Implicative Pseudo Valuations on Hoops

(School of Mathematics,Northwest University,Xi’an,710127,P.R.China)

§1. Introduction

It is well known that non-classical logic has become a formal and useful tool for computer science to deal with uncertain information and fuzzy information.Various logical algebras have been proposed as the semantic systems of non-classical logic systems.Among these logical algebras,hoops[1]are very basic and important algebraic structures.In the last few years,the theory of hoops has been enriched with deep structure theorems[2-6].Many of these results have a strong impact with fuzzy logics.In particular,from the structure theorem of finite basic hoops,one obtains an elegant short proof of the completeness theorem for propositional basic logic,which introduced by Hájek[7].It was proved that a hoop has the divisibility condition and it is a meet-semilattice,so a bounded R‘-monoid can be viewed as a bounded hoop together with the join-semilattice property.In other words,a hoop is a meet-semilattice ordered residuated,integral and divisible monoid[8].As a more general structure,hoops contains all algebraic structures that induce by continuous t-norms and their residue[9].Therefore,hoops play an important role in studying fuzzy logics and the related algebraic structures.

Bu¸sneag[10]de fined a pseudo valuation on a Hilbert algebra,and proved that every pseudo valuation induces a pseudo metric on a Hilbert algebra.Also,Bu¸sneag[11]provided several theorems on extensions of pseudo valuations.Bu¸sneag[12]introduced the notions of pseudo valuations on residuated lattices,and proved some theorems of extension for pseudo valuation on residuated structures.Using the Bu¸sneags model,Jun introduce the notion of pseudo valuations on BCK-algebras and its related structures[13-17],and induce a pseudo metric by using a pseudo valuation on BCK-algebras.Based on the notion of pseudo valuation,they show that the binary operation in BCK-algebras is uniformly continuous.After then,Zhan[18]introduce the notion of pseudo valuations on R0-algebras and obtain some important results.

As we have mentioned in the above paragraph,pseudo valuations has been widely studied on R0-algebras,residuated lattices and BCK-algebras,etc.All the above mentioned algebraic structures except for BCK-algebras are the special case of hoops.In fact,hoops are the widest possible residuated structure.Therefore,it is interesting to study the pseudo valuations on hoops for treating a variant of the concept of pseudo valuations within the framework of universal algebras and provide a more general algebraic foundation for pseudo valuations theory on residuated structure.This is the motivation for us to investigate implicative pseudo valuations on hoops.

Based on the above considerations,using the Bu¸sneags model,we extend the notion of pseudo valuations on hoops and proved some theorems of extension for implicative pseudo valuation on this widest residuated structures.The main of this paper is to investigate implicative pseudo valuations on hoops.After then,the relationship between a pseudo valuation and an implicative pseudo valuation is provided.In particular,we show that a pseudo valuation on regular hoops is implicative if and only if it satis fiesThis result will provide a more general algebraic foundation for pseudo valuations theory on algebraic structures based on substructure logic.

§2. Preliminaries

Deni fition 2.1[1]An algebraof type(2,2,0)is called a hoop if it satis fies the following conditions:for any x,y,z∈H,

(H1)is a commutative monoid,

(H2)

(H3)

(H4)

In the sequel we will also refer to the hoopby its universe H,unless otherwise stated.

On the hoop H we de fine x≤y i ffx→y=1 and”≤”is a partial order on H.A hoop is bounded if there is an element 0∈H such that 0≤x for all x∈H.For any bounded hoopfor n≥1,x0=x→ 0.One can see that not every hoop has a lattice.In fact,hoops are exactly the join free reducts of BL-algebras.Moreover,regular hoop can be considered as a bounded hoop with the axiom x00=x.For more on hoops the reader can refer to[2-6]and their references.

Proposition 2.2[2−5]In a hoop H the following properties hold:for any x,y,z∈ H,

(a1)(H,≤)is a∧-semilattice with

(a2)x≤y→x,

(a3)x→1=1,

(a4)1→x=x,

(a5)

(a6)

(a7)

(a8)

(a9)

(a10)

(a11)x→y≤(z→x)→(z→y),

(a12)x→y≤(y→z)→(x→z),

(a13)

Proposition 2.3[3−4]Let H be a bounded hoop.Then we have,for any x,y,z∈ H,

(a14)00=1,10=0,

(a15)x≤y⇒y0≤x0,

(a16)

(a17)If H is a regular hoop,then x→y=y0→x0.

Proposition 2.4[3−4,8]In a hoop H,we de finefor any x,y∈H.Then the following conditions are equivalent:

(1)is an associative operation on H,

(2)x≤y impliesfor any x,y,z∈H,

(3),for any x,y,z∈H,

(4)is the supremum operation on H.

