consider F a filter of L. Then F is a nonempty upset and by (ii) we have 1 ∊ Da(F). Let x, xyDa(F), then there are m, n ∊ ℕ such that (a**)nx** ∊ F and (a**)m → (xy)** ∊ F. By (r5) and (r17) we obtain successively ((a**)nx**)((a**)n+my**)=((a**)nx**)((a**)n(a**)my**)=(r5)(a**)m[((a**)nx**)((a**)ny**)](r39)(a**)m(x**y**)(r17)(a**)m(xy)**F.

Since F is a filter and (a**)nx** ∊ F, then (a**)n+my** ∊ F, hence yDa(F). We deduce that Da (F) is a filter of L.

(vii). Consider as and xDs(X), then there exists n ∊ ℕ such that (s**)nx** ∊ X. By (r4) we obtain successively as, a** ≤ s**, (a**)n ≤ (s**)n, (s**)nx** ≤ (a**)nx**. Since (s**)nx** ∊ X, then (a**)nx** ∊ X, hence xDa(X). We deduce that Ds(X) ⊆ Da(X).

(viii). Consider xL. Then we have the equivalences:

xDa(∪{Xα|αI}) iff anx ∊ ∪{Xα|αI} iff

anxXα, for some n ∊ ℕ, and α s I iff

xDa(Xα), for some n ∊ ℕ and αI iff

x ∊ ∪(Da({Xα|αI})). Hence Da(∪{Xα|αI}) = ∪(Da({Xα|αI})).

(ix). Following (iv), since ∩{Xi| i = 1, 2,..., n} ⊆ Xi, for any i ∊ {1, 2,..., n}, then Da| i = 1, 2, n}) ⊆ Da(Xi), for any i ∊ {1, 2, n}, hence Da(∩{Xi| i = 1, 2, n}) ⊆ ∩{Xi| i = 1, 2, n}. Consider x ∊ ∩{Da(Xi)| i = 1, 2, n}. Then there exist m1, m2, mn ∊ ℕ such that (a**)mix** ∊ Xi, for any i ∊ { 1 , 2 , ... , n}.

Consider now p = max{m1, m2,...,mn}. By (r4), since mip, for any i ∊ {1, 2, n}, then (a**)mix** ≤ (a**)px**. Since (a**)mix** ∊ Xi, then (a**)px** ∊ Xi, for any i ∊ {1, 2,..., n}. It follows that (a**)px** ∊ ∩{Xi = 1, 2, n}, hence xDa(∩{Xi = 1, 2, n}). We deduce that Da(∩{Xi| i = 1, 2,..., n}) = ∩{Da (Xi)| i = 1, 2,..., n}.

(x). We have successively:

Da (Dx (X)) = {tL| antDx (X), for some n ∊ ℕ} =

= {tL| xm ⊖ (ant) ∊ X, for some m, n ∊ ℕ}

=id.(4){tL|(x**)m(ant)**X,for somem,n}

=(r36){tL|(x**)m(ant)X,for somem,n}

=id.(4){tL|(x**)m[(a**)nt**]X,for somem,n}

=(r5){tL|(a**)n[(x**)mt**]X, for some m,n}

=id.(4){tL|an(xmt)X,for somem,n}

= {tL|xmtDa (X), for some m ∊ ℕ} = Dx (Da (X)).

(xi). It follows from (viii) and (ix).

In universal algebra, for a nontrivial lattice A, a unary mapping f : P(A) → P(A) is called laticeal modal operator on A if it satisfies the conditions: For all A1, A2A,

(1) A1f(A1);

(2) f (f(A1)) = f (A1);

(3) f (A1A2) = f (A1) ∩ f (A2).

A laticeal modal operator is called monotone ifit satisies:

(m) If A1A2, then f (A1) ⊆ f (A2).

Following Theorem 3.12 (ii), (iv), (v), and (viii) we deduce:

Corollary 3.13

Let aL. Then the map Da : U(L) → U(L) is a laticeal monotone modal operator.

Proof

Following Theorem 3.12 (ii), (iv), (v), and (viii) we deduce that the map Da : U(L) → U(L) is a laticeal monotone modal operator.

We recall that, for a residuated lattice L, by U(L) we denote the set of all upsets of L.

Corollary 3.14

For any aL, the map Da : U(L) → U(L) is a closure operator and Da = Da**.

Proof

Following Theorem 3.12 (i), (ii), (iv) and (v), we deduce that Da is a closure operator.

Consider X an upset of L. Following Theorem 3.12 (vii), since by (r16) we have aa**, then Da** (X) ⊆ Da(X). Consider now xDa(X). Then there exists n ∊ ℕ such that (a**)nx** ∊ X. By (r15) we have a**** = a**, then (a****)nx** ∊ X, hence xDa**(X). We deduce that Da(·) = Da**(·).

Lemma 3.15

Let aL and F be a filter of L. Then Da (F) = D1(F) iff a** ∊ F.

Proof

” ⇐ ”. Consider aL and F a filter of L such that Da (F) = D1(F). Following Theorem 3.12 (ii), we have aDa(F) = D1(F), then aD1 (F), and we have successively aD1(F), (1**)na** ∊ F, 1na** ∊ F, 1 → a**F, a**F.

” ⇐ ”. Suppose a** ∊ F. By Theorem 3.12 (vii), since a ≤ 1, then D1(F) ⊆ Da(F). Consider now xDa (F). Then there exists n ∊ ℕ such that (a**)nx** ∊ F. Since a** ∊ F and F is a filter of L, then (a**)nF, for any n ∊ ℕ. By (a**)nF, (a**)nx**F, we deduce x**F. Since x** = (1**)nx**F, then xD1(F), hence Da (F) ⊆ D1(F). We deduce that Da (F) = D1(F).

Corollary 3.16

Let L be an involutive residuated lattice, aL and F be a filter of L. Then Da (F) = F iff aF.

Proof

” ⇐ ”. Consider aL and F a filter of L such that Da(F) = F. Following Theorem 3.12 (ii), we have aDa (F) = F, then aF.

” ⇐ ”. Suppose aF. By Theorem 3.12 (ii), aFDa(F). Consider now xDa(F). Then there exists n ∊ ℕ such that (a**)nx** ∊ F. Since aa** ∊ F and F is a filter of L, then (a**)nF, for any n ∊ ℕ. By (a**)nF, (a**)nx** ∊ F, we deduce x** ∊ F, hence xF. Therefore, Da(F) ⊆ F. We deduce that Da (F) = F.

Remark 3.17

(i). By Theorem 3.12 (vi), if F is a proper filter, then Da (F) is a proper filter or Da (F) = L. We consider the lattice L = {0, a, c, d, m, 1} with 0 < a < m < 1, 0 < c < d < m < 1, but a incomparable with c and d.

Then ([12], page 233) L becomes a residuated lattice relative to the following operations:

We havea〉 = {a, m, 1} is a proper filter of L, Dm (〈a〉) = {xL | (m**)nx** ∊ 〈a〉, for some n ∊ ℕ} = {xL | 1 → x** ∊ 〈a〉} = 〈aand Dc (〈a〉) = {xL | (c**)nx** ∊ 〈a〉, for some n ∊ ℕ} = {xL | cx**} = L.

Moreover, 〈ais a maximal filter of L and Dc (〈a〉) = L ≠ 〈a〉. So, if M is a maximal filter, then Dx (M) = M or Dx(M) = L, for any xL.

(ii). There are residuated lattices L such that for a proper filter F of L, thereis aL and Da (F) = L. Indeed, we consider L = {0, a, b, c, d, e, f, m, 1} with 0 < a < c < m < 1, 0 < a < e < m < 1, 0 < b < c < m < 1, 0 < b < f < m < 1, 0 < d < e < m < i and elements {a, b}, {a, f}, {a, d}, {b, d}, {b, e}, {d, c}, {c, e}, {c, f} and {e, f} are pairwise incomparable.

