Using Buşneag's model, the notion of pseudovaluations (valuations) on a WFI algebra is introduced, and a pseudometric is induced by a pseudovaluation on WFI algebras. Given a valuation with additional condition, we show that the binary operation in WFI algebras is uniformly continuous.
1. Introduction
In 1990, Wu [1] introduced the notion of fuzzy implication algebras (FI algebra, for short) and investigated several properties. In [2], Li and Zheng introduced the notion of distributive (regular, and commutative, resp.) FI algebras and investigated the relations between such FI algebras and MV algebras. In [3], Jun discussed several aspects of WFI algebras. He introduced the notion of associative (normal and medial, resp.) WFI algebras and investigated several properties. He gave conditions for a WFI algebra to be associative/medial, provided characterizations of associative/medial WFI algebras, and showed that every associative WFI algebra is a group in which every element is an involution. He also verified that the class of all medial WFI algebras is a variety. Jun et al. [4] introduced the concept of ideals of WFI algebras, and gave relations between a filter and an ideal. Moreover, they provided characterizations of an ideal, and established an extension property for an ideal. Buşneag [5] defined pseudovaluation on a Hilbert algebra and proved that every pseudovaluation induces a pseudometric on a Hilbert algebra. Also, Buşneag [6] provided several theorems on extensions of pseudovaluations. Buşneag [7] introduced the notions of pseudovaluations (valuations) on residuated lattices, and proved some theorems of extension for these (using the model of Hilbert algebras ([6])).
In this paper, using Buşneag’s model, we introduce the notion of pseudovaluations (valuations) on WFI algebras, and we induce a pseudometric by using a pseudovaluation on WFI algebras. Given a valuation with additional condition, we show that the binary operation in WFI algebras is uniformly continuous.
2. Preliminaries
Let K(τ) be the class of all algebras of type τ=(2,0). By a WFI algebra, we mean an algebra (X;⊖,θ)∈K(τ) in which the following axioms hold:
(∀x∈X)(x⊖x=θ),
(∀x,y∈X)(x⊖y=y⊖x=θ⇒x=y),
(∀x,y,z∈X)(x⊖(y⊖z)=y⊖(x⊖z)),
(∀x,y,z∈X)((x⊖y)⊖((y⊖z)⊖(x⊖z))=θ).
For the convenience of notation, we will write [x,y1,y2,…,yn] for(⋯((x⊖y1)⊖y2)⊖⋯)⊖yn.
We define [x,y]0=x, and for n>0,[x,y]n=[x,y,y,…,y], where y occurs n-times.
Proposition 2.1 (see [3]).
In a WFI algebra X, the following are true:
x⊖[x,y]2=θ,
θ⊖x=θ⇒x=θ,
θ⊖x=x,
x⊖y=θ⇒(y⊖z)⊖(x⊖z)=θ,(z⊖x)⊖(z⊖y)=θ,
(x⊖y)⊖θ=(x⊖θ)⊖(y⊖θ),
[x,y]3=x⊖y.
We define a relation “⪯’’ on X by x⪯y if and only if x⊖y=θ. It is easy to verify that a WFI algebra is a partially ordered set with respect to ⪯. A nonempty subset S of a WFI algebra X is called a subalgebra of X if x⊖y∈S whenever x,y∈S. A nonempty subset F of a WFI algebra X is called a filter of X if it satisfies:
θ∈F,
(∀x∈F)(∀y∈X)(x⊖y∈F⇒y∈F).
A filter F of a WFI algebra X is said to be closed (see [3]) if F is also a subalgebra of X. A nonempty subset I of a WFI algebra X is called an ideal of X (see [4]) if it satisfies the condition (c1) and
(∀x,y∈X)(∀z∈I)((x⊖z)⊖y∈I⇒x⊖y∈I).
Proposition 2.2 (see [3]).
Let F be a filter of a WFI algebra X. Then F is closed if and only if x⊖θ∈F for all x∈F.
Proposition 2.3 (see [3]).
In a finite WFI algebra, every filter is closed.
Note that every ideal of a WFI algebra is a closed filter (see [4, Theorem 4.3]). For a WFI algebra X, the set
S(X):={x∈X∣x⪯θ}
is called the simulative part of X.
3. WFI Algebras with Pseudovaluations
In what follows, let X denote a WFI algebra unless otherwise specified.
Definition 3.1.
A mapping f:X→ℝ is called a pesudovaluation on X if it satisfies the following two conditions:
f(θ)=0,
(∀x,y∈X)(f(x⊖y)+f(x)≥f(y)).