De finition 2.5[8]A hoop is called aif it satis fies one of the equivalent conditions of Proposition 2.4.

Proposition 2.6[8]Let H be aThen for any x,y,z∈H,

(1)

(2)

A non-empty set F of H is called a filter of H if it satis fies:(1)x,y∈F implies(2)x∈F,y∈H and x≤y implies y∈F.It is proved that a non-empty set F of H is a filter if and only if it satis fies x,x→y∈F implies y∈F,for any x,y∈H.Moreover,a non-empty set F of H is called an implicative filter of H if it satis fies x→(y→z)∈F,for any x,y,z∈H[9,19,20].

De finition 2.7[21]Let H be a bounded hoop.A Bosbach state on H is a function s:L−→[0,1]such that the following conditions hold:

(1)s(0)=0,s(1)=1,

(2)s(x)+s(x→y)=s(y)+s(y→x).

§3.Pseudo Valuations on Hoops

De finition 3.1[22]A real-valued function ϕ :H → R,where R is the set of all real numbers,is called a pseudo valuation on H if it satis fies the following conditions,for all x,y∈H,

(pv1)ϕ(1)=0,

(pv2)ϕ(y)≤ϕ(x→ y)+ϕ(x).

A pseudo valuation ϕ is called a valuation on H if it satis fies the following condition:(pv3)∀x ∈ H,ϕ(x)=0 implies x=1.

Example 3.2Let H={0,a,b,c,1},the order of the elements in H is as the following Hasse diagram:

The operations fland→are de fined as following,respectively.

Then(H, fl,→,0,1)is a hoop.De fine two real-valued functions ϕ1and ϕ2on H by

Then ϕ1is a pseudo valuation on H while ϕ2is not a pseudo valuation on H since 2=

The following example shows that there is a connection between a pseudo valuation and the state.

Example 3.3Let s be a Bosbach state on H and de fine ϕ(x)=1−s(x)for all x ∈ H.Then ϕ is a pseudo valuation on H.Indeed,ϕ(y)=1−s(y)=1−(s(x)+s(x → y)−s(y→x))=1−s(x)−s(x→y)+s(y→x)=(1−s(x))+(1−s(x→y))−(1−s(y→x)6(1−s(x))+(1−s(x → y))= ϕ(x)+ϕ(x → y).It is clear that ϕ(1)=0.Hence ϕ is a pseudo valuation.

Proposition 3.4[22]Let ϕ be a pseudo valuation on H.Then the following properties hold,for any x,y,z∈H,

(1)ϕ is order reversing,

(2)ϕ(x)≥0,

(3)ϕ(x→y)≤ ϕ(y).

Theorem 3.5Let ϕ be a pseudo valuation on H.The we have,for all x,y,z ∈ H,

ProofLet ϕ be a pseudo valuation on H,then by(pv2),we have ϕ(z)≤ ϕ(y)+ϕ(y → z),and ϕ(y→ z)≤ ϕ(x→ (y→ z))+ϕ(x).If x→ (y→ z)=1,then ϕ(y→ z)≤ ϕ(1)+ϕ(x)=ϕ(x).So,ϕ(z)≤ ϕ(y)+ϕ(y→ z)≤ ϕ(y)+ϕ(x).Hence,x→ (y→ z)=1⇒ ϕ(z)≤ϕ(x)+ϕ(y).

In the following,conditions for a real-valued function to be a pseudo valuation on hoops are given.

Theorem 3.6Let ϕ :H → R be a real-valued function.Then the following conditions are equivalent:

(1)ϕ is pseudo valuation on H,

(2)ϕ satis fies the conditions(pv1)and(pv4),

(3)ϕ satis fies the conditions(pv1)and(pv5),where(pv5)isfor any x,y,z∈H.

ProofIf ϕ is a pseudo valuation on H,then from De finition 3.1 and Theorem 3.5,we know(pv1)and(pv4)holds.

(2)Note that x→(y→z)=1 i ffx≤y→z i ffThen by(pv4),we have

(3)Since(x→y)→(x→y)=1,thus x→y≤x→y,and then byUsing(pv5),ϕ(y)≤ ϕ(x → y)+ϕ(x).This proves that(pv2)holds.Hence ϕ is pseudo valuation on H.