Then ([12]) L is a residuated lattice with the following operations:

Let F = 〈1〉 = {1}. Since Dm (〈1〉) = {xL|(m**)nx** ∊ 〈1〉, for some n ∊ ℕ} = {xL|0 → x** ∊ 〈1〉} = {xL|1 ∊ 〈1〉} = L.

(iii). There are residuated lattices L such that for a prime filter F, there is aL such that Da (F) is not prime. Indeed, we consider L = {0, a, b, n, c, d, 1} with 0 < a < n < d < 1, 0 < b < n < c < 1, but (a, b) and (c, d) are pairwise incomparable.

Then ([12], page 191) L becomes a residuated lattice relative to the operations:

Clearly, 〈c〉 = {c, 1} is aprime filter of L. Since Dn (〈c〉) = {xL|(n**)mx** ∊ 〈c〉, for some m ∊ ℕ} = {xL|1 → x** ∊ 〈c〉} = {xL|x** ∊ 〈c〉} = {n, c, d, 1}, hence Dn(〈c〉) = {n, c, d, 1}. Since n = ab, but aDn (〈c〉) and bDn (〈c〉), we deduce Dn (〈c〉) is not prime.

(iv). There are residuated lattices L such that for a filter F, there is aL such that Da(F) is prime, but F is not prime. Indeed, we consider L = {0, a, b, c, d, e, f, 1} with 0 < a < c < e < 1, 0 < b < e < f < 1 0 < b < c < e < 1, 0 < b < d < e < 1, and elements {a, b}, {a, d}, {a, f}, {c, d}, {c, f}, and {e, f} are pairwise incomparable.

Then [12] L becomes a MTL-algebra with the following operations:

Sincee〉 = {e, 1}, with e = cd, but c ∉ 〈eand d ∉ 〈e〉, theneis not aprime filter. Since Dd (〈e〉) = {xL|(d**)nx** ∊ 〈e〉, for some n ∊ ℕ} = {xL|dx** ∊ 〈e〉} = {d, e, f, 1}. Hence Dd(〈e〉) = {d, e, f, 1}, which is a prime filter of L, buteis not prime.

(v). If MMaxi (L) is a maximal filter and Da (M) is a proper filter, then Da (M) = M. Indeed, by Theorem 3.12 (ii), MDa (M) and Da (M) is a proper filter. We get that if M is a maximal filter, then M = Da (M).

There are residuated lattices L such that Da (M) is a maximal filter, but MMaxi (L). We consider L = {0, n, a, b, c, d, 1} with 0 < n < a < b < c, d < 1, but c and d are incomparable.

Then ([12]) L becomes a distributive residuated lattice relative to the followingoperations:

Clearly, 〈a〉 = {a, b, c, d, 1} is the unique maximal filter of L andb〉 = {b, c, d, 1} is a filter of L. Since Dc(〈b〉) = {xL|(c**)nx** ∊ 〈b〉, for some n ∊ ℕ} = {xL|1 → x** ∊ 〈b〉} = {xL|x** ∊ 〈b〉} = {a, b, c, d, 1} = 〈a〉. Therefore, Dc(〈b〉) = 〈ais the unique maximal filter of L, butbis not maximal.

Proposition 3.18

Let aL and F be a filter of L. Then:

(i) if M is a maximal filter of L and a** ∊ M, then Da (M) = M;

(ii) D1(F) is a prime filter iff Da (F) is prime;

(iii) (an)* ∊ F, for some n ∊ ℕ iff Da (F) = L;

(iv) if M is a maximal filter of L, then Da (M) = L iff aL \ M.

Moreover, if M is a maximal filter of L, then Da(M) = M iff aM.

Proof

(i). Let M be a maximal filter of L and a** ∊ M. Following Theorem 3.12 (ii) and (vi) we have Da(M) is a filter and MDa(M). We must to prove that Da(M) is a proper filter of L. If 0 ∊ Da(M), then by (r5) and (r15) we obtain successively 0 ∊ Da(M), (a**)n → 0** ∊ M, (a**)n → 0 ∊ M, (a**)n-1a** → 0 ∊ M, (a**)n-1 → [a** → 0] ∊ M, (a**)n-1a*** ∊ M, (a**)n-1a* ∊ M. Since a** ∊ M and M is a filter of L, then (a**)n-1M. Since (a**)n-1M, (a** )n-1a* ∊ M, then a* ∊ M. Since a*, a** ∊ M and M is a filter of L, then 0 = a* ⊙ a** ∊ M, a contradiction. Hence Da(M) ≠ L.

By hypothesis, M is a maximal filter and MDa(M) ≠ L, then Da(M) = M.

(ii). Let F beafilter of L. Following Theorem 3.12 (vi), D1(F) and Da (F) are filters of L. Suppose D1(F) is a prime filter of L and xyD1(F). If xyD1(F), then (1**)n → (xy)** = (xy)** ∊ F, for any n ∊ ℕ. Since F is a filter and (xy)** ≤ (a**)n → (xy)**, then (a**)n → (xy)** ∊ F, hence xyDa(F).

Since xyD1(F) and D1(F) is prime, then xD1(F) or yD1(F). It follows that x** ∊ F or y** ∊ F. We have successively x** ∊ F, x** ≤ (a**)nx** ∊ F or y** ∊ F, y** ≤ (a**)my** ∊ F, then xDa(F) or yDa(F), for some m, n ∊ ℕ. Hence Da(F) is prime.

Now, suppose Da(F) is prime and for a = 1 we deduce Di(F) is prime, too.

(iii). Suppose (an)* ∊ F, for some n ∊ ℕ. Following Theorem 3.12 (ii), F ⊆ Da(F), then (an)* ∊ Da(F). By Theorem 3.12 (ii) and (vi), we obtain aDa (F) and Da (F) is a filter, then anDa (F), for any n ∊ ℕ. Since anDa(F), (an)* ∊ Da(F) and Da(F) is a filter, then 0 = an ⊙ (an)* ∊ Da(F). Hence Da(F) = L.

Now, suppose Da(F) = L. Then 0 ∊ Da(F), (a**)→ 0**F, (a**)n → 0 ∊ F, for some n ∊ ℕ.

We prove by induction that (a**)n+1 → 0 = an+1 → 0, for n ∊ ℕ. Clear, for n = 0. Suppose (a**)n → 0 = an → 0, for n ∊ ℕ. By (r5), (a**)n+1 → 0 = ((a**)na**) → 0 = a** → ((a∊**)n → 0) = a** → (an → 0) = ana*** = an → (a → 0) = an+1 → 0.

Since (a**)n → 0 ∊ F, then an → 0 ∊ F, that is (an)* ∊ F.

(iv) Consider M a maximal filter of L.

”(⇒)”. Consider Da(M) = L. If aM, then we get a**M, and by (i) we obtain L = Da(M) = M, a contradiction. We deduce aM, that is, aL \ M.

” ⇐ ”. Consider a ∊ L \ M. Since M is a maximal filter, then following Proposition 2.8 there is n ∊ ⊙ such that (an)*M, and by (iii) we obtain Da(M) = L.

Now, the fact that Da(M) = M iff aM is routine.

Georgescu et al. (2015)[15] called Gelfand residuated lattices those residuated lattices in which any prime filter is included in a unique maximal ilter. They are also called normal residuated lattices. Examples of Gelfand residuated lattices are Boolean algebras, BL-algebras and Stonean residuated lattices (see [8]).

