Pseudovaluations on WFI Algebras

Abstract

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:(a1)(∀𝑥∈𝑋)(𝑥⊖𝑥=𝜃),(a2)(∀𝑥,𝑦∈𝑋)(𝑥⊖𝑦=𝑦⊖𝑥=𝜃⇒𝑥=𝑦),(a3)(∀𝑥,𝑦,𝑧∈𝑋)(𝑥⊖(𝑦⊖𝑧)=𝑦⊖(𝑥⊖𝑧)),(a4)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥⊖𝑦)⊖((𝑦⊖𝑧)⊖(𝑥⊖𝑧))=𝜃).

For the convenience of notation, we will write [𝑥,𝑦₁,𝑦₂,…,𝑦_𝑛] for⋯𝑥⊖𝑦₁⊖𝑦₂⊖⋯⊖𝑦_𝑛. (2.1) 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:(b1)𝑥⊖[𝑥,𝑦]²=𝜃,(b2)𝜃⊖𝑥=𝜃⇒𝑥=𝜃,(b3)𝜃⊖𝑥=𝑥,(b4)𝑥⊖𝑦=𝜃⇒(𝑦⊖𝑧)⊖(𝑥⊖𝑧)=𝜃,(𝑧⊖𝑥)⊖(z⊖𝑦)=𝜃,(b5)(𝑥⊖𝑦)⊖𝜃=(𝑥⊖𝜃)⊖(𝑦⊖𝜃),(b6)[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:(c1)𝜃∈𝐹,(c2)(∀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(c3)(∀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} (2.2) 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:(i)𝑓(𝜃)=0,(ii)(∀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 (3.1) is called a valuation on 𝑋.

Obviously, a mapping 𝑓∶𝑋⟶ℝ,𝑥⟼0 (3.2) 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)={𝜃},0if𝑥=𝜃,𝑘if𝑥∈𝑋⧵ (3.3) 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, (3.4) 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: (3.5) 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:(1)(∀𝑥,𝑦∈𝑋)(𝑥⪯𝑦⇒𝑓(𝑥)≥𝑓(𝑦)),(2)(∀𝑥,𝑦,𝑧∈𝑋)(𝑓(𝑥⊖𝑧)≤𝑓(𝑥⊖𝑦)+𝑓(𝑦⊖𝑧)),(3)(∀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 𝑓𝑦≤𝑓𝑥⊖𝑦+𝑓(𝑥)=𝑓(𝜃)+𝑓(𝑥)=0+𝑓(𝑥)=𝑓(𝑥). (3.6)
(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 𝑓𝑥⊖𝑦≥𝑓𝑦⊖𝑧⊖(𝑥⊖𝑧)≥𝑓(𝑥⊖𝑧)−𝑓𝑦⊖𝑧, (3.7) 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. (3.8) 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). (3.9)

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) that𝑓𝑥₃≥𝑓𝑥₂⊖𝑥₁⊖𝑥≥𝑓𝑥₁⊖𝑥−𝑓𝑥₂≥𝑓(𝑥)−𝑓𝑥₁−𝑓𝑥₂, (3.10) 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,…,𝑥_𝑛 of 𝑋, if 𝑥_𝑛⊖(⋯⊖(𝑥₂⊖(𝑥₁⊖𝑥))⋯)=𝜃, then 𝑓(𝑥)≤∑^𝑛_𝑘=1𝑓(𝑥_𝑘).

Theorem 3.9. Let 𝐹 be a filter of 𝑋, and let 𝑓_𝐹∶𝑋→ℝ be a mapping defined by 𝑓_𝐹(𝑥)=0if𝑥∈𝐹,𝑘if𝑥∉𝐹, (3.11) where 𝑘 is a positive real number. Then, 𝑓_𝐹 is a pseudovaluation on 𝑋. In particular, 𝑓_𝐹 is a valuation on 𝑋 if and only if 𝐹={𝜃}.

Proof. Straightforward.

We say 𝑓_𝐹 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 𝐹_𝑓∶=𝑥∈𝑋∣𝑓(𝑥)≤0 (3.12) is a filter of 𝑋.

Proof. Obviously, 𝜃∈𝐹_𝑓. Let 𝑥,𝑦∈𝑋 be such that 𝑥∈𝐹_𝑓 and 𝑥⊖𝑦∈𝐹_𝑓. 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∈𝐹_𝑓. Hence, 𝐹_𝑓 is a filter of 𝑋.

We say 𝐹_𝑓 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 𝐹_𝑓∶=𝑥∈𝑋∣𝑓(𝑥)≤0 (3.13) 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, F_𝑓={𝜃}∪{𝑘∈ℤ∣𝑘<𝜃} which is a filter but not a subalgebra of ℤ, since (−3)⊖(−1)=−1−(−3)=2∉𝐹_𝑓. Hence, 𝐹_𝑓 is not a closed filter of ℤ.

