Fuzzy Logic and Zadeh Algebra

In this work we create a connection between AFS (Axiomatic Fuzzy Sets) fuzzy logic systems and Zadeh algebra. Beginning with simple concepts we construct fuzzy logic concepts. Simple concepts can be interpreted semantically. The membership functions of fuzzy concepts form chains which satisfy Zadeh algebra axioms. These chains are based on important relationship condition (1) represented in the introduction where the binary relation m R of a simple concept m is defined more general in Definition 2.10. Then every chain of membership functions forms a Zadeh algebra. It demands a lot of preliminaries before we obtain this desired result.


Introduction
Starting with simple concepts such as "young people" or "tall people" it is possible to form AFS logic system ( ) ( , , , EM ′ ∨ ∧ .The elements are fuzzy concepts constructed by simple concepts.Notice that ( ) , , EM ∨ ∧ is a comple- tely distributive lattice and is called the EI (expanding one set M) algebra over M .So the AFS logic system is a completely distributive lattice equipped with the logical negation ' .Let X be a non-empty set.For where ζ is a fuzzy concept "old".The exact definition is represented in Definition 2.10.In Section 2 all the results are known and can be found from [1].
Also the used examples are there.For Zadeh algebra axioms we refer to [2].In Section 3 it is proved new results.But all the preliminaries represented in Section 2 are necessary to know for understanding these results and their proofs.The crucial condition is for simple concepts m and for all the pairs ( )

( ) ( )
, , In the conclusion it is illustrated the research motivation and contribution of this paper.
Let X be a subset of a poset P .Denote the least upper bound of X by l.u.b. i.e. sup X and the greatest lower bound of X by g.l.b. i.e. inf X .Definition 2.2.A lattice L is a poset P where any two of whose elements x and y have g.l.b. or a meet denoted by x y ∧ , and l.u.b. or a join denoted by x y ∨ .A lattice L is complete if each of its subsets has l.u.b. and g.l.b. in L .
It is clear that any nonvoid complete lattice contains a least element 0 and a greatest element 1.
In any lattice L (or a poset), the operations ∧ and ∨ satisfy the following laws, whenever the expressions are refered to exist: Conversely, any system L with the two binary operations satisfying (L1) - (L4) is a lattice.) ( ) ( ) In fact, these conditions are equivalent if they are valid.Definition 2.4.[1], pages 77, 116 or [3], page 119 Let L be a complete lattice.Then L is called a completely distributive lattice if it satisfies the extended distributive laws: for any family where I and i J are non-empty indexing sets, the following equations are valid

A Survey to Simple Concepts and Their Operations
In this subsection we approach to simple concepts and their operations because it is necessary to form the idea what do simple concepts mean.The exact definition will be represented in Definition 2.13.All these are based on [1], pages 113,114.
Consider the set of four people 1 2 3 4 , , , x x x x and a simple concept "hair colour".By intuition, we may set: 1 x has "hair black" with number 6 and , , x x x with numbers 4,6,3.So, the numbers imply the order 4 2 3 1 x x x x > > = which can be interpreted as follows: Moving from right to left, the relationship states how strongly the hair colour resembles black colour.More exactly, i j x x > means that the hair of i x is closer to the black colour than the colour of the hair which j x has.
Let M be a set of fuzzy or Boolean concepts on the set X .For each m M ∈ we associate to a single feature.For example 1 m : "old people" is a fuzzy concept but 2 m : "male" is a Boolean concept.In fact, M is a set of simple concepts.In general let A M Example 2.5.Let 1 m : "old people", 2 m : "male", 3 m : "tall people".Then m m : "old males" and 1 3 m m + : "old or tall people".Further, m m m is always less than or equal to the degree of x belonging to the fuzzy concept represented by 1 2 m m or 2 3 m m .Therefore the former 1 2 3 m m m is including in both of the latter ones 1 2 m m or 2 3 m m .