Proposition 3.19

Let a ∊ L and P ∊ Speci (L) be aprime filter. If P = Da (P), then L is Gelfand (normal) residuated lattice. The converse does not hold.

Proof

Let P be a prime filter of L such that P = Da (P). Using Zorn’s Lemma we deduce that P is contained in a maximal filter. Suppose that there are two distinct maximal filters M1 and M2 such that PM1 and PM2. Since M1 ≠ M2, there is aM1 such that aM2. By Theorem 3.12 (ii) and (vi) we have that aDa(P) and Da(P) is a filter, then (an)**Da(P) = P, for any n ∊ ℕ. Hence (an)**P, for any n ∊ ℕ.

Following Proposition 2.8, there is n ≥ 1 such that (an)*M2. Then (an)**M2, hence (an)**P, a contradiction.

For the converse we consider the residuated lattice L from Remark 3.17 (v). The prime filter of L are (a) = {a, b, c, d, 1}, 〈b〉 = {b, c, d, 1}, 〈c〉 = {c, 1} and 〈d〉 {d, 1}. The maximal filter of L is 〈a〉, which include the all other prime filters. Hence L is Gelfand, but Dc(〈b〉) = 〈a〉 ≠ 〈b〉.

Proposition 3.20

Let F ∊ ℱi (L) and a ∊ F. For x ∊ Da(F) the following assertions are equivalent:

(i)Da(F) = F;

(ii)x**F, then x ∊ F.

Proof

(i) ⇒ (ii). Consider Da(F) = F. By hypothesis F is a filter and aF, then a**F, (a**)nF, for every n ∊ ℕ. Since F is a ilter and (a**)nF, (a**)nxF, then xF. We obtain successively F = Da(F) = {xL | (a**)nx**F, for some n ∊ ℕ} = {x ∊ L | x**F}, that is, if x**F, then xF, for all xL.

(ii) ⇒ (i). By hypothesis F is a filter and aF, then a**F, (a**)nF, for every n ∊ ℕ. Since F is a filter and (a**)nF, (a**)nx**F, then x**F. We obtain successively Da(F) = {xL | (a**)nx**F, for some n ∊ ℕ} = {x ∊ L | x**F} = F.

Theorem 3.21

Let a ∊ L and F be a filter of L. For x ∊ Da (F) the following assertions are equivalent:

(i)Da(F) = F

(ii)x** ∊ F iff xF.

Proof

(i) ⇒ (ii). Consider Da(F) = F. By hypothesis F is a filter and xDa(F) = F, by (r16) we obtain xx**F, so, if xF, then x**F. Now, we prove that if x**F, then xF. Since F is a filter and x**F, x** ≤ (a**)nx**F, for any n ∊ N, then xDa(F) = F.

(ii) ⇒ (i). By Theorem 3.12, (ii) we have FDa(F). Now, we consider xDa(F) such that x** ∊ F iff xF. Since x** ∊ F and x** ≤ (a**)nx** ∊ F, for any n ∊ ℕ, then we obtain successively Da(F) = {xL | (a**)nx** ∊ F, for some n ∊ ℕ} ⊆ {xL | x** ∊ F} ⊆ F. Therefore, Da(F) = F.

Remark 3.22

Examples of residuated lattices which satisfy the conditions from Proposition 3.20 and Theorem 3.21 are Boolean algebras, MV-algebras and involutive residuated lattices.

Corollary 3.23

The set τa = {Da(X)|X ϵ U(L)} is a topology on L and (L, τa) is a topological space.

Proof

Let a ϵ L. Clearly, Da) = ϕ and Da(L) = L. Following Theorem 3.12 (viii) and (ix) the set τa = {Da(X)|X ϵ U(L)} is a topology on L and (L, τa) is a topological space.

Proposition 3.24

Theset βa = {Da (↑ x)|x ϵ L)} is a base for the topology τa on L.

Proof

Let Z be an open subset of (L, τa). Then there is X ϵ U(L) such that Z = Da(X). Since X is an upset, then X = u{↑ x|x ϵ X} and by Theorem 3.12 (viii), we deduce that Da(X) = ∪{Da (↑ x)| x ϵ X}. Hence βa is a base for the topology τa on L.

Corollary 3.25

Every open set X relative to the topology τa is an upset. But the converse does not hold.

Proof

Following Corollary 3.23 and Proposition 3.24 we have (L, τa) is a topological space with β a a base for the topology τa. Since every union of elements of βa is an upset and every open set X relative to τa can be written as a union of elements of βa, then X is an upset.

For the converse we consider the residuated lattice L from Remark 3.17 (v), where ↑ n = {n, a, b, c, d, 1} is an upset of L. The upsets of L are ↑ 0 = L, ↑ n = {n, a, b, c, d, 1}, ↑ a = {a, b, c, d, 1}, ↑ b = {b, c, d, 1}, ↑ c = {c, 1}, ↑ d = {d, 1} and ↑ 1 = {1}, so U (L) = {↑ 0, ↑ n, ↑ a, ↑ b, ↑ c, ↑ d, ↑ 1}. Since the base of topology τp on L is the set βp = {Dp(↑ x) | p ϵ L and ↑ x ϵ U(L)} = {z ϵ L| (p**)nz** ϵ↑ x for some n ϵ ℕ}, then it is easy to verify that ↑ n can not be written as a union of elements of βp. Therefore, ↑ n is not an open set relative to the topology τp.

Proposition 3.26

If u, v ϵ L such that u ≤ v, then the topology τv is finer than topology τu.

Proof

We denote by βu, βv basis of the topology τu, respectively τv.

Consider t ϵ L and Du (↑ x) ⊆ βu an element of the basis βu such that t ϵ Du(↑ x). Then there exists n ϵ N such that (u**)n → t** ϵ↑ x, that is, x ≤ (u**)n → t**.

Following Theorem 3.12 (ii) we have t ϵ Dv(↑ t). Following Theorem 3.12 (vii), since u ≤ v, then Dv (↑ t) ⊆ Du(↑ t). We prove that Du (↑ t) ⊆ Du(↑ x). Now, we consider s ϵ Du (↑ t). Then there exists m ϵ N such that (u**)m → s** ϵ ↑ t and so t ≤ (u**)m → s**. By (r4), (r35) and (r36) we obtain successively t ≤ (u**)m → s**, t**(r4)[(u**)ms**]**=(r35),(r36)(u**)ms**. Since x(u**)nt**(u**)n[(u**)ms**]=(r5)(u**)m+ns**, then x ≤ (u**)m+n → s**, hence (u**)m+n→ s** ϵ ↑ x and so s ϵ Du (↑ x).

Since Du (↑ t) ⊆ Du (↑ x) and Dv (↑ t) ⊆ Du (↑ t), it follows that Dv (↑ t) ⊆ Du (↑ t) ⊆ Du (↑ x).

Clearly, Dv (↑ t) ⊆ βv is an element of the basis βv. We deduce that for any t ϵ Du (↑ x) ⊆ βu, there is a basis element Dv (↑ t) ⊆ βv such that t ϵ Dv (↑ t) ⊆ Du(↑ t) ⊆ Du (↑ x). Following Lemma 3.4 we deduce that the topology τv is finer than topology τu.

Lemma 3.27

If X is a nonempty subset of L and a ϵ L, then X is a compact subset of (L, τa) iff XDa (↑ {xi1, xi2, xin}), for some xi1, xi2, xin ϵ X and i ϵ I.