Proposition 3.7Let ϕ be a pseudo valuation on H.Then we have,for all x,y,z ∈ H,

(pv6)

(pv7)

(pv8)

(pv9)

Proof(pv6)Puttingin(pv5),we have

(pv7)By(a10)and Proposition 3.4(1),we haveUsing(pv6),

(pv8)By(a2)and(a12),we have 1=y→(x→y)≤((x→y)→z)→(y→z),that is,((x→y)→z)→(y→z)=1,and so(x→y)→z≤y→z.Thus ϕ(y→z)≤ϕ((x→y)→z).Since y→z≤x→(y→z),we have ϕ(x→(y→z))≤ϕ(y→z)≤ϕ((x→y)→z).Hence,(pv8)holds.

(pv9)Sinceby(a13),soIt follows from(pv6)thatThus(pv9)holds.

§4. Implicative Pseudo Valuations on Hoops

De finition 4.1A real-valued function ϕ on H is called an implicative pseudo valuation if it satis fies(pv1)and(pv10):ϕ(x→z)≤ϕ(x→(y→z))+ϕ(x→y),for any x,y,z∈H.

Example 4.2Let H={0,a,b,c,d,1},where 0≤a≤b≤c≤d≤1.De fineand→as follows:

Thenis a hoop.De fine a real-valued function ϕ on H by

Then ϕ is an implicative pseudo valuation on H.

Proposition 4.3Every implicative pseudo valuation on H is a pseudo valuation on H.

ProofPutting x=1 in(pv10),we obtain ϕ(1→ z)≤ϕ(1→ (y→z))+ϕ(1→y),that is,ϕ(z)≤ ϕ(y→ z)+ϕ(y),thus(pv2)holds.Hence,every implicative pseudo valuation is a pseudo valuation on H.

The following example shows that the converse of Proposition 4.3 may not be true in general.

Example 4.4Let H={0,a,b,c,1},where 0≤a≤b≤c≤1.De fineand→as follows:

Thenis a hoop.De fine a real-valued function ϕ on H by

Then ϕ is a pseudo valuation on H while it is not an implicative pseudo valuation on H,since

Theorem 4.5Let H be a regular hoop and ϕ be a pseudo valuation on H.Then the following conditions are equivalent:

(1)ϕ is an implicative pseudo valuation on H,

(2)(pv11):ϕ(x→z)≤ϕ(x→(z0→y))+ϕ(y→z)holds,for all x,y,z∈H.

proofAssume that ϕ is an implicative pseudo valuation on H.For any x,y,z∈ H,we have ϕ(x→ z)= ϕ(z0→ x0)≤ ϕ(z0→ (y0→ x0))+ϕ(z0→ y0)= ϕ(z0→ (x→y))+ϕ(y→z)=ϕ(x→(z0→y))+ϕ(y→z).Thus,(pv11)holds.

(2)Assume that ϕ is a pseudo valuation on H and satis fies(pv11).Then ϕ(x → z)=ϕ(z0→x0)≤ϕ(z0→ (x00→y0))+ϕ(y0→x0)=ϕ(z0→(x→y0))+ϕ(x→y)=ϕ(x→(z0→y0))+ϕ(x→y)=ϕ(x→(y→z))+ϕ(x→y),that is,ϕ(x→z)≤ϕ(x→(y→z))+ϕ(x→y).Thus,(pv10)holds,and so ϕ is an implicative pseudo valuation on H.

Theorem 4.6Let H be a regular hoop and ϕ be a pseudo valuation on H.Then the following conditions are equivalent:

(1)ϕ is an implicative pseudo valuation on H,

(2)ϕ(x→z)≤ϕ(x→(z0→z)),for any x,z∈H,

(3)ϕ(x→z)≤ϕ(y→(x→(z0→z)))+ϕ(y),for any x,y,z∈H.

ProofAssume that ϕ is an implicative pseudo valuation on H.Putting y=z in(pv11),we have ϕ(x→ z)≤ ϕ(x→ (z0→ z))+ϕ(z→ z)= ϕ(x→ (z0→ z))+ϕ(1)=ϕ(x→(z0→z)).That is,ϕ(x→z)≤ϕ(x→(z0→z)).

(2)For any x,y,z∈ H,we have ϕ(x→ (z0→ z))≤ ϕ(y→ (x→ (z0→ z)))+ϕ(y).Using(2),we obtain ϕ(x→ z)≤ ϕ(x→ (z0→ z))≤ ϕ(y→ (x→ (z0→ z)))+ϕ(y).Hence,(3)holds.

(3)Let ϕ be a pseudo valuation on H and satis fies the condition(3).Then by(pv9),we haveBy(H4),we havePutting y=1 in(3),we havethat is,ϕ(x→z)≤ϕ(x→(z0→y))+ϕ(y→z).Thus,(pv11)holds.It follows from Theorem 4.5 that ϕ is an implicative pseudo valuation on H.