Theorem 3.13. For any pseudovaluation 𝑓∶𝑋→ℝ, if 𝐹 is a filter of 𝑋, then 𝐹_𝑓𝐹=𝐹.

Proof. We have 𝐹_𝑓𝐹={𝑥∈𝑋∣𝑓_𝐹(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(1)𝑓 is not a pseudovaluation on 𝑋,(2)𝐹_𝑓∶={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: (3.14) Then (𝑋;⊖,𝜃) is a WFI algebra. Let 𝑓∶𝑋→ℝ be a mapping defined by 𝑓=𝜃12𝑎𝑏0−43−25. (3.15) Then, 𝐹_𝑓={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). (3.16)

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 𝐹⁼_𝑓∶=𝑥∈𝑋∣𝑓(𝑥)=0 (3.17) is a filter of 𝑋.

Proof. Clearly, 𝜃∈𝐹⁼_𝑓. Let 𝑥,𝑦∈𝑋 be such that 𝑥∈𝐹⁼_𝑓 and 𝑥⊖𝑦∈𝐹⁼_𝑓. Then, 𝑓(𝑥)=0 and 𝑓(𝑥⊖𝑦)=0. Since 𝑓 is positive, it follows from Definition 3.1(ii) that 0≤𝑓𝑦≤𝑓𝑥⊖𝑦+𝑓(𝑥)=0, (3.18) so that 𝑓(𝑦)=0, that is, 𝑦∈𝐹⁼_𝑓. Hence, 𝐹⁼_𝑓 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=𝜃12𝑎𝑏00435ℎ=𝜃12𝑎𝑏00423 (3.19) Then, 𝑔 and ℎ are pseudovaluations on 𝑋, and 𝐹_𝑔={𝜃,1}=𝐹_ℎ.

For a mapping 𝑓∶𝑋→ℝ, define a mapping 𝑑_𝑓∶𝑋×𝑋→ℝ by 𝑑_𝑓(𝑥,𝑦)=𝑓(𝑥⊖𝑦)+𝑓(𝑦⊖𝑥) for all (𝑥,𝑦)∈𝑋×𝑋. Note that 𝑑_𝑓(𝑥,𝑦)≥0 for all (𝑥,𝑦)∈𝑋×𝑋.

Theorem 3.18. If 𝑓∶𝑋→ℝ is a pseudovaluation on 𝑋, then 𝑑_𝑓 is a pseudometric on 𝑋, and so (𝑋,𝑑_𝑓) is a pseudometric space.

We say 𝑑_𝑓 is called the pseudometric induced by pseudovaluation 𝑓.

Proof. Let 𝑥,𝑦,𝑧∈𝑋. Then, 𝑑_𝑓(𝑥,𝑦)=𝑓(𝑥⊖𝑦)+𝑓(𝑦⊖𝑥)≥0 by Proposition 3.5(3), and obviously, 𝑑_𝑓(𝑥,𝑦)=𝑑_𝑓(𝑦,𝑥) and 𝑑_𝑓(𝑥,𝑥)=0. Now, 𝑑_𝑓𝑥,𝑦+𝑑_𝑓=𝑓+𝑓=𝑓+𝑓()()𝑦,𝑧𝑥⊖𝑦+𝑓𝑦⊖𝑥𝑦⊖𝑧+𝑓𝑧⊖𝑦𝑥⊖𝑦+𝑓𝑦⊖𝑧𝑧⊖𝑦+𝑓𝑦⊖𝑥≥𝑓𝑥⊖𝑧+𝑓𝑧⊖𝑥=𝑑_𝑓(𝑥,𝑧). (3.20) Therefore, (𝑋,𝑑_𝑓) is a pseudometric space.