A pseudovaluation f on X satisfying the following condition:
(∀x∈X)(x≠θ⟹f(x)≠0)
is called a valuation on X.
Obviously, a mapping
f:X⟶R,x⟼0
is a pseudovaluation on X, which is called the trivial pseudovaluation.
Example 3.2.
Let f:X→ℝ be a mapping defined by
f(x)={0ifx=θ,kifx∈X∖{θ},
where k is a positive real number. Then, f is a pseudovaluation on X. Moreover, it is a valuation on X.
Example 3.3.
Let ℤ be the set of integers. Then, (ℤ;⊖,θ) is a WFI algebra, where θ=0 and x⊖y=y-x for all x,y∈ℤ (see [8]). Let f:ℤ→ℝ be a mapping defined by
f(x)={0ifx=θ,ax+botherwise,
for all x∈ℤ, where a and b are real numbers with a≠0 and b≥0. Then, f is a pseudovaluation on ℤ.
Example 3.4.
Let X={θ,a,b} be a set with the following Cayley table:
Then, (X;⊖,θ) is a WFI algebra (see [3]). Define a mapping f:X→ℝ by f(θ)=0,f(a)=2 and f(b)=9. Then, f is a pseudovaluation on X. Also, it is a valuation on X.
Proposition 3.5.
Every pseudovaluation f on X satisfies the following conditions:
(∀x,y∈X)(x⪯y⇒f(x)≥f(y)),
(∀x,y,z∈X)(f(x⊖z)≤f(x⊖y)+f(y⊖z)),
(∀x,y∈X)(f(x⊖y)+f(y⊖x)≥0).
Proof.
(1) Let x,y∈X be such that x⪯y. Then, x⊖y=θ, and so
f(y)≤f(x⊖y)+f(x)=f(θ)+f(x)=0+f(x)=f(x).
(2) Using (a4), we have x⊖y⪯(y⊖z)⊖(x⊖z) for all x,y,z∈X. It follows from (1) and Definition 3.1(ii) that
f(x⊖y)≥f((y⊖z)⊖(x⊖z))≥f(x⊖z)-f(y⊖z),
so that f(x⊖z)≤f(x⊖y)+f(y⊖z) for all x,y,z∈X.
(3) Let x,y∈X. Using Definition 3.1(ii), we have f(x⊖y)+f(x)≥f(y) and f(y⊖x)+f(y)≥f(x); that is, f(x⊖y)≥f(y)-f(x) and f(y⊖x)≥f(x)-f(y). It follows that f(x⊖y)+f(y⊖x)≥0.
Corollary 3.6.
Let f:X→ℝ be a pseudovaluation on X. Then, f(x)≥0 for all x∈𝒮(X).
Proof.
Since x⪯θ for all x∈𝒮(X), we have f(x)≥f(θ)=0 by Proposition 3.5(1) and Definition 3.1(i).
The following example shows that the converse of Corollary 3.6 may not be true.
Example 3.7.
Let X be a WFI algebra which is considered in Example 3.4. Let g:X→ℝ be a mapping defined by
g=(θab0-32).
Then, 𝒮(X)={θ,b},g(θ)=0 and g(b)=2≥0. But g is not a pseudovaluation on X, since
g(a⊖θ)+g(a)=g(θ)+g(a)=-3≱0=g(θ).
Let f:X→ℝ be a pseudovaluation on X. If x1⊖x=θ, that is, x1⪯x, for all x,x1∈X, then f(x)≤f(x1) by Proposition 3.5(1). If x2⊖(x1⊖x)=θ for all x,x1,x2∈X, then x2⪯x1⊖x, and so, f(x2)≥f(x1⊖x)≥f(x)-f(x1) by Proposition 3.5(1) and Definition 3.1(ii). Hence, f(x)≤f(x1)+f(x2). Now, if x3⊖(x2⊖(x1⊖x))=θ for all x,x1,x2,x3∈X, then x3⪯x2⊖(x1⊖x). It follows from Proposition 3.5(1) and Definition 3.1(ii) thatf(x3)≥f(x2⊖(x1⊖x))≥f(x1⊖x)-f(x2)≥f(x)-f(x1)-f(x2),
so that f(x)≤f(x1)+f(x2)+f(x3). Continuing this process, we have the following proposition.
Proposition 3.8.
Let f:X→ℝ be a pseudovaluation on X. For any elements x,x1,x2,…,xn of X, if xn⊖(⋯⊖(x2⊖(x1⊖x))⋯)=θ, then f(x)≤∑k=1nf(xk).