Theorem 4.7Let H be a regular hoop and ϕ be a pseudo valuation on H.Then the following conditions are equivalent:

(1)ϕ is an implicative pseudo valuation on H,

(2)ϕ(x)≤ϕ(x0→x),for any x∈H,

(3)ϕ(x)≤ϕ((x→y)→x),for any x,y∈H,

(4)ϕ(x)≤ϕ(z→((x→y)→x))+ϕ(z),for any x,y,z∈H.

ProofFrom Theorem 4.6(2),we have ϕ(x)= ϕ(1→ x)≤ ϕ(1→ (x0→ x))=ϕ(x0→ x),that is,ϕ(x)≤ ϕ(x0→ x).

(2)Since x0≤x→y by(a6),we have(x→y)→x≤x0→x from(a5).Since ϕ is a pseudo valuation on H,we have ϕ(x0→ x)≤ ϕ((x → y)→ x).Thus,from(2),we deduce that ϕ(x)≤ ϕ(x0→ x)≤ ϕ((x→ y)→x).Hence(3)holds.

(3)Since ϕ is a pseudo valuation on H,we have ϕ(x)≤ ϕ((x → y)→ x)≤ ϕ(z→ ((x →y)→x))+ϕ(z).Thus(4)holds.

(4)Since z≤x→z by(a2),we have(x→z)0≤z0and z0→(x→z)≤(x→z)0→(x→z).Thus,ϕ((x→z)0→(x→z))≤ϕ(z0→(x→z)).It follows from(4)that ϕ(x→ z)≤ ϕ(1→ (((x→ z)→ 0)→ (x→ z)))+ϕ(1)= ϕ((x→ z)0→(x→z))≤ϕ(z0→(x→ z))= ϕ(x→ (z0→ z)),that is,ϕ(x→ z)≤ ϕ(x→ (z0→ z)).Thus,from Theorem 4.6(2),ϕ is an implicative pseudo valuation on H.

Theorem 4.8Let H be a regular hoop and ϕ be a pseudo valuation on H.Then the following conditions are equivalent:

(1)ϕ is an implicative pseudo valuation on H,

(2)for all x∈H.

ProofAssume that ϕ is an implicative pseudo valuation on H.Since x0→(((x0→x)→x)→(x0→x)0)=((x0→x)→x)→(x0→(x0→x)0)=((x0→x)→x)→((x0→x)→x)=1,and x0→((x0→x)→x)=(x0→x)→(x0→x)=1,we have ϕ((x0→x)→x)=ϕ(x0→(x0→x)0)≤ϕ(x0→(((x0→x)→x)→(x0→x)0))+ϕ(x0→ ((x0→ x)→ x))= ϕ(1)+ϕ(1)=0,that is,ϕ((x0→ x)→ x)=0 by Proposition 3.4(2).Similarly,we can obtain ϕ((x → x0)→ x0)=0.Thus,by(pv7),we haveand so

(2)Since ϕ be a pseudo valuation on H.Thenthat is,ϕ(x→ y)= ϕ(x→ (y0→ y)).Thus,from Theorem 4.6(2),ϕ is an implicative pseudo valuation on H.

Theorem 4.9Let ϕ be an implicative pseudo valuation on H.Then the set F=kerϕ =is an implicative filter of H.

proofSince ϕ(1)=0,we have 1∈F.Let x→(y→z)∈F,x→y∈F,then ϕ(x→ (y→ z))=0,ϕ(x→ y)=0.Then by(pv10),we have ϕ(x→ z)≤ ϕ(x→ (y→z))+ϕ(x→y)=0,and so ϕ(x→z)=0,that is,x→z∈F.Hence,F is an implicative filter of H.

The following example shows that the converse of Theorem 4.9 may not be true.

Example 4.10Consider a hoop H as in Example 4.2.De fine a real-valued function ϕ on H by

Then,F=kerϕ ={x ∈ H|ϕ(x)=0}={c,d,1}is an implicative filter of H while ϕ is not an implicative pseudo valuation on H since 2=ϕ(0)=ϕ(1→0)?ϕ(1→(a→0))+ϕ(1→a)=ϕ(d)+ϕ(a)=1.

§5. Conclusion

In this paper,motivated by the previous research of pseudo valuations on R0-algebras,we extended the concept of implicative pseudo valuations to hoops.Also,we give a characterizations of implicative pseudo valuations on hoops and discuss the relationship between kinds of pseudo valuations on hoops.This result will provide a more general algebraic foundation for pseudo valuations theory on algebraic structures based on substructure logic.Based on these results,we will consider some its applications to knowledge based information systems in the future.

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