Proposition 3.19. Every pseudometric 𝑑_𝑓 induced by pseudovaluation 𝑓 satisfies the following inequalities:(1)𝑑_𝑓(𝑥,𝑦)≥𝑑_𝑓(𝑥⊖𝑎,𝑦⊖𝑎), (2)𝑑_𝑓(𝑥,𝑦)≥𝑑_𝑓(𝑎⊖𝑥,𝑎⊖𝑦), (3)𝑑_𝑓(𝑥⊖𝑦,𝑎⊖𝑏)≤𝑑_𝑓(𝑥⊖𝑦,𝑎⊖𝑦)+𝑑_𝑓(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 𝑑_𝑓⊖()()⊖𝑥,𝑦=𝑓𝑥⊖𝑦+𝑓𝑦⊖𝑥≥𝑓𝑦⊖𝑎𝑥⊖𝑎+𝑓𝑥⊖𝑎𝑦⊖𝑎=𝑑_𝑓𝑥⊖𝑎,𝑦⊖𝑎. (3.21)
(2) It is similar to the proof of (1).
(3) Using Proposition 3.5(2), we have 𝑓⊖()⊖⊖(),𝑓)⊖)⊖⊖,𝑥⊖𝑦𝑎⊖𝑏≤𝑓𝑥⊖𝑦𝑎⊖𝑦+𝑓𝑎⊖𝑦𝑎⊖𝑏(𝑎⊖𝑏𝑥⊖𝑦≤𝑓(𝑎⊖𝑏𝑎⊖𝑦+𝑓𝑎⊖𝑦𝑥⊖𝑦 (3.22) for all 𝑥,𝑦,𝑎,𝑏∈𝑋. Hence, 𝑑_𝑓⊖()()⊖≤𝑓⊖⊖()+𝑓()⊖⊖=𝑓⊖⊖+𝑓()⊖⊖()𝑥⊖𝑦,𝑎⊖𝑏=𝑓𝑥⊖𝑦𝑎⊖𝑏+𝑓𝑎⊖𝑏𝑥⊖𝑦𝑥⊖𝑦𝑎⊖𝑦+𝑓𝑎⊖𝑦𝑎⊖𝑏𝑎⊖𝑏𝑎⊖𝑦+𝑓𝑎⊖𝑦𝑥⊖𝑦𝑥⊖𝑦𝑎⊖𝑦+𝑓𝑎⊖𝑦𝑥⊖𝑦𝑎⊖𝑏𝑎⊖𝑦+𝑓𝑎⊖𝑦𝑎⊖𝑏=𝑑_𝑓𝑥⊖𝑦,𝑎⊖𝑦+𝑑_𝑓𝑎⊖𝑦,𝑎⊖𝑏 (3.23) for all 𝑥,𝑦,𝑎,𝑏∈𝑋.

Theorem 3.20. Let 𝑓∶𝑋→ℝ be a pseudovaluation on 𝑋 such that 𝐹_𝑓={𝑥∈𝑋∣𝑓(𝑥)≤0} is a closed filter of 𝑋. If 𝑑_𝑓 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∈𝐹_𝑓 and so 𝑥⊖𝜃∈𝐹_𝑓, since 𝐹_𝑓 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. (3.24) Hence, 𝑓(𝑥⊖𝜃)=0, and thus 𝑑_𝑓(𝑥,𝜃)=𝑓(𝑥⊖𝜃)+𝑓(𝜃⊖𝑥)=𝑓(𝑥⊖𝜃)+𝑓(𝑥)=0. Thus, 𝑥=𝜃 since 𝑑_𝑓 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, (3.25) for all 𝑥∈ℤ, and 𝐹_𝑓={𝑥∈ℤ∣𝑏≤𝑥}∪{𝜃} which is not a closed filter of ℤ. Since 𝑓 is a pseudovaluation on ℤ, we know that (ℤ,𝑑_𝑓) is a pseudometric space by Theorem 3.18. If 𝑥≠𝑦 in ℤ, then𝑑_𝑓𝑥,𝑦=𝑓𝑥⊖𝑦+𝑓𝑦⊖𝑥=𝑓𝑦−𝑥+𝑓𝑥−𝑦=−𝑦+𝑥+𝑏−𝑥+𝑦+𝑏=2𝑏≠0. (3.26) Hence, (ℤ,𝑑_𝑓) 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 𝐹_𝑓 is not a closed filter of 𝑋.

Theorem 3.21. For a mapping 𝑓∶𝑋→ℝ, if 𝑑_𝑓 is a pseudometric on 𝑋, then (𝑋×𝑋,𝑑^∗_𝑓) is a pseudometric space, where 𝑑^∗_𝑓𝑥,𝑦,(𝑎,𝑏)=max𝑑_𝑓(𝑥,𝑎),𝑑_𝑓𝑦,𝑏 (3.27) for all (𝑥,𝑦),(𝑎,𝑏)∈𝑋×𝑋.

