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Residuated lattices play an important role in the study of fuzzy logic based on

Since Hájek introduced his basic fuzzy logics, (BL-logics in short) in 1998 [

A close analysis of the situation reveals that the main drive in all the previously mentioned works resides in the existence of an adjoint pair of operations. Just as the foldness theory for filters in BL-algebras generalizes filters introduced by Hájek, our foldness theory for filters in residuated lattices builds on recently published works on filters in residuated lattices by Haveshki et al. [

More specifically, we introduce the notions of

Diagram of

Diagrams of

A

L-1:

L-2:

L-3:

A

L-4:

A

L-5:

A

L-6:

L-7:

A

L-8:

A

L-9:

In this work, unless mentioned otherwise,

For any element

The following properties hold in a residuated lattice.

Let

(F1): for every

(F2): for every

It is known that in a residuated lattice, filters and deductive systems coincide [

The residuated lattices listed below are not BL-algebras and will be used to illustrate the concepts treated in the paper.

Let

Let

Let

Let

It is well known that the class of residuated lattices is a variety. So from the above examples, we may obtain infinite residuated lattices which are not

Let

Clearly, a locally finite residuated lattice is local.

Given a filter

One can easily verify the following result.

For any filter

Consequently, it is straightforward to see that

Let

A proper filter

(i) Prime filters are prime filters of the second kind. The converse is true if

(ii) Prime filters are prime filters of the third kind.

(iii) Boolean filters of the second kind are boolean filters.

(iv) If

(v) If a filter is prime of the second kind and boolean, then it is boolean of the second kind.

(vi) Maximal filters are semi maximal filters.

Maximal filters are prime filters of the second kind.

Assume that

Any prime filter of

Using Definition

Now, unless mentioned otherwise,

A class

Let

A subset

By taking

Like in the case of

Let

A filter

The following conditions are equivalent for a filter

Moreover, the class of

Since

Firstly, for

Secondly, for

Finally for

By repeating the process

Using Propositions

Given a filter

The following conditions are equivalent for a filter

From Propositions

The following conditions are equivalent for a residuated lattice

Any 1-fold implicative residuated lattice is a Heyting lattice, and any Heyting lattice is an

Assume that

An

A filter of

In particular, any 1-fold Boolean filter is a Boolean filter. [

From this definition, the class of

The following conditions are equivalent for any filter

A filter

Every filter of

Let

The converse of Proposition

A residuated lattice

Let

Since

It is easy to check that the residuated lattice of Example

As in the case of

The following conditions are equivalent for a residuated lattice

So,

To end this section, we note that any 1-fold Boolean residuated lattice is a Boolean lattice, and any Boolean lattice is an

However, for the residuated lattice of Example

A filter

Let

A residuated lattice

(i) By simple computation, one can show that the lattice of Example

(ii) The lattice of Example

It is easy to prove the following result.

If

Let

Now we give the definition of an

As in the case of

Note that the residuated lattice of Example

Let

In particular a 1-fold fantastic filter is a fantastic filter.

Let

Let

For the residuated lattice of Example

The following result gives a simple characterization of

Let

Thus the class of

Assume that

Since

Conversely, assume that

Let

Let

Assume that

We have

We also have

Since

Let

Assume that

Now,

Moreover,

Let us note that if

Let

Since

From the observation above, we have

A residuated lattice

Let

The residuated lattice of Example

The residuated lattice of Example

Here is a characterization of

The residuated lattice

We have

The following conditions are equivalent for a residuated lattice

On the other hand,

So, a filter

Combining Propositions

Let

To end this section, we note that:

1-fold fantastic residuated lattices are

however, an

Let

In particular a 1-fold obstinate filter is an obstinate filter.

The following result gives a characterization of

Let

(i) An

(ii) Since

Let

The filter

The filter

The following conditions are equivalent for any proper filter

On the other hand, let

Let

We note that this is an improvement of [

A residuated lattice

This means that

The following conditions are equivalent for any proper filter

Since

Let

Thus, an

Let

The lattice of Example

The lattice of Example

Let

Let

In Example

The residuated lattice of