Proof

“ ⇐ “. SupposeX ⊆ Da(↑ {xi1, xi2,..., xin}), for some xi1, xi2,..., xin ϵ X and {Da(Xi)|i ϵ I} be a family of open subsets of L whose union contains X. For any j ϵ {1, 2,..., n}, there is ij ϵ I such that xij ϵ Da (Xij). Following Theorem 3.12 (i) and (iv), we have Da(xij) ⊆ Da(Xij), for any j ϵ {1,2,..., n}. By Theorem 3.12 (viii) weobtain X ⊆ Da (↑ {xi1, xi2xin}) = Da (xi1 ) ∪ Da (xi2)∪...∪Da (xin) ⊆ Da (Xi1)∪Da (Xi2)∪...∪Da (Xin). Hence X is compact.

“ ⇒ “. Now, suppose X be a compact subset of L. Since X ⊆ ∪{↑ x|x ϵ X}, then by Theorem 3.12(ii), Xu{Da(↑ x)|x ϵ X}. Hence {Da(↑ x)|x ϵ X} is a family of open subsets of L whose union contains X.

By hypothesis, there are x1, x2,..., xn ∊ X such that X ⊆ Da(↑ x1) ∪ Da(↑ x2) ∪ ... ∪ Da(↑ xn). By Theorem 3.12(viii) we obtain X ⊆ Da(↑ {x1, x2,..., xn}).

Theorem 3.28

The topological space (L, τa) is connected.

Proof

Consider X a non-empty subset of L such that is both closed and open relative to the topology Ta. If X is an open set, by Corollary 3.25, then X is an upset. If 0 ∊ X and X is an upset, then X = L. If 0 ∊ L -X, since X is closed, we get L - X is open, by Corollary 3.25, L - X is an upset of L, since 0 ∊ L - X, then L - X = L. Hence X = ∅, a contradiction. We conclude that {∅, L} is the set of all subsets of L which are both closed and open. That is, (L, τa) is connected.

4 Some Properties of Quotient Topology on Residuated Lattices

Proposition 4.1

Let P, Q ∊ ℱi(L) and a ∊ L such that Q ⊆ P. Then D[a](P/Q) = Da(P)/Q, where [a] = {a/Q | aL}.

Proof

Consider x ∊ L. Then

D[a](P/Q) = {[x] ∊ L/Q/ [a]m ⊖ [x] ∊ P/Q, for some m ∊ ℕ} = {[x] ∊ L/Q/ [amx] ∊ P/Q} = {[x] ∊ L/Q\ amxP} = Da(P)/Q.

Following Proposition 3.24 we obtain the following result.

Remark 4.2

(i). Let F ∊ Fi(L) and a ∊ L. Then (L/F, τa/F) is a topological space, where the set τa/F = {Da/F (X/F)\ XU (L)} is a topology on L/F and the set {Da/F (↑ x/F)\ x/FL/F} is a base for the topology Ta/F on L/F.

If π : L → L/F is the canonical morphism defined by π(x) = [x], then π is a continuous map. Indeed, if O is an open set with respect the topology τa/F, then O = ∪{Da/F(↑ xi/F)\for some xi/F ∊ L/F}, and π-1(O) = π-1(∪ (Da/F(t xi/F)) = ∪ π-1 (Da/F(↑ xi/f)) = ∪ Da(π’1(↑ xi/F)), which is an open set on (L, τa).

(ii) Let F ∊ ℱi(L) and a ∊ L. Then (L/F,τa), is a topological space, where the set τa={Da(X)/F|XU(L)} is a topology on L/F and the set {Da(↑ x)/F | x/F ∊ L/F} is a base for the topology τa.

If π : L → L/F is the canonical morphism defined by π(x) = [x], then π is a continuous map. Indeed, if O is an open set with respect the topology τa, then O = ∪{Da(↑ xi)/F| for some xi/F ∊ L/F}, and π-1) = π-1(∪Da(∧ xi)/F) = ∪π-1(Da(↑ xi)/F) = ∪Da(π’1 (↑xi/F)), which is an openseton (L, τa).

Theorem 4.3

Let F be a filter of L and a, x ∊ L. Then:

(i) Da(↑ x)/F ⊆ Da/F(↑ x/F).

(ii) The topology τa/F is finer than τa, where τa is the quotient topology of L/F.

Proof

(i). Da/F(↑ x/F) = {u/F ∊ L/F\ x/F ≤ ((a/F)**)n → (u/F)**} = {u/F ∊ L/F| x → [(a**)nu**] ∊ F} and Da(↑ x)/F = {v/F : v ∊ Da(↑ x)}. Let u/F ∊ Da(↑ x)/F. Then there exists v ∊ Da(↑ x) such that u/F = v/F and so x → [(a**)n v**] = 1, for some n ∊ ℕ. Since u ≡F v, then x → [(a**)n → u**] = x → [(a**)n → v**] = 1, that is x → [(a**)n → u**] ∊ F, hence u/F ∊ Da/F(↑ x/F). Therefore, Da(↑ x)/F ⊆ Da/F(↑x/F).

(ii). By Proposition 3.24 the set {Da/F (↑ x/F): x/F ∊ L/F} is a base for the topology τa/F on L/F. Moreover, the set {Da (↑ x)/F: x ∊ L} is abase for the topology τa. Now, the proof of (ii) is straightforward by (i).

Let L, L′ be residuated lattices. On L × L′ we consider the relation of order (x, y) ≤ (x′, y′) iff x ≤ x′ and y ≤ y′ and the operations

(x, y) ∧ (x′, y′) = (x ∧x′ , y∧ y′),

(x, y) ∨ (x′, y′) = (x ∨x′, y∨y′),

(x, y) ⊙ (x′ , y′) = (x ⊙ x′, y ⊙ y),

(x, y) → (x′ , y′) = (x → x′ , y→y′) for all x, y ∊ L and x′, y′ ∊ L .

Then L, L′ with the above operations is a residuated lattice called direct product of L and L′.

Lemma 4.4

([17]). Let L, L′ be residuated lattices. Then K is a filter of L χ L′ iff there exist P ϵ Fi(L) and Q ϵ Fi (L′) such that K = P × Q.

Proof. “ ⇒ “. If K ϵ Fi(L × L′), we consider P = {x ϵ L : (x, x′) ϵ K for some x′ ϵ L′} and Q = {x′ ϵ L′ : (x, x′) ϵ K for some x ϵ L}.

Clearly, K = P × Q. Since (i, i) ϵ K we get i ϵ P. Let x,y ϵ P. Then there exist x′,y′ ϵ L′ such that (x, x′), (y, y′) ϵ K. Thus, (x, x′) ⊙ (y, y′) = (x ⊙ y, x′ ⊙ y′) ϵ K, so x ⊙ y ϵ P.

Consider x < y and x ϵ P. Then there exists x′ ϵ L′ such that (x, x′) ϵ K. Since (x, x′) < (y, x′), then y ϵ P, hence P ϵ Fi (L). Similarly, Q ϵ Fi(L′).

“ ⇐ ”=. Let K = P × Q for some P ϵ Fi(L′) and Q ϵ Fi(L′). Clearly, K ϵ L × L′. We consider (x, y), (p, q) ϵ K. Then x, p ϵ P and y, q ϵ Q, that is x ⊙ p ϵ P, y ⊙ q ϵ Q. Therefore, (x, y) ⊙ (p, q) = (x ⊙ p, y ⊙ q) ϵ P × Q = K.

Now, we consider (x, y) ϵ K such that (x, y) < (p, q). Then x < p with x ϵ P and y < q with y ϵ Q. Since P ϵ Fi(L) and Q ϵ Fi(L′), then p ϵ P and q ϵ Q, that is (p, q) ϵ K. Hence K ϵ Fi(L × L′).

Lemma 4.5

Let P, Q ϵ Fi(L) and a ϵ L. Then Da(P) × Da(Q) = Da(P × Q).