Theorem 3.9.
Let F be a filter of X, and let fF:X→ℝ be a mapping defined by
fF(x)={0ifx∈F,kifx∉F,
where k is a positive real number. Then, fF is a pseudovaluation on X. In particular, fF is a valuation on X if and only if F={θ}.
Proof.
Straightforward.
We say fF is a pseudovaluation induced by a filter F.
Theorem 3.10.
If a mapping f:X→ℝ is a pseudovaluation on X, then the set
Ff:={x∈X∣f(x)≤0}
is a filter of X.
Proof.
Obviously, θ∈Ff. Let x,y∈X be such that x∈Ff and x⊖y∈Ff. Then, f(x)≤0 and f(x⊖y)≤0. It follows from Definition 3.1(ii) that f(y)≤f(x⊖y)+f(x)≤0 so that y∈Ff. Hence, Ff is a filter of X.
We say Ff is a filter induced by a pseudovaluation f on X.
Corollary 3.11.
If a mapping f:X→ℝ is a pseudovaluation on a finite WFI algebra X, then the set
Ff:={x∈X∣f(x)≤0}
is a closed filter of X.
Proof.
It follows from Proposition 2.3 and Theorem 3.10.
Remark 3.12.
A filter induced by a pseudovaluation on X may not be closed. Indeed, in Example 3.3, if we take a=1 and b=0, then f:ℤ→ℝ,x↦x, is a pseudovaluation on ℤ. Then, Ff={θ}∪{k∈ℤ∣k<θ} which is a filter but not a subalgebra of ℤ, since (-3)⊖(-1)=-1-(-3)=2∉Ff. Hence, Ff is not a closed filter of ℤ.
Theorem 3.13.
For any pseudovaluation f:X→ℝ, if F is a filter of X, then FfF=F.
Proof.
We have FfF={x∈X∣fF(x)≤0}={x∈X∣x∈F}=F.
The following example shows that the converse of Theorem 3.10 may not be true; that is, there exist a WFI algebra X and a mapping f:X→ℝ such that
f is not a pseudovaluation on X,
Ff:={x∈X∣f(x)≤0} is a filter of X.
Example 3.14.
Let X={θ,1,2,a,b} be a set with the following Cayley table:
Then (X;⊖,θ) is a WFI algebra. Let f:X→ℝ be a mapping defined by
f=(θ12ab0-43-25).
Then, Ff={θ,1,a} is a filter of X. But f is not a pseudovaluation on X, since
f(a⊖b)+f(a)=1≱5=f(b).
Definition 3.15.
A pseudovaluation (or, valuation) f on X is said to be positive if f(x)≥0 for all x∈X.
The pseudovaluation f on X which is given in Example 3.4 is positive.
Theorem 3.16.
If a pseudovaluation f on X is positive, then the set
Ff=:={x∈X∣f(x)=0}
is a filter of X.
Proof.
Clearly, θ∈Ff=. Let x,y∈X be such that x∈Ff= and x⊖y∈Ff=. Then, f(x)=0 and f(x⊖y)=0. Since f is positive, it follows from Definition 3.1(ii) that
0≤f(y)≤f(x⊖y)+f(x)=0,
so that f(y)=0, that is, y∈Ff=. Hence, Ff= is a filter of X.
The following example shows that two distinct pseudovaluations induce the same filter.
Example 3.17.
Consider a WFI algebra X={θ,1,2,a,b}which is given in Example 3.14. Let g and h be mappings from X to ℝ defined by
g=(θ12ab00435),h=(θ12ab00423).
Then, g and h are pseudovaluations on X, and Fg={θ,1}=Fh.
For a mapping f:X→ℝ, define a mapping df:X×X→ℝ by df(x,y)=f(x⊖y)+f(y⊖x) for all (x,y)∈X×X. Note that df(x,y)≥0 for all (x,y)∈X×X.
Theorem 3.18.
If f:X→ℝ is a pseudovaluation on X, then df is a pseudometric on X, and so (X,df) is a pseudometric space.
We say df is called the pseudometric induced by pseudovaluation f.
Proof.
Let x,y,z∈X. Then, df(x,y)=f(x⊖y)+f(y⊖x)≥0 by Proposition 3.5(3), and obviously, df(x,y)=df(y,x) and df(x,x)=0. Now,
df(x,y)+df(y,z)=[f(x⊖y)+f(y⊖x)]+[f(y⊖z)+f(z⊖y)]=[f(x⊖y)+f(y⊖z)]+[f(z⊖y)+f(y⊖x)]≥f(x⊖z)+f(z⊖x)=df(x,z).