AFS Fuzzy Logic System
All the definitions and the propositions with their proofs are represented in [1], pages 115-123.For a moment we give up the assumption that M consists only of simple concepts.Let M be a non-empty set.The set EM  is defined by | , , is any non-empty indexing set where the elements of EM  are expressed semantically with "equivalent to", "or" (disjunction) and "and" (conjunction).
Definition 2.6.Let M be a non-empty set.A binary relation R on EM  is defined as follows: for R is an equivalence relation and we define EM as the quotient set EM R  .Proposition 2.7.Let M be a non-empty set.Then ( ) , , EM ∨ ∧ forms a completely distributive lattice under the binary compositions ∨ and ∧ defined as follows: for any ( ) ( ) where the disjoint union I J  means that every element in I and every element in J are always regarded as different elements in The proof of the proposition can be found from [1].
To be a distributive lattice means that for any , , , , EM ∨ ∧ is such a lattice it guarantees the existance of the EM elements ( ) . We can also define the order in ( ) , , EM ∨ ∧ as follows: , , , such that .
Further, as a (distributive) completely lattice ( ) The lattice ( ) , , EM ∨ ∧ is called the EI (expanding one set M ) algebra over M .Proposition 2.8.Let M be a set and : g M M → be a map satisfying ( ) ( ) Then for any , EM α β ∈ , g has the following properties: (1) ( ) ( ) , ( ) Therefore the operator g is an order reversing involution in the EI algebra ( ) The operator g defines the negation m′ of the concept m : ( ) is called an AFS fuzzy logic system.
The AFS fuzzy logic system ( ) , , , EM ' ∨ ∧ can be regarded as a completely distributive lattice.It is also a complete lattice.But this lattice is equipped with the logical negation.
We conclude that the complexity of human concepts is a direct result of the combinations of a few relatively simple concepts.In fact, some suitable simple concepts play the same role as used in linear vector spaces and we can regard them as a "basis".

Relations, Simple and Complex Concepts
For this subsection we refer to [1] Example 2.12.Let fuzzy concept ζ : "hair black" and define R ζ in the corresponding way as above.By human intuition, we assume that for the three persons , 1, 2, 3 the degree of ζ is the following: but the fourth person 4 x has no hairs.Then ( ) ( ) ( ) , , , , , , Definition 2.13.Let X be a set and R be a binary relation on X .R is called a sub-preference relation on X if for , , , x y z X x y ∈ ≠ , R satisfies the following conditions: (1) if ( ) , , x x x where R ζ is the binary relation defined in Example 2.11.Observe that the latter of the assumptions of (2) in Definition 2.13 is not valid and so the condition (2) is valid.In general it is known that all elements belonging to a simple concept at some degree are comparable and are arranged in a linear order, that is, they form a chain.In above we can think shortly that 1 2 3 x x x > > .
Further, there exists a pair of different elements belonging to a complex concept at some degree such that their degrees in this complex concept are incomparable.
Example 2.15.The set X consists of disjoint sets Y : "males" and Z : "females".The concept ζ : "beautiful" is simple on Y and on Z : However, if we apply ζ to the whole set X it is a complex concept because the degrees of the elements x X ∈ and y X ∈ may be incomparable: In this case ( ) . The condition (4) in 2.13 not satisfied and so ζ is a complex concept.

The AFS Fuzzy Logic and Coherence Membership Functions
For introduction to characteristic and membership functions we refer to [4], page 255, and [5], pages 12-18.Definitions 2.16 and 2.17 can be found from [1], pages 128, 130.We first become acquainted with concepts fuzzy sets and membership functions.
Let X ≠ ∅ and x X ∈ , A X ⊂ .Define a characteristic function for the set A as follows: Consider an extended case ( ) < is also possible.We call for the set A X ⊂ (a) a crisp set, if its characteristic function is Therefore for every element x X ∈ there is a membership degree The set of pairs determines completely the fuzzy set A .The characteristic function of a crisp set A is a special case of a membership function Definition 2.16.[1] Let X , M be sets and 2 M be the power set of M .
Let : is called an AFS structure if τ satisfies the following axioms: (1) ( ) ( ) ( ) We again return to the case that M is a set of simple concepts.Let X be a set of objects and M be a set of simple concepts on X .: 2 M X X τ × → is defined as follows: for any ( ) where m R is the binary relation of simple concepts m M ∈ defined in Definition 2.10 (it was defined more general than for simple concepts).
It is proved in [1] that ( ) , , M X τ is an AFS structure.
Definition 2.17.[1] Let ( ) , , M X τ be an AFS structure of a data set X .For , x X A M ∈ ⊆ , the set is defined as follows: (2) For ( ) (3) For , , , Remark: It is important to see that M consists of simple elements m .
Definition 2.18.Suppose that ( ) be the set of all functions from X to L .Assume that the lattice operations the least upper bound ∨ and the greatest lower bound ∧ on L are extended pointwise for the functions on X L .Further, define the where ⊥ and  are the least and the greatest elements of L , respectively.A unary operation η on L satisfies the involution property for any a L ∈ , and η is extended pointwise for the functions on X L , i.e., ( ) ( ) and for any , x y X ∈ , ( ) ( ) ( ) ( ) , where µ ¬ and ν ¬ are the logical negations of µ and ν , respectively.