Proof

By Lemma 4.4, Da(P) × Da(Q) = {x ϵ L|an θ x ϵ P, for some n ϵ {y s L|am θ y ϵ Q, for some m ϵ ℕ} = {x × y ℕ L × L|(at θ x, at θ y) ϵ P × Q, for some t > max{m, n} ϵ ℕ} = Da(P × Q).

Consider the topological spaces (L, τb) and (L, τ1).

Theorem 4.6

Let P, Q ϵ Fi (L) and a, b ϵ L. Then there exists a homeomorphism from L×LDa(P×Q) to LDa(P)×LDa(Q)

Proof

We define ϕ:L×LLDa(P)×LDa(Q), by ϕ(x,y)=(xDa(P),yDa(Q)) Clearly, ϕ is onto. Let (x, y) ϵ L × L. Then (x, y) ϵ ker(ϕ) iff x/Da (P) = 1/Da (P) and yϵDa(Q) = 1/Da (Q) iff x ϵ Da (P) and y ϵ Da (Q). Hence ker(ϕ) = Da(P) × Da(Q). It follows easily by Lemma 4.5 that Da(P) × Da(Q) = Da(P × Q). Consider the map h:L×LDa(P×Q)LDa(P)×LDa(Q),, defined by h((x,y)Da(P)×Da(Q))=(x/Da(P),y/Da(Q)) then by the first isomorphism theorem h becomes an isomorphism. If we suppose that X is an open subset of LDa(P)×LDa(Q) then there exist U, Vϵ τb open subsets such that X=LDa(P)×LDa(Q) Clearly, h1=U×VDa(P×Q) is an open subset of L×LDa(P×Q) Hence h is a continuous map. In the same manner we can prove that h-i is a continuous map. Therefore, h is a homeomorphism.

4.1 Uniform topology on quotient residuated lattice L/J

In this section based on the work of Ghorbani and Hasankhani (2010)[18] we define and study a uniform topology τΛ¯ on the quotient residuated lattice L/J, where J is a filter of L.

If X is a non-empty set, U and V are subsets of X × X. Then:

U o V = {(x, y) ϵ X × X : (z, y) ϵ U and (x, z) ϵ V for some z ϵ X},

U-1 = {(x,y) ϵ X × X : (y, x) ϵ U},

∆= {(x, x) ∆ X × X : x ∆ X}.

In universal algebra if X is a non-empty set, then a non-empty collection K of subsets of X × X is called a uniformity on X if it satisfies the following conditions:

(U1)∆ ⊆ U for any U ϵ K,

(U2) if U ∆ K, then U-1K,

(U3) if U ϵ K, then there exists a set V ∆ K such that V o V ⊆ U,

(U4) if U, V ϵ K, then U∩ V ⊆ K,

(U5) if U ϵ; K and U ⊆ V ⊆ X × X, then V ϵ K.

The pair ( X, K) is called a uniform structure.

In Ghorbani and Hasankhani (2010)[18], Theorem 2.10, it was proved that if Λ is a family of filters of a residuated lattice L which is closed under intersection. If we consider the sets UF = {(x, y) ϵ L × L : x =Fy} for every F ϵ Λ and K* = {UF : F ϵ Λ}, then K = {U ⊆ L × L : UF ⊆ U for some UF ϵ K} is a uniformity on L. Let U ϵ K, define U[x] = {y ϵ L : (x, y) ϵ U}.

Then τΛ = {O ⊆ L : ∀x ϵ O, ∃U ϵ K such that U[x] ⊆ O} is a topology on L and is called the uniform topology on L induced by Λ.

In [18], Theorem 3.1, it was proved that if J is a filter of L. For each F ϵ Λ, let F¯=FJ/J. Then Λ*={F¯:FΛ} gives a uniform topology on L/J, where UF¯={([x],[y])L/J×L/J:[x]F¯[y]}.

The set τ = {O ⊆ L/J : π—1(O) ϵ τΛ}, where π : L → L/J is the canonical epimorphism, τ becomes the quotient topology on L/J, and τΛ* becomes the uniform topology induced by Λ* on L/J.

Lemma 4.7

Let F, Q ϵ Fi(L) and a ϵ L. Then:

  • (i) ({a} ∪ F) ⊆ Da (F), but the converse does nothold;

  • (ii) ({a} ∪ Q ∪ F) ⊆ Da ((Q ∪ F)), but the converse does nothold.

Proof

(i). Let F ϵ Fi(L) and a ϵ L. If x ϵ ({a} ∪ F) then x ≥ a ⊙ f, for some f ϵ F. We obtain successively (a**)nx**(r4)(a**)n(af)**(r4)a**(af)**=(r14)a(af)**=a[(af)*0]=(r5)[a(af)*]0=(r5)[a(af*)]0f*0=f**fF, that is x ϵ Da(F). Thus ({a} ∪ F) ⊆ Da(F).

For the converse we consider L = {0, n, a, b, c, d, e, f, m, i} with 0 < n < a < c < e < m < i, 0 < n < b < d < f < m < i and the elements {a, b}, {c, d}, {e, f} are pairwise incomparable.

Then([12]) L becomes a distributive residuated lattice relative to the following operations:

Let F = {1, m, f}, then ({d} ∪ F) = {x ∪ L : x ≥ d ⊙ t for some t ϵ F} = {d, e, f, m.1}. But Dd(F) = {x ϵ L : (d**)n → x** ϵ F} = {L}, hence 〈{b} ∪ F 〉 ⊂ Db(〈QF〉).

(ii). Let F, Q ϵ Fi(L) and a ϵ L. If x ϵ ({a} ∪ Q ∪ F) then x ≥ a ⊙ q ⊙ f, for some q ϵ Q, f ϵ F. We obtain successively (a**)nx**(r4)(a**)n(aqf)**(r4)a**(aqf)**=(r14)a(aqf)**=a[(aqf)*0]=(r5)[a(aqf)*]0=(r5)[a(a(qf)*)]0(qf)*0=(qf)**(qf)QF, that is x ϵ Da(〈Q ∪ F〉). Thus 〈{a} ∪ Q ∪ F〉 ⊂ Da(〈Q ⊂ F〉).

For the converse we consider the residuated lattice L from (i), let Q = {1, m} and F = {1, m, f} , then ({b} ∪ Q ∪ F) = {x ϵ L : x ≥ b ⊙ q ⊙ t for some q ϵ Q, t ϵ F} = {b, c, d, e, f, m.1}. But Db(〈Q ∪ F〉) = {x ϵ L : (d**)n - x** ϵ 〈Q ∪ F〉} = {{L}, hence 〈{b} ∪ F〉 ⊂ Db(〈Q u F〉).

Clearly, if a ϵ L and J, F ϵ Fi(L), then 〈J ⊙ F〉 ϵ Fi(L) and Da(〈J ⊙ F〉) ϵ Fi(L), too.

Theorem 4.8

Let a ϵ L. Thelattice (Da (Fi (L)), ⊆) is a complete Brouwerian lattice (hence distributive).

Proof

Clearly, if {Xi : i ϵ I} is a family of filters from L, then the infimum of this family is ΛiDa(Xi) = ∩iϵI(Xi) and the supremum is viDa(Xi) = ∩i∩IDa(Xi), that is the lattice (Da(Fi(L)), ∨i, Λi, ⊆, {1}, L) is complete.

By Lemma 3.27 we have that an upset X of L is compact iff X ⊆ Da (↑ {xi1, xi2,..., xin}), for some xi1, xi2, ... , xin ϵ X and i ϵ I. Since any ilter is an upset we deduce that the compact elements of Da(Fi(L)) are exactly the upsets generated by the principal filters of L, which are filters, too.

Following Theorem 3.12, (xi) we deduce that the lattice (Da(Fi(L))), ς) is distributive.