Therefore, (X,df) is a pseudometric space.
Proposition 3.19.
Every pseudometric df induced by pseudovaluation f satisfies the following inequalities:
df(x,y)≥df(x⊖a,y⊖a),
df(x,y)≥df(a⊖x,a⊖y),
df(x⊖y,a⊖b)≤df(x⊖y,a⊖y)+df(a⊖y,a⊖b),
for all x,y,a,b∈X.
Proof.
(1) Let x,y,a∈X. Since (x⊖y)⊖((y⊖a)⊖(x⊖a))=θ and (y⊖x)⊖((x⊖a)⊖(y⊖a))=θ, it follows from Proposition 3.5(1) that f(x⊖y)≥f((y⊖a)⊖(x⊖a)) and f(y⊖x)≥f((x⊖a)⊖(y⊖a)) so that
df(x,y)=f(x⊖y)+f(y⊖x)≥f((y⊖a)⊖(x⊖a))+f((x⊖a)⊖(y⊖a))=df(x⊖a,y⊖a).
(2) It is similar to the proof of (1).
(3) Using Proposition 3.5(2), we have
f((x⊖y)⊖(a⊖b))≤f((x⊖y)⊖(a⊖y))+f((a⊖y)⊖(a⊖b)),f((a⊖b)⊖(x⊖y))≤f((a⊖b)⊖(a⊖y))+f((a⊖y)⊖(x⊖y)),
for all x,y,a,b∈X. Hence,
df(x⊖y,a⊖b)=f((x⊖y)⊖(a⊖b))+f((a⊖b)⊖(x⊖y))≤[f((x⊖y)⊖(a⊖y))+f((a⊖y)⊖(a⊖b))]+[f((a⊖b)⊖(a⊖y))+f((a⊖y)⊖(x⊖y))]=[f((x⊖y)⊖(a⊖y))+f((a⊖y)⊖(x⊖y))]+[f((a⊖b)⊖(a⊖y))+f((a⊖y)⊖(a⊖b))]=df(x⊖y,a⊖y)+df(a⊖y,a⊖b)
for all x,y,a,b∈X.
Theorem 3.20.
Let f:X→ℝ be a pseudovaluation on X such that Ff={x∈X∣f(x)≤0} is a closed filter of X. If df is a metric on X, then f is a valuation on X.
Proof.
Suppose that f is not a valuation on X. Then, there exists x∈X such that x≠θ and f(x)=0. Thusθ,x∈Ff and so x⊖θ∈Ff, since Ff is a closed filter of X. It follows that f(x⊖θ)≤0 so that
0=f(θ)≤f(x⊖θ)+f(x)=f(x⊖θ)≤0.
Hence, f(x⊖θ)=0, and thus df(x,θ)=f(x⊖θ)+f(θ⊖x)=f(x⊖θ)+f(x)=0. Thus, x=θ since df is a metric on X. This is a contradiction. Therefore, f is a valuation on X.
Consider the pseudovaluation f on ℤ which is described in Example 3.3. If a=-1, thenf(x)={0ifx=θ,-x+botherwise,
for all x∈ℤ, and Ff={x∈ℤ∣b≤x}∪{θ} which is not a closed filter of ℤ. Since f is a pseudovaluation on ℤ, we know that (ℤ,df) is a pseudometric space by Theorem 3.18. If x≠y in ℤ, thendf(x,y)=f(x⊖y)+f(y⊖x)=f(y-x)+f(x-y)=-y+x+b-x+y+b=2b≠0.
Hence, (ℤ,df) is a metric space. But f(b)=0, and so, f is not a valuation on ℤ. This shows that Theorem 3.20 may not be true when Ff is not a closed filter of X.
Theorem 3.21.
For a mapping f:X→ℝ, if df is a pseudometric on X, then (X×X,df*) is a pseudometric space, where
df*((x,y),(a,b))=max{df(x,a),df(y,b)}
for all (x,y),(a,b)∈X×X.
Proof.
Suppose df is a pseudometric on X. For any (x,y),(a,b)∈X×X, we have
df*((x,y),(x,y))=max{df(x,x),df(y,y)}=0,df*((x,y),(a,b))=max{df(x,a),df(y,b)}=max{df(a,x),df(b,y)}=df*((a,b),(x,y)).