Connection between Coherence Membership
for pairs ( ) x z and ( ) This is a contradiction.According to discussion after Proposition 2.7 we have . We will prove that β δ ≥ , that is, Repeating the proof of (b) we obtain the following: Let

( ) ( )
, , . These i ζ exist because EM is a completely lattice.By Definition 2.17 ( ) ( ) be a set of membership functions of the AFS fuzzy logic system ( ) ( ) ( ) ( ) ( ) and for all x X ∈ there exists a unary opera- where the operation : g EM EM → is defined by Let m M ∈ be simple concepts.Observe that we need this assumption for Definition 2.17 used bellow: in Definition 2.16 and Definition 2.17 the definition of ( ) Therefore η is an involution.Here ζ is not necessary simple.Let α β ≤ be elements in EM , and since g is order reversing, ( ) ( ) Using Definition 2.17 (1) it is

Conclutions
Simple concepts form chains.The elements of any chain form a "basis" in AFS fuzzy logic system ( ) x z R y z R ∈ ⇒ ∈ .Here X is a non-empty set and , , x y z on X and m are simple concepts.Then the conditions consti- P. Kukkurainen

,
of membership functions of the concept ζ of the AFS fuzzy logic system ( ) Zadeh algebra axioms and then forms some Zadeh algebra, Proposition 3.5.These are the two main results.Observing that by Lemma 3.1 (a) the condition (1) implies the condition (2) needed in Proposition 3.4.

Definition 2 . 1 .
A partially ordered set or a poset is a set in which a binary relation ≤ is defined satisfying the following conditions (P1)-(P3): (P1) For all x , x x ≤ .(P2) If x y ≤ and y x ≤ , then x y = .(P3) x y ≤ and y z ≤ , then x z ≤ .Let (P4) Given x and y , either x y ≤ or y x ≤ .

Moreover, x y ≤ is equivalent to each of the conditions and x
If a poset P (or a lattice) has an 0, then 0 0 x ∧ = and 0 x x ∨ = for all x P ∈ .If P has a universal upper bound I , then x I x ∧ = and x I I ∨ = for all x P ∈ .Definition 2.3.A lattice L is distributive if and only if the conditions

(
is because for any person x the degree of x belonging to the fuzzy concept represented by 1 2 3

0 1 ,
is called Zadeh algebra if it satiesfies the following conditions: (Z1) The operations ∨ and ∧ are commutative on X L .(Z2) The operations ∨ and ∧ are associative on X L .(Z3) The operations ∨ and ∧ are distributive on X L .(Z4) The neutral elements of the operations ∨ and ∧ are 0 and 1 , respectively, i.e., for all for all x X ∈ , there exists ( ) i.e., η is order reversing.(Z6) ≠ 0 1.(Z7) Zadeh algebra fulfils the Kleene condition: for any function , EMwhere concepts m are simple and let X be a non-empty set.If every relation

∈
EM such that the condition (a) is satisfied for all pairs .Because m is simple, m R is a sub- preference relation and by Definition 2.13 (3) it is transitive.This implies that

C
. This proves β δ ≥ .In the same way we can prove that α β ≥ for a pair ( ) , α β and then we conclude that α β δ ≥ function of the concept ζ .Proof.Let m M ∈ be simple concepts and Proof.By Lemma 3.1 (b), α β ≤ .On the other hand, by Lemma 3.2 binary relations of simple concepts m .If the condition is valid for every simple concept , i m A i I ∈ ∈ and , , x y z X ∈ then the membership functions ζ µ satisfy the conditions -(Z7) of Zadeh algebra in Definition 2.18.Proof.We verify the Zadeh algebra axioms: The first three axioms (Z1) -(Z3) are clear.(Z1) The operations ∨ and ∧ are commutative on [ ] Z2) The operations ∨ and ∧ are associative on [ ] The operations ∨ and ∧ are distributive on [ ] The neutral elements of the operations ∨ and ∧ are be simple.According to Proposition 2.8 g is an order reversing involution and ( )()g g m m =but in this case m need not be simple.We obtain

R
The EM elements ζ are of the form be binary relations of simple concepts m .If the condition By Lemma 3.1 the elements ζ forms a chain ( ) operations disjunction ∨ , conjunction ∧ and the logical negation.The elements are of the form ∨ ∧ is a completely distributive lattice.By means of the binary relations m R X X ∈ × defined in Definition 2.10 we construct the condition ( ) ( ) , pages 124, 125.Definition 2.10.Let ζ be any concept on the universe of discourse X . is called the binary relation of the concept ζ if R ζ satisfies:if and only if x belongs to concept ζ at some extent or x is a member of ζ and the degree of x belonging to ζ is larger or equal to that of y , or x belongs to concept ζ at some degree and y does not at all.
means that x belongs to ζ at some degree and that ( )meansthat x does not belong to ζ at all.If for the two persons x and y , 30 x age = and 20 y age = then ( ) on X .Example 2.14.Let ζ : "old people".The concept is simple.For example if for the membership function of the concept