Following Lemma 4.7 and Theorem 3. 1 from [18] we obtain:

Lemma 4.9

Consider J a filter of L and a ϵ L. For each F ϵ Λ, let Da(F)¯=Da(JF)/J. Then Λ¯={Da(JF)/JL/J:FΛ} gives a uniform topology on L/J, where UDa(F)¯={([x],[y])L/J×L/J:[x]Da(F)¯[y]}.

That is, L/J isa topological space with the uniform topology τΛ¯ induced by ⊼.

For a residuated lattice L and a topology τ defined on the set L, the pair (L, τ) is called a topological residuated lattice if the operations Λ, ∨, ⊙ and → are continuous with respect to τ.

In the same manner as Theorem 3.2 from [18] the following result can be proved:

Theorem 4.10

For each x ϵ L and F ϵ Λ, π(UF[x])=UDaF¯[[x]]. Hence each UDaF¯[[x]] is open in the quotient topology and τΛ¯τ. Moreover, τΛ*τΛ¯τ.

Following Lemma 4.7, Lemma 4.9, Theorem 4.10 and Theorem 3.3 from [18] we obtain:

Theorem 4.11

τ=τΛ*=τΛ¯ in L/J. Moreover, if J is a filter, then (L/J,τΛ¯) is a topological residuated lattice.

We recall that for any non-empty set X of L, we denote by Xc = L \ X. In the same manner as Theorem 3.9 and Theorem 3.10, from [18] the following result can be proved:

Theorem 4.12

(L/J,τΛ¯) is Hausdorff iff J is closed and (L/J,τΛ¯) is discrete iff J is open.

Theorem 4.13

Let J be a filter of L. If J is closed relative to the uniform topology τΛ¯, then (L/J,τΛ¯) is a regular space.

Proof

Consider J a filter of L such that J is closed relative to the uniform topology τΛ¯, then (L/J,τΛ¯) is Hausdorf. Since every Hausdor and locally compact topological space is regular, we have to prove that (L/J,τΛ¯) is locally compact.

Let [x] ϵ L/J. Since L is a locally compact space by uniform topology induced by ⊼, there exists an open neighborhood O of x and a compact set K such that x ϵ O ⊆ K. Since π is open and continuous map, we have that π(0) is open and π(K) is compact such that [x] s π(0) ⊆ π(K). Thus L/J is locally compact at [x] and hence is locally compact.

5 Some Classes of Semitopological Residuated Lattices

By definition the class of BL-algebras is a subclass of divisible residuated lattices. In [11], O. Zahiri and R. A. Borzooei studied the (semi)topological algebras and for a BL-algebra L they proved that (see Theorem 3.17,[11]) (L, {∨, ∧, ⊙) is a semitopological BL-algebra and (L, {→}, τa) is a right semitopological BL-algebra, for an element a ϵ L. We prove that their result works in the case of divisible residuated lattices, which is a larger class than BL-algebras.

We notice the following rules of calculus:

Lemma 5.1

If L is a residuated lattice, then for a, p, q ∊ L and m ≥ 1 we have:

(ε1) (a**)m — (q ∨ p)** = [(a**)m — (q ∨ p)**]**;

(ε2) q — [(a**)m — (q ∨ p)**] = 1;

(ε3) ((a**)m — (p**)) — [(a**)m — (q ∨ p)**] = 1;

(ε4) [q ∧ ((a**)m — (p**))] — [((a**)m — (q**)) ∧ ((a**)m — (p**))] = 1;

(ε5)[q Θ ((a**)m ∧ (p**))] — [(q ⊙ p)**] = 1.

Proof

(ε1). Following (r35) we have am ⊖ (qp) = (a**)m → (qp)**, by (r3) we obtain am ⊖ (qp) = (am ⊖ (qp))** = [(a**)m — (qp)**]**. Hence (a**)m — (qp)** = [(a**)m → (qp)**]**.

The rule of calculus (ε1) holds if instead of (qp) we have (qp), (qp), (qp) or p.

(ε2). Since qqp ≤ (qp)** ≤ (a**)m → (qp)**, we obtain q → [(a**)m → (qp)**] = 1.

(ε3). Since p** ≤ (qp)** and by (r4), we obtain (a**)mp** ≤ (a**)m → (qp)**, that is ((a**)m → (p**)) → [(a**)m → (qp)**] = 1.

(ε4). Since qq** ≤ (a**)m → (q**), then [q ∧ ((a**)m → (p**))] ≤ [((a**)m → (q**)) ∧ ((a**)m → (p**))]. Hence [q ∧ ((a**)m → (p**))] → [((a**)m → (q**)) ∧ ((a**)m → (p**))] = 1.

(ε5). Since qq**, (a**)m ∧ (p**) ≤ (p**), then q ⊙ ((a**)m ∧ (p**)) ≤ (q**) ⊙ (p**). By (r18), (q**) ⊙ (p**) ≤ (qp)**, then [(q**) ⊙ (p**)] → (qp)** = 1. Hence [q⊙((a**)m ∧ (p**)] → [(qp)**] = 1.

Definition 5.2

Let τ be a topology on the divisible residuated lattice L. If (L, {*i}, τ), where {*i} ⊆ {∨, ∧, ⊙, →} is a (semi)topological algebra, then (L, {*i}, τ) is a (semi)topological divisible residuated lattice. For simplicity, if {*i} ⊆ {∨, ∧, ⊙, →}, we consider (L, τ) instead of (L, {∨, ∧, ⊙, →}, τ).

Theorem 5.3

Let L be a divisible residuated lattice and a ϵ L. Then:

(i) (L, {∨, ∧, ⊙}, τa) is a semitopological divisible residuated lattice;

(ii) (L, {→}, τa) is a right semitopological divisible residuated lattice.

Proof

(i). Consider q an arbitrary element of L.

(1). We prove that (L, {∨}, τa) is a semitopological divisible residuated lattice.

Consider the map φq : LL, defined by φq(i) = qx, for any xL. Following Proposition 3.24 the set βa = {Da (↑ x) | xL)} is abase for the topology τa on L. Then it suffices to prove that φq1(Da(x))τa, for any xL.

Let xL, then we obtain successively φq1(Da(x))={pL|φq(p)Da(x)}={pL|(a**)n(qp)**x,for some n}={pL|x(a**)n(qp)**,for somen}.

Consider the set A = {pL|x ≤ (a**)n → (qp)**, for some n ∊ ℕ} and we show that the set A is an upset of L. Let pA and ps, for some sL. Then there is nN such that x ≤ (a**)n → (qp)**. By (r4), (a**)n → (qp)** ≤ (a**)n → (q ∨ s)**, and so sA. Hence A is an upset. Now, we prove that Da(A) = A, then we deduce that Aτa, that is φq1(Da(x)τa, hence φq is a continuous map.

Let pDa (A). Then there is m ∊ ℕ such that (a**)m → (p**) ∊ A, that is x ≤ (a**)n → (q ∨ ((a**)m → (p**)))**, for some n ∊ ℕ.

We obtain successively

[(a**)n → (q ∨ ((a**)m — (p**)))**] →

[(a**)m+n(qp)**]=(r5)

[(a**)n - (q ∨ ((a**)m → (p**)))**] → [(a**)n((a**)m(qp)**)](r39) [q((a**)m(p**))]**[(a**)m(qp)**]=(ε1) [q((a**)m(p**))]**[(a**)m(qp)**]**=(r22) ([q((a**)m(p**))][(a**)m(qp)**])**=(r22) ([q((a**)m(p**))][(a**)m(qp)**])**=(r7)

([q → [(a**)m →(qp)**]]∧[((a**)m → (p**)) → [(a**)m(qp)**]])**=(ε2) (1[((a**)m(p**))[(a**)m(qp)**]])**=(ε3) (1 ∧ 1)** = 1** = 1.

