Fuzzy filters and their generalized types have been extensively studied
in the literature. In this paper, a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences is established, a quotient residuated lattice with respect to generalized fuzzy filter is induced, and several types of generalized fuzzy

Residuated lattices, introduced by Ward and Dilworth in [

Filters are tools of extreme importance in studying these logical algebras and the completeness of nonclassical logics. A filter is also called a deductive system [

At present, there are two branches to generalize the existing types of filters. One is the folding theory and the other is fuzzy sets theory. In the folding approach, in [

This paper continues the study of generalized fuzzy

Here, we recall some basic concepts and results which will be used in the sequel. Throughout this paper,

An algebra

In the rest of the paper by

Let

Let

However, a filter can also be alternatively described as for all

Let

Let

a generalized fuzzy implicative (or Boolean) filter if for all

a generalized fuzzy positive implicative filter if for all

a generalized fuzzy fantastic filter if for all

In this section, we develop new notions about the thresholds and investigate their properties. Particularly, by these notions, we will verify some properties of generalized fuzzy filters similar to the classical or fuzzy cases.

Here, for all

However, the following lemmas will be useful in the sequel.

Let

We only prove (1), (2) can be proven in a similar way. Consider the following:

Let

We only prove (1)

Conversely, assume that

Let

Since

Let

It follows immediately from Lemmas

For

Let

We only prove (1). Assume that

Here, the notions of generalized fuzzy filters can be rewritten in residuated lattices as follows.

Let

Let

The items in Definition

Moreover, the following properties of generalized fuzzy filters will be used in the sequel.

Let

Here, we define the concept of generalized fuzzy congruence to determine the relationships between generalized fuzzy filters and generalized fuzzy congruences.

Let

For a given generalized fuzzy filter

Obviously, it follows from Proposition

Let

It is trivial.

Let

Let

Consider the following special case of Definition

Thus

The following lemma is useful.

Let

It is obvious that

On the other hand, using Lemma

Let

Using Lemma

Let

It is trivial.

There is a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences.

It follows immediately from Propositions

Let

Assume that

Conversely, it follows from Lemma

Let

Let

Routine proofs show that

The above residuated lattice

In this section, we introduce some types of generalized fuzzy

Let

Let

Let

(1)

(2)

(3)

Associated with the alternative definitions of generalized fuzzy

Let

Each generalized fuzzy

Assume that

It is easy to prove by induction that every fuzzy generalized fuzzy

The converse of the above proposition does not always hold.

In Example

Let

Using Theorem

Let

In Example

The following lemma will be useful.

Let

Applying Lemma

Let

(1)

(2)

(3)

Associated with the alternative definitions of generalized fuzzy

In Example

Each generalized fuzzy

Assume that

It is easy to prove by induction that every generalized fuzzy

The converse of the above proposition does not always hold.

Let

Let

Using Theorem

Let

Let

The following lemma will be useful.

Let

By Lemma

Let

(1)

(2)

(3)

Associated with the alternative definitions of generalized fuzzy

Each generalized fuzzy

Assume that

It is easy to prove by induction that every generalized fuzzy

The converse of the above proposition does not always hold.

In Example

In order to investigate the relationships among these three types defined above, the following lemma is useful.

Let

Using Lemma

Let

Assume that

Using Lemma

Conversely, it follows from Theorem

Obviously, the generalized fuzzy filters in Examples

Let

Using Theorem

Generalized fuzzy filters have been extensively studied in the literature. In this paper, we researched the properties of generalized fuzzy filters and introduced some types of generalized fuzzy

In our future work, we will introduce some other types of generalized fuzzy

This research was supported by AMEP of Linyi University, the Natural Science Foundation of Shandong Province (Grant no. ZR2013FL006).