# Researches on Six Lattice-Valued Logic.

1. Introduction

Lattice-valued logic is an important case of multi-valued logic, and it plays more and more important roles in artificial intelligence and automated reasoning. Six lattice-valued is a kind of common lattice, which can express logic in real world, such as language values, and evaluation values. It can deal with not only comparable information but also non-comparable information. Therefore, theoretical researches and logic and reasoning systems based on six lattice-valued logic are of great significance.

2. The Structure of Lattice [L.sub.6]

The set of L = {O, a, b, c, d, I} is a lattice, and the order relation of L is shown in Figure 1. The complement operator '""and implication operation "[right arrow]" are defined in Table 1 respectively.

L means an lattice implication algebra.

Then set A = {O, I}, B = {O, m, I}. As A is the true set of classical binary logic, the operation rules of the complement operation and the implication operation are the same with the classical two-valued logic systems. B is the true-value set of Lukasiewicz system with three-valued logic, and complement operations and implication operations are defined in Table 2.

Let L* = A X B, the order relations, disjunctive, conjunctive, complement operation and implication operation on L are defined as follows:

For any (a, b)[member of] L (*), (c, d)[member of] L (*):

(1) (x,y)[less than or equal to] (z,r), if and only if x[less than or equal to]z and y [less than or equal to] r.

(2) (x,y) = (z,r), if and only if x = z and y = r.

(3) Under other circumstances, (x, y) cannot be compared with (z, r).

(4) (x,y)[conjunction](z,r) = (x[conjunction]z,y[conjunction]r), (x,y)[disjunction](z,r) = (x[disjunction]z,y[disjunction]r).

(5) (x, y) [right arrow] (z, r) = (x [right arrow] z, y [right arrow] r).

(6) (x, y)'=(x', y').

The L* constitute a six element lattice and its operation diagram is shown in Hasse Figure 2.

Theorem 1. L is isomorphic lattice implication of L*.

Proof:

Obviously, we can construct a upward one-to-one mapping from L to L*: f : L [right arrow] L*, making

f (O) = (O, O), f (a) = (O, I), f (b) = (I, m) f (c) = (I, O), f (d) = (O, m), f (I) = (I, I)

Clearly f is conjunctive homomorphic mapping and disjunctive homomorphism mapping. Here is the proof that f is complement homomorphic mapping and implication homomorphism mapping. According to the definition of implication operations and complement operations, it can be easily obtained in Table 3.

It can be seen from the Table 3, f is the implication operations and the complement operations homomorphic.

In summary, we proofed that:

For any x, y [member of] L, f (x') = (f (x)), f (x * y) = f (x)* f (y), where * is one of disjunctive, conjunctive, complement operation.

Thus L and L* is isomorphic lattice implication.

3. The Property and Language of Lattice [L.sub.6]

Due to [L.sub.6] is a lattice implication algebra, it not only has all the properties of lattice implication algebra but also properties as follows.

Theorem 2. As shown the six-valued lattice [L.sub.6] in Figure 1, the implication operation satisfies the following properties: For any x,y, z [member of] [L.sub.6]:

(1) z [less than or equal to] y [right arrow] x iff y [less than or equal to] z [right arrow] x.

(2) z [right arrow] (y [right arrow] x)[greater than or equal to](z [right arrow] y) [right arrow] (z [right arrow] x).

(3) (x [right arrow] y) [disjunction] ((x [right arrow] y) [right arrow] (x' [disjunction] y)) = I.

(4) y [right arrow] z [less than or equal to](x [right arrow] y) [right arrow](x [right arrow] z).

(5) x [right arrow] (y [right arrow] z) = y [right arrow] (x [right arrow] z).

(6) x [right arrow] y [less than or equal to](x [disjunction] z) [right arrow](y [disjunction] z).