By (r2), (a**)n → (q ∨ ((a**)m → (p**)))** ≤ (a**)m+n → (qp)**.

We deduce that x ≤ (a**)n → (q ∨ ((a**)m → (p**)))** ≤ (a**)m+n → (qp)**, hence x ≤ (a**)m+n → (qp)**, that is pA. Following Theorem 3.12 (ii), we deduce that A = Da (A), that is ϕq is a continuous map.

(2). We prove that (L, {∧}, τa) is a semitopological divisible residuated lattice.

Consider the map ϕq : LL defined by ϕq(x) = qx, for any xL. Following Proposition 3.24 the set βa = {Da(↑ x)|xL)} is abase for the topology τa on L. Then it suffices to prove that ϕq1(Da(x))τa, for any xL.

Let xL, then we obtain successively ϕq1(Da(x))={pL|ϕq(p)Da(x)}={pL|(a**)n(qp)**x, for some n ∈ ℕ} = {p ∈ L|x ≤ (a**)n → (q ∧ p)**, for some n ∈ ℕ}.

Consider the set B = {p ∈ L|x < (a** )n → (q ∧ p)**, for some n ∈ ℕ} and we show that the set B is an upset of L. Let p ∈ B and p ≤ s, for some s ∈ L. Then there is n ∈ ℕ such that x ≤ (a**)n → (q∧p)**. By (r4), (a**)n → (q∧ p)** < (a**)n → (q∧s)**, and so s ∈ B. Hence B is an upset. Now, we prove that Da(B) = B, then we deduce that B ∈ τa, that is ϕq1(Da(x)τa, ϕq is a continuous map.

Let pDa(B). Then there is m ∈ ℕ such that (a**)m → (p**) ∈ B, that is x ≤ (a**)n → (q ∧ ((a**)m →(p**)))**, for some n ∈ ℕ.

We obtain successively

[(a**)n(q((a**)m(p**)))**][(a**)m+n(qp)**]=(r5)

[(a**)n(q((a**)m(p**)))**][(a**)n((a**)m(qp)**)](r39)

[q((a**)m(p**))]**[(a**)m(qp)**]=(ε1)

[q((a**)m(p**))]**[(a**)m(qp)**]**=(r22)

([q((a**)m(p**))][(a**)m(qp)**])**=(r9)

([q((a**)m(p**))][((a**)m(q**))((a**)m(p**))])**=(ε4)1**=1.

By (r2),(a**)n → (q ∧ ((a**)m →(p**)))**≤(a**)m+n →(q ∧ p)**.

We deduce that x ≤ (a**)n → (q ∧ ((a**)m - (p**)))** < (a**)m+n - (q ∧p)**, hence x ≤ (a**)m+n →(q ∧ p)**, that is p ∈ B. Following Theorem 3.12 (ii), we deduce that B = Da(B), that is ϕq is a continuous map.

(3). We prove that (L, {⊙}, τa) is a semitopological divisible residuated lattice.

Consider the map ψq : L → L, defined by ψq(x) = q ⊙ x, for any x ∈ L. Following Proposition 3.24 the set βa = {Da(↑ x)| x ∈ L)} is abase for the topology τa on L. Then it suffices to prove that ψq1(Da(x))τa for any x s L.

Let x ∈ L, then we obtain successivelyψq1(Da(x))={pL|ψq(p)Da(x)}={pL|(a**)n(qp)**x, for some n ∈ ℕ} = {pL|x ≤ (a**)n → (qp)**  , for some n ∈ N}.

Consider the set C = {p ∈ L|x ≤ (a**)n → (q ⊙ p)**, for some n ∈ ℕ} and we show that the set C is an upset of L. Let p ∈ C andp ≤ s, for some s ∈ L. Then there is n ∈ N such that x ≤ (a**)n → (q ⊙ p)**. By (r4), (a**)n → (qp)** ≤ (a**)n → (qs)**, and so sC. Hence C is an upset. Now, we prove that Da(C) = C, then we deduce that Cτa, that is ψq1(Da(x)τa, hence ψpq is a continuous map.

Let pDa(C). Then there is m ∊ ℕ such that (a**)m → (p**) ∊ C, that is x ≤ (a**)n → (q ⊙ ((a**)m → (p**)))**, for some n ∊ ℕ.

We obtain successively

[(a**)n(q((a**)m(p**)))**][(a**)m+n(qp)**]=(r5)

[(a**)n - (q ∨ ((a**)m → (p**)))**] →

[(a**)n(q((a**)m(p**)))**][(a**)n((a**)m(qp)**)](r39)

[q((a**)m(p**))]**[(a**)m(qp)**]=(ε1)

[q((a**)m(p**))]**[(a**)m(qp)**]**=(r22)

([q((a**)m(p**))][(a**)m(qp)**])**=(r5)

([q(a**)m((a**)m(p**))][(qp)**])**=(i1) ([q((a**)m(p**))][(qp)**])**=(ε5)1**=1.

By (r2), (a**)n → (q ∨ ((a**)m → (p**)))** ≤ (a**)m+n → (qp)**.

We deduce that x ≤ (a**)n → (q ∨ ((a**)m → (p**)))** ≤ (a**)m+n → (qp)**, hence x ≤ (a**)m+n → (qp)**, that is pA. Following Theorem 3.12 (ii), we deduce that A = Da (A), that is ϕq is a continuous map.

(2). We prove that (L, {∧}, τa) is a semitopological divisible residuated lattice.

Consider the map ϕq : LL defined by ϕq(x) = qx, for any xL. Following Proposition 3.24 the set βa = {Da(↑ x)|xL)} is abase for the topology τa on L. Then it suffices to prove that ϕq1(Da(x))τa, for any xL.

Let xL, then we obtain successively ωq1(Da(x))={pL|ωq(p)Da(x)}={pL|(a**)n(qp)**x,for somen}={pL|x(a**)n(qp)**,for some n} for some n ∈ ℕ} = {p ∈ L|x ≤ (a**)n → (q ∧ p)**, for some n ∈ ℕ}.

Consider the set B = {p ∈ L|x < (a** )n → (q ∧ p)**, for some n ∈ ℕ} and we show that the set B is an upset of L. Let p ∈ B and p ≤ s, for some s ∈ L. Then there is n ∈ ℕ such that x ≤ (a**)n → (q∧p)**. By (r4), (a**)n → (q∧ p)** < (a**)n → (q∧s)**, and so s ∈ B. Hence B is an upset. Now, we prove that Da(B) = B, then we deduce that B ∈ τa, that is ϕq1(Da(x)τa, ϕq is a continuous map.

Let p ∈ Da(B). Then there is m ∈ ℕ such that (a**)m → (p**) ∈ B, that is x ≤ (a**)n → (q ∧ ((a**)m →(p**)))**, for some n ∈ ℕ.

We obtain successively

(a**)n[q((a**)m(p**))]**=(r22)(a**)n[(q**)((a**)m(p**))**]=(ε1)

(a**)n[(q**)((a**)m(p**))]=(r5)(a**)n[((q**)(a**)m)(p**)]=(r5)

(a**)n[(a**)m((q**)(p**))]=(r5)[(a**)n(a**)m][(q**)(p**)]=

(a**)n+m[(q**)(p**)]=(r22)(a**)n+m(qp)**.

We deduce that x ≤ (a**)n → (q ∧ ((a**)m - (p**)))** < (a**)m+n - (q ∧p)**, hence x ≤ (a**)m+n →(q ∧ p)**, that is p ∈ B. Following Theorem 3.12 (ii), we deduce that B = Da(B), that is ϕq is a continuous map.