Now, let (x,y),(a,b),(u,v)∈X×X. Then,
df*((x,y),(u,v))+df*((u,v),(a,b))=max{df(x,u),df(y,v)}+max{df(u,a),df(v,b)}≥max{df(x,u)+df(u,a),df(y,v)+df(v,b)}≥max{df(x,a),df(y,b)}=df*((x,y),(a,b)).
Therefore, (X×X,df*) is a pseudometric space.
Corollary 3.22.
If f:X→ℝ is a pseudovaluation on X, then (X×X,df*) is a pseudometric space.
It is natural to ask that if f:X→ℝ is a valuation on X, then is (X,df) a metric space. But, we see that it is incorrect in the following example.
Example 3.23.
For a WFI algebra (ℤ;⊖,θ), a mapping f:ℤ→ℝ defined by f(x)=(1/2)x for all x∈ℤ is a valuation on ℤ. Then, df is a pseudometric on ℤ. Note that df(-2,3)=f(-2⊖3)+f(3⊖(-2))=0, but -2≠3. Hence, (X,df) is not a metric space.
Theorem 3.24.
If f:X→ℝ is a positive valuation on X, then (X,df) is a metric space.
Proof.
Suppose that f is a positive valuation on X. Then, (X,df) is a pseudometric space by Theorem 3.18. Let x,y∈X be such that df(x,y)=0. Then, 0=df(x,y)=f(x⊖y)+f(y⊖x), and so f(x⊖y)=0 and f(y⊖x)=0, since f is positive. Also, since f is a valuation on X, it follows that x⊖y=θ and y⊖x=θ so from (a2) that x=y. Therefore, (X,df) is a metric space.
Corollary 3.25.
If f:X→ℝ is a valuation on X such that Ff={θ}, then (X,df) is a metric space.
Theorem 3.26.
If f:X→ℝ is a positive valuation on X, then (X×X,df*) is a metric space.
Proof.
Note from Corollary 3.22 that (X×X,df*) is a pseudometric space. Let (x,y),(a,b)∈X×X be such that df*((x,y),(a,b))=0. Then,
0=df*((x,y),(a,b))=max{df(x,a),df(y,b)},
and so df(x,a)=0=df(y,b), since df(x,y)≥0 for all (x,y)∈X×X. Hence,
0=df(x,a)=f(x⊖a)+f(a⊖x),0=df(y,b)=f(y⊖b)+f(b⊖y).
Since f is positive, it follows that f(x⊖a)=0=f(a⊖x) and f(y⊖b)=0=f(b⊖y) so that x⊖a=θ=a⊖x and y⊖b=θ=b⊖y. Using (a2), we have a=x and b=y, and so (x,y)=(a,b). Therefore, (X×X,df*) is a metric space.
Theorem 3.27.
If f is a positive valuation on X, then the operation ⊖:X×X→X is uniformly continuous. (Suppose that (X,d) and (Y,ρ) are metric spaces and f:X→Y. We say that f is uniformly continuous provided that for every ɛ>0, there exists δ>0 such that for any points x1 and x2 in X, if d(x1,x2)<δ, then ρ(f(x1),f(x2))<ɛ.)
Proof.
For any ɛ>0, if df*((x,y),(a,b))<ɛ/2, then df(x,a)<ɛ/2, and df(y,b)<ɛ/2. Using Proposition 3.19, we have
df(x⊖y,a⊖b)≤df(x⊖y,a⊖y)+df(a⊖y,a⊖b)≤df(x,a)+df(y,b)<ɛ2+ɛ2=ɛ.
Therefore, the operation ⊖:X×X→X is uniformly continuous.
Corollary 3.28.
If f is a valuation on X such that Ff={θ}, then the operation ⊖:X×X→X is uniformly continuous.
WuW. M.Fuzzy implication algebras1990415663LiZ.ZhengC.Relations between fuzzy implication algebra and MV algebra2001912012051822328ZBL0981.06008JunY. B.Weak fuzzy implication algebras20037141521981604ZBL1036.03054JunY. B.ParkC. H.RohE. H.Characterizations of filters and ideals on WFI-algebras20062844714842284868BuşneagD.Hilbert algebras with valuations20032631124BuşneagD.On extensions of pseudo-valuations on Hilbert algebras20032631–3112410.1016/S0012-365X(02)00552-61955711ZBL1018.03050BuşneagC.Valuations on residuated lattices20073421282517868ZBL1174.03353JunY. B.Weak and concrete filters of WFI-algebras200826925932