Theorem 3. As the true subset of [L.sub.6], [L.sub.0] = {O, I, a, c} is a sub lattice implication algebra. What's more, [L.sub.0] is a Boolean algebra, and the implication arithmetic of it meets that: for any x, y [member of] [L.sub.0], x [right arrow] y = x' [disjunction] y.

Proof: It is clearly that [L.sub.0] is a sub lattice of [L.sub.6]. For any x, y [member of] [L.sub.0], x' [member of] [L.sub.0], x [right arrow] y [member of] [L.sub.0], therefore when regarding [L.sub.6], the operation of [L.sub.0] is closed, that is to say, [L.sub.0] is a sub lattice implication algebras of [L.sub.6].

It can be verified easily: for any x, y [member of] [L.sub.0], x [right arrow] y = x' [disjunction] y. Meeting the of Boolean algebra axiom, [L.sub.0] is a

Boolean algebra.

Any sub-set of power set lattice in a collection is called the set lattice for the collection. The isomorphism from a lattice L to a set lattice B(X) in collection X is named as a isomorphic representation L by B(X), which can be denoted as L for abbreviation. Through establishing the lattice representation, lattice language can be simplified, which is very important for studying the structure and properties of the lattice.

Definition 1 . Let L is a lattice, an element x [member of] L is called as an join-irreducible element, if

(1) x [not equal to] O (when there is a minimum of O when L);

(2) For any a,b [member of] L, if x = a [disjunction] b, then x = a or x = b.

Assume L is a finite distributive lattice, [??](L) denotes the set of all join-irreducible element in the collection, and all the join-irreducible element in L can form under set lattice (i.e. ideal Lattice) according to the order relation which can be indicated as O([??](L)). Then we have the following conclusions:

Theorem 4 . Let L is a finite distributive lattice, and mapping can be constructed as follows:

[mathematical expression not reproducible]

The [eta] is the lattice isomorphism from L to O([??](L)).

Theorem 5 . Let L is a finite distributive lattice, then the following equivalent hold:

1) L is a distributive lattice;

2) L [congruent to] O([??](L));

3) L is isomorphic to a set lattice;

4) For any n [greater than or equal to] 0, L is isomorphic to 2n sub lattice.

According to Theorem 5, theorem representation of six lattice-valued [L.sub.6] can be got easily.

Theorem 6. As shown the six-valued lattice [L.sub.6] in Figure 1, conclusions as follows can be got:

(1) The set of join-irreducible element in [L.sub.6] is [??]([L.sub.6]) = {a,b,c}, and its order relation are shown in Figure 3.

(2) The under set lattice (i.e. ideal lattice), which is the set of all the join-irreducible element and forms according to its order relation, is O([??]([L.sub.6])) = {[phi] {c},{a},{b,c},{a,c},{a,b,c}}.

(3) The Hasse diagram O([??]([L.sub.6])) of the ideal lattice of [L.sub.6], which forms through inclusion relation, is shown in Figure 4. Form the figure, we can see that [L.sub.6] is isomorphic of lattice implication to its ideal lattice O([??]([L.sub.6])). Lattice implication isomorphism [eta] is defined as follows:

[eta]: L [right arrow] O([??](L))

h(O) = [phi], [eta](a) = {b,c}, [eta](b) = {a,c}, [eta](c) = {a}

[eta] (d) = {c}, [eta] (I) = {a, b,c}

4. The Filter of Lattice [L.sub.6]

Since all Lukasiewicz algebras are lattice implication algebra , it can be proved that Lukasiewicz algebra filters are trivial.

Theorem 7.

(1) The finite chain of Lukasiewicz only contains trivial filters.

(2) Lukasiewicz algebra [0,1] only contains trivial filters.

Proof: (1) Let's set [mathematical expression not reproducible]. Specific operations are as follows:

For any x, y [member of] L,

x [disjunction] y = max {x, y}, x [conjunction] y = min {x, y}

x' = 1 - x, x [right arrow] y = min {1,1 - x + y}

It is clearly that set {1} and L are trivial filters in L. we can proof that L don't contain any other trivial filters.