(3). We prove that (L, {⊙}, τa) is a semitopological divisible residuated lattice.

Consider the map ψq : L → L, defined by ψq(x) = q ⊙ x, for any x ∈ L. Following Proposition 3.24 the set βa = {Da(↑ x)| x ∈ L)} is abase for the topology τa on L. Then it suffices to prove that ωq1(Da(x))τa for any x s L.

Let x ∈ L, then we obtain successively ωq1(Da(x))={pL|ωq(p)Da(x)}={pL|(a**)n(qp)**x,for somen}={pL|x(a**)n(qp)**,for some n}.

Consider the set D = {pL|x ≤ (a**)n → (qp)**, for some n ∊ ℕ} and we show that the set D is an upset of L. Let pD and ps, for some sL. Then there is m ∊ ℕ such that x ≤ (a**)n → (qp)**. By (r4), (a**)n → (qp)** ≤ (a**)n → (qs)**, and so sD. Hence D is an upset. Now, we prove that Da(D) = D, then we deduce that DTa, that is ωq1(Da(x)τa, hence ωq is a continuous map.

Let pDa(D). Then there is m ∊ ℕ such that (a**)m → (p**) ∊ D, that is x ≤ (a**)n → (q → ((a**)m → (p**)))**, for some n ∊ ℕ.

We obtain successively

(a**)n[q((a**)m(p**))]**=(r22)(a**)n[(q**)((a**)m(p**))**]=(ε1)

(a**)n[(q**)((a**)m(p**))]=(r5)(a**)n[((q**)(a**)m)(p**)]=(r5)

(a**)n[(a**)m((q**)(p**))]=(r5)[(a**)n(a**)m][(q**)(p**)]=

(a**)n+m[(q**)(p**)]=(r22)(a**)n+m(qp)**.

We deduce that x ≤ (a**)n → (q → ((a**)m → (p**)))** = (a**)m+n → (qp)**, hence x ≤ (a**)m+n → (qp)**, that is pD. Following Theorem 3.12 (ii), we deduce that D = Da(D), that is ωq is a continuous map.

Since → is not commutative we deduce that (L, {→}, τa) is a right semitopological divisible residuated Lattice.

We recall ([12]) that a residuated Lattice L with double-negation property (x** = x, for all xL) is called involutive. It is known that the classes of divisible residuated Lattices and involutive residuated Lattices are different (see [12]). The following result is an easy consequence of Lemma 2.2, Lemma 5.1 and Theorem 5.3.

Corollary 5.4

Let L be an involutive residuated Lattice and a ∊ L. Then:

(i) (L, {∨, ∧, ⊖}, τa) is a semitopological involutive residuated Lattice;

(ii) (L, {→}, τa) is a right semitopological involutive residuated Lattice.

In [3] it was proved that L is an MV-algebra iff L is an involutive BL-algebra. Therefore, following Theorem 5.3 and Corollary 5.4 we deduce that:

Corollary 5.5

Let L be an MV-algebra and a ∊ L. Then:

(i) (L, {∨, ∧, ⊙}, τa) is a semitopological MV-algebra;

(ii) (L, {→}, τa) is a right semitopological MV-algebra.

Corollary 5.6

Let L be a divisible residuated lattice. For any filter F of L and a, xL, then:

(i) Da(↑ x)/F = Da/F(↑ x/F);

(ii) The topologies τa/F and τa on L/J are the same, where τa is the quotient topology of L/F.

Proof

(i). By Theorem 4.3 (i) we get Da(↑ x)/FDa/F(↑ x/F), then it remains to prove Da/F(↑ x/F) ⊆ Da(↑ x)/F. Let u/FDa/F(↑ x/F). Then there exists n ∊ ℕ such that x → [(a**)nu**] ∊ F. Let f = x → [(a**)nu**]. Then by (r5) we obtain x → ((a**)n → [fu**]) = f → [x → ((a**)nu**)] = ff = 1 and so by (r15) and (r22) we get 1 = x → ((a**)n → [fu**]) ≤ x → ((a**)n → [fu**]**) = x → ((a**)n → [f** → u**]) = x → ((a**)n → [fu]**). Hence fuDa(↑ x). Since fF, then f/F = 1/F and so (fu)/F = f/Fu/F = 1/Fu/F = (1 → u)/F = u/F. That is (fu) ≡ F u. Therefore, u/FDa(↑ x)/F.

(ii). By Proposition 3.24 the set {Da/F (↑ x/F): x/FL/F} is a base for the topology τa/f on L/F. Moreover, the set {Da (↑ x)/F : xL} is abase for the topology τa. Now, the proof of (ii) is straightforward by (i).

Corollary 5.7

Let L be an involutive residuated lattice. For any filter F of L and a, x ∊ L, then:

(i) Da(↑ x)/F = Da/F(↑ x/F);

(ii) The topologies τa/f and τa on L/F are the same, where τa is the quotient topology of L/F.

Proof

(i). By Theorem 4.3 (i) we get Da(↑ x)/FDa/F(↑ x/F), then it remains to prove Da/F(↑ x/F) ⊆ Da(↑ x)/F. Let u/FDa/F(↑ x/F). Then there exists n ∊ ℕ such that x → [(a**)nu**] ∊ F. Let f = x → [(a**)nu**]. Then by (r5) we obtain x → ((a**)n → [fu**]) = f → [x → ((a**)nu**)] = ff = 1 and so by the involutive property (that is, x = x**, for all xL) we get 1 = x → ((a**)n → [fu**]) = x → ((a**)n → [fu]**). Hence fuDa(↑ x). Since fF, then f/F = 1/F and so (fu)/F = f/Fu/F = 1/Fu/F = (1 → u)/F = u/F. That is (fu) ≡F u. Therefore, u/FDa(↑ x)/F.

(ii). By Proposition 3.24 the set {Da/F (↑ x/F): x/FL/F} is a base for the topology τa/f on L/F. Moreover, the set {Da(↑ x)/F : xL} is abase for the topology τa. Now, the proof of (ii) is straightforward by (i).

6 Conclusions

In Borzooei and Zahiri (2014)[11] the definition of double complemented elements for any ilter F in BL-algebras, initially, introduced by A. Borumand Saeid and S. Motamed (2009)[10], was generalized to the concept of Dy(F), for any upset F of the BL-algebra L. Their aim was to show that any BL-algebra L with that topologyis a semitopological BL-algebra, so in that way they could construct many semitopological BL-algebras. Our goal is to extend the study in the case of residuated Lattices. Therefore, we work on a special type of topology induced by a modal and closure operator denoted by Da(X), for an upset X of a residuated Lattice L, where a is an element of L.

We discuss briefly the properties and applications of the operator Da(·) in residuated Lattices. We obtain some of the important topological aspects of these structures such as connectivity and compactness. We study some properties of quotient topologies on residuated Lattices by considering two types of quotient topologies denoted by τa/f and τa. We study the uniform topology τΛ¯ and we obtain important characterizations as (L/J,τΛ¯) becomes a Hausdorff space iff J is closed relative to the uniform topology. Also, we study some properties of the direct product of residuated Lattices.

Finally, we apply our results on classes of residuated Lattices such as divisible residuated Lattices, MV-algebras and involutive residuated Lattices and we find that any of this subclasses of residuated Lattices with respect to these topologies form semitopological algebras.

In this way we can construct many semitopological algebras on residuated Lattices.

Acknowledgement

The author is very grateful to the anonymous referees for their useful remarks and suggestions.

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About the article

Received: 2017-03-08

Revised: 2018-08-16

Published Online: 2018-10-19


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1104–1127, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0092.

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© 2018 Holdon, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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