_{1}

^{*}

Based on the direct product of Boolean algebra and Lukasiewicz algebra, six lattice-valued logic is put forward in this paper. The algebraic structure and properties of the lattice are analyzed profoundly and the tautologies of six-valued logic system L6P(X) are discussed deeply. The researches of this paper can be used in lattice-valued logic systems and can be helpful to automated reasoning systems.

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.

The set of

L means an lattice implication algebra.

Then set

Let

x | x' | ® | 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 |

x | x' | ® | O | m | I | ||
---|---|---|---|---|---|---|---|

O | I | O | I | I | I | ||

m | m | m | m | I | I | ||

I | O | I | O | m | I |

on L are defined as follows:

For any

(1)

(2)

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

(4)

The L^{*} constitute a six element lattice and its operation diagram is shown in Hasse

Theorem 1. L is isomorphic lattice implication of L^{*}.

Proof:

Obviously, we can construct a upward one-to-one mapping from L to L^{*}:

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

x | x' | ® | (O,O) | (O,I) | (I,m) | (I,O) | (O,m) | (I,I) | |
---|---|---|---|---|---|---|---|---|---|

(O,O) | (I,I) | (O,O) | (I,I) | (I,I) | (I,I) | (I,I) | (I,I) | (I,I) | |

(O,I) | (I,O) | (O,I) | (I,O) | (I,I) | (I,m) | (I,O) | (I,m) | (I,I) | |

(I,m) | (O,m) | (I,m) | (O,m) | (O,I) | (I,I) | (I,m) | (O,I) | (I,I) | |

(I,O) | (O,I) | (I,O) | (O,I) | (O,I) | (I,I) | (I,I) | (O,I) | (I,I) | |

(O,m) | (I,m) | (O,m) | (I,m) | (I,I) | (I,I) | (I,m) | (I,I) | (I,I) | |

(I,I) | (O,O) | (I,I) | (O,O) | (O,I) | (I,m) | (I,O) | (O,m) | (I,I) |

It can be seen from the

In summary, we proofed that:

For any

Thus L and L^{*} is isomorphic lattice implication.

Due to L_{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_{6} in

(1)

Theorem 3. As the true subset of L_{6}, _{0} is a Boolean algebra, and the implication arithmetic of it meets that: for any x,

Proof: It is clearly that L_{0} is a sub lattice of L_{6}. For any_{6}, the operation of L_{0} is closed, that is to say, L_{0} is a sub lattice implication algebras of L_{6}.

It can be verified easily: for any_{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 [

(1) x ¹ O (when there is a minimum of O when L);

(2) For any

Assume L is a finite distributive lattice, Á(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(Á(L)). Then we have the following conclusions:

Theorem 4 [

The h is the lattice isomorphism from L to O(Á(L)).

Theorem 5 [

1) L is a distributive lattice;

2)

3) L is isomorphic to a set lattice;

4) For any n ³ 0, L is isomorphic to 2^{n} sub lattice.

According to Theorem 5, theorem representation of six lattice-valued L_{6} can be got easily.

Theorem 6. As shown the six-valued lattice L_{6} in

(1) The set of join-irreducible element in L_{6} is

(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

(3) The Hasse diagram O(Á(L_{6})) of the ideal lattice of L_{6}, which forms through inclusion relation, is shown in _{6} is isomorphic of lattice implication to its ideal lattice O(Á(L_{6})). Lattice implication isomorphism h is defined as follows:

Since all Lukasiewicz algebras are lattice implication algebra [

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

For any

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

And

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_{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

This shows that F is a closed interval.

Secondly, assume

For any x, making

thereby

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.

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

{I}, L_{6},

Proof: According to Theorem 1, L_{6} can be seen as the direct product of C_{2} and L_{3}. According to Corollary 1,

The filters of

The filters of

It is easy to know, the filters of L_{6} are the direct products of the filters of C_{2} and the filters of L_{3}. So the filters of L_{6} are as followed:

_{6} itself.

In other words: The six element lattice L_{6} only contains the following four filters:

{I}, L_{6},

Here we take the lattice-valued logic system L_{6}P(X) into consideration, and discuss its tautologies and F-tauto- logies, the true value domain is L_{6}.

It is easy to verify:

where C_{2} is a Boolean algebra_{3} is a Lukasiewicz algebra

Theorem 9. (The definition of tautologies in L_{6}P(X) [_{6}P(X) process the following relationship:

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_{6} only contains four filters as followed:

{I}, L_{6},

Therefore, its non-trivial filters are

We can get the definition of F-tautologies in six lattice-valued logic system L_{6}P(X) as Theorem 8 similarly.

Theorem 10. (The definition of F-tautologies in L_{6}P(X) [_{6}P(X) process the following relationship:

Proof:

Since

Clearly T is a surjection. For any

Thus G is an isomorphic functor of À(L).

As isomorphic relationship means an equivalence relation, so SÀ(L) and Á(L) are isomorphic.

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.

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

HuaLi, (2015) Researches on Six Lattice-Valued Logic. Journal of Computer and Communications,03,36-42. doi: 10.4236/jcc.2015.310005