From Theorem 6 we can see that filters in L are ideal dual filters of L, and the set of ideal dual filters of L are upper set of L.

If [mathematical expression not reproducible] (where k[greater than or equal to] 1) is a filter of L, then [mathematical expression not reproducible], And [mathematical expression not reproducible], so it can be seen that the definition of filters: [mathematical expression not reproducible].

This shows that F = L, so it demonstrated that L don't contain any other trivial filters.

(2) Let L = [0,1], its upper operation is the same as defined [C.sub.2].

It is clearly that set {1} and L are trivial filters in L. we can proof that L don't contain any other trivial filters.

We can see that filters in L are ideal dual filters of L, and the set of ideal dual filters of L are upper set of L. So the filter of L must be an interval containing greatest element 1.

Firstly, we can proof that the filter of L must be a closed interval.

Let us set F = (u,1) is a filter of L, where 0 < u < 1, for any x, satisfies u < x < 1, then x [member of] F, and x [right arrow] u = min {1,1 - u + x} = 1 - u + x > u [member of] F, conclusion can get u [member of] F.

This shows that F is a closed interval.

Secondly, assume F = [u,1] is a filter of L, where 0 < u < 1.

For any x, making u < x < 1 and x + u [greater than or equal to] 1, then u [right arrow] (x + u -1) = min {1 - u +(x + u -1),1} = x [member of] F

thereby x + u-1 [member of] F, that is contradictory, because 0[less than or equal to] x + u-1 < u.

So F is an interval.

This proves that Lukasiewicz interval only have trivial filters.

As a special case of Theorem 7, we have the following corollary.

Corollary 1. [C.sub.2] ={O, I} and [L.sub.3] ={O, m, I} only contain trivial filters.

Theorem 8. The six element lattice only contains the following four filters:

{I}, [L.sub.6], [F.sub.a] ={I, a}, [F.sub.bc] ={I, b,c}.

Proof: According to Theorem 1, [L.sub.6] can be seen as the direct product of [C.sub.2] and [L.sub.3]. According to Corollary 1, [C.sub.2] = {O, I} and [L.sub.3] = {O, m, I} only contain trivial filters. As followed:

The filters of [C.sub.2] ={O, I} are {I} and {O, I}.

The filters of [L.sub.3] ={O, m, I} are {I} and {O,m,I}.

It is easy to know, the filters of [L.sub.6] are the direct products of the filters of [C.sub.2] and the filters of [L.sub.3]. So the filters of [L.sub.6] are as followed:

{(I, I)}, {(I, I), (O, I)}, {(I, I), (I, m), (I, O)} and [L.sub.6] itself.

In other words: The six element lattice [L.sub.6] only contains the following four filters:

{I}, [L.sub.6], {I, a}, {I,b, c}.

5. The Tautologies of Lattice-Valued Logic System [L.sub.6]P(X)

Here we take the lattice-valued logic system [L.sub.6]P(X) into consideration, and discuss its tautologies and F-tautologies, the true value domain is [L.sub.6].

It is easy to verify:

[L.sub.6] = [C.sub.2] X [L.sub.3]

where [C.sub.2] is a Boolean algebra {O, I}, L3 is a Lukasiewicz algebra {O, m, I}.

Theorem 9. (The definition of tautologies in [L.sub.6]P(X) ) The tautologies in six lattice-valued logic system [L.sub.6]P(X) process the following relationship:

(1) [mathematical expression not reproducible].

(2) [mathematical expression not reproducible].

(3) [mathematical expression not reproducible].

(4) [mathematical expression not reproducible].

(5) [mathematical expression not reproducible].

Proof: It is noticed that the tautologies in Lukasiewicz three-valued logic system process the following relationship:

Proof of this theorem can be obtained.

From Theorem 7, the six element lattice [L.sub.6] only contains four filters as followed:

{I}, [L.sub.6], [F.sub.a] ={I, a}, ={I, b, c}.