Proof. Suppose 𝑑_𝑓 is a pseudometric on 𝑋. For any (𝑥,𝑦),(𝑎,𝑏)∈𝑋×𝑋, we have 𝑑^∗_𝑓𝑥,𝑦,𝑥,𝑦=max𝑑_𝑓(𝑥,𝑥),𝑑_𝑓𝑑𝑦,𝑦=0,_∗𝑓𝑥,𝑦,(𝑎,𝑏)=max𝑑_𝑓(𝑥,𝑎),𝑑_𝑓𝑑𝑦,𝑏=max_𝑓(𝑎,𝑥),𝑑_𝑓𝑏,𝑦=𝑑^∗_𝑓(𝑎,𝑏),𝑥,𝑦. (3.28) Now, let (𝑥,𝑦),(𝑎,𝑏),(𝑢,𝑣)∈𝑋×𝑋. Then, 𝑑^∗_𝑓𝑥,𝑦,(𝑢,𝑣)+𝑑^∗_𝑓((𝑢,𝑣),(𝑎,𝑏))=max𝑑_𝑓(𝑥,𝑢),𝑑_𝑓𝑦,𝑣+max𝑑_𝑓(𝑢,𝑎),𝑑_𝑓(𝑑𝑣,𝑏)≥max_𝑓(𝑥,𝑢)+𝑑_𝑓(𝑢,𝑎),𝑑_𝑓𝑦,𝑣+𝑑_𝑓(𝑑𝑣,𝑏)≥max_𝑓(𝑥,𝑎),𝑑_𝑓𝑦,𝑏=𝑑^∗_𝑓𝑥,𝑦,(𝑎,𝑏). (3.29) Therefore, (𝑋×𝑋,𝑑^∗_𝑓) is a pseudometric space.

Corollary 3.22. If 𝑓∶𝑋→ℝ is a pseudovaluation on 𝑋, then (𝑋×𝑋,𝑑^∗_𝑓) is a pseudometric space.

It is natural to ask that if 𝑓∶𝑋→ℝ is a valuation on 𝑋, then is (𝑋,𝑑_𝑓) 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, 𝑑_𝑓 is a pseudometric on ℤ. Note that 𝑑_𝑓(−2,3)=𝑓(−2⊖3)+𝑓(3⊖(−2))=0, but −2≠3. Hence, (𝑋,𝑑_𝑓) is not a metric space.

Theorem 3.24. If 𝑓∶𝑋→ℝ is a positive valuation on 𝑋, then (𝑋,𝑑_𝑓) is a metric space.

Proof. Suppose that 𝑓 is a positive valuation on 𝑋. Then, (𝑋,𝑑_𝑓) is a pseudometric space by Theorem 3.18. Let 𝑥,𝑦∈𝑋 be such that 𝑑_𝑓(𝑥,𝑦)=0. Then, 0=𝑑_𝑓(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,𝑑_𝑓) is a metric space.

Corollary 3.25. If 𝑓∶𝑋→ℝ is a valuation on 𝑋 such that 𝐹_𝑓={𝜃}, then (𝑋,𝑑_𝑓) is a metric space.

Theorem 3.26. If 𝑓∶𝑋→ℝ is a positive valuation on 𝑋, then (𝑋×𝑋,𝑑^∗_𝑓) is a metric space.

Proof. Note from Corollary 3.22 that (𝑋×𝑋,𝑑^∗_𝑓) is a pseudometric space. Let (𝑥,𝑦),(𝑎,𝑏)∈𝑋×𝑋 be such that 𝑑^∗_𝑓((𝑥,𝑦),(𝑎,𝑏))=0. Then, 0=𝑑^∗_𝑓𝑥,𝑦,(𝑎,𝑏)=max𝑑_𝑓(𝑥,𝑎),𝑑_𝑓𝑦,𝑏, (3.30) and so 𝑑_𝑓(𝑥,𝑎)=0=𝑑_𝑓(𝑦,𝑏), since 𝑑_𝑓(𝑥,𝑦)≥0 for all (𝑥,𝑦)∈𝑋×𝑋. Hence, 0=𝑑_𝑓(𝑥,𝑎)=𝑓(𝑥⊖𝑎)+𝑓(𝑎⊖𝑥),0=𝑑_𝑓𝑦,𝑏=𝑓𝑦⊖𝑏+𝑓𝑏⊖𝑦. (3.31) 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,𝑑^∗_𝑓) 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 𝜀>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))<𝜀.)

Proof. For any 𝜀>0, if 𝑑^∗_𝑓((𝑥,𝑦),(𝑎,𝑏))<𝜀/2, then 𝑑_𝑓(𝑥,𝑎)<𝜀/2, and 𝑑_𝑓(𝑦,𝑏)<𝜀/2. Using Proposition 3.19, we have 𝑑_𝑓𝑥⊖𝑦,𝑎⊖𝑏≤𝑑_𝑓𝑥⊖𝑦,𝑎⊖𝑦+𝑑_𝑓𝑎⊖𝑦,𝑎⊖𝑏≤𝑑_𝑓(𝑥,𝑎)+𝑑_𝑓<𝜀𝑦,𝑏2+𝜀2=𝜀. (3.32) Therefore, the operation ⊖∶𝑋×𝑋→𝑋 is uniformly continuous.

Corollary 3.28. If 𝑓 is a valuation on 𝑋 such that 𝐹_𝑓={𝜃}, then the operation ⊖∶𝑋×𝑋→𝑋 is uniformly continuous.