Therefore, its non-trivial filters are [F.sub.a] = {I, a}, [F.sub.bc] = {I,b, c}.

We can get the definition of F-tautologies in six lattice-valued logic system [L.sub.6]P(X) as Theorem 8 similarly.

Theorem 10. (The definition of F-tautologies in [L.sub.6]P(X) ) The F-tautologies in six lattice-valued logic system [L.sub.6]P(X) process the following relationship:

(1) [mathematical expression not reproducible].

(2) [mathematical expression not reproducible].

Proof:

Since T(f)(A)[not equal to] T(g)(A), so T is an injection.

Clearly T is a surjection. For any [mathematical expression not reproducible], [mathematical expression not reproducible], G has the inverse image.

Thus G is an isomorphic functor of [??](L).

As isomorphic relationship means an equivalence relation, so S[??](L) and [??](L) are isomorphic.

6. Conclusion

In this paper, the six element lattice is built by the direct product of Boolean algebra and Lukasiewicz algebra; the operation of the lattice is defined; the structures, properties and filters are studied; finally the tautologies and F-tautologies of the six lattice-valued logic system are discussed. The results of this paper can be applied to lattice-valued logic systems and automated reasoning applications.

Acknowledgements

The work is supported by the project of Zhejiang province education department of China, Grant No. Y201326675.

References

 Yang, X. and Yun, Q.K. (1995) Fuzzy Lattice Implication Algebra. Southwest Jiaotong University, 2, 121-127.

 Xu, Y., Ruan, D. and Liu, J. (2004) Progress and Prospect in Lattice-Valued Logic Systems Based on Lattice Implication Algebras. Proceedings of the 6th International FLINS Conference Applied Computational Intelligence, 29-34. http://dx.doi.org/10.1142/97898127026610009

 Min, H.T. (1996) Georgia, Sequencing Primer Theory and Its Application. Southwest Jiaotong University Press.

 Fang, S. and Mei, Z.F. (2010) Six Yuan-Based Language of Logic Attributed True Value Method. Guangxi Normal University, 3, 118-122.

Hua Li

Department of Information Engineering, Hangzhou Polytechnic College, Hangzhou, China

Email: zjlihua@126.com

Received 2 September 2015; accepted 26 October 2015; published 29 October 2015

http://dx.doi.org/10.4236/jcc.2015.310005
```Table 1. Computing of the six-valued lattice [L.sub.6].

x  x'  [right arrow]  O  a  b  c  d  I

O  I   O              I  I  I  I  I  I
a  c   a              c  I  b  c  b  I
b  d   b              d  a  I  b  a  I
c  a   c              a  a  I  I  a  I
d  b   d              b  I  I  b  I  I
I  O   I              O  a  b  c  d  I

Table 2. Computing of [L.sub.3].

x  x'  [right arrow]  O  m  I

O  I   O              I  I  I
m  m   m              m  I  I
I  O   I              O  m  I

Table 3. Six-valued lattice generated by the direct product.

x      x'     [right arrow]  (0,0)  (0,I)  (I,m)  (I,0)  (0,m)  (I,I)

(0,0)  (II)   (0,0)          (I,I)  (I,I)  (I,I)  (I,I)  (I,I)  (I,I)
(0,I)  (I,0)  (0,I)          (I,0)  (I,I)  (I,m)  (I,0)  (I,m)  (I,I)
(I,m)  (0,m)  (I,m)          (0,m)  (0,I)  (I,I)  (I,m)  (0,I)  (I,I)
(I,0)  (0,I)  (I,0)          (0,I)  (0,I)  (I,I)  (I,I)  (0,I)  (I,I)
(0,m)  (I,m)  (0,m)          (I,m)  (I,I)  (I,I)  (I,m)  (I,I)  (I,I)
(II)   (0,0)  (I,I)          (0,0)  (0,I)  (I,m)  (I,0)  (0,m)  (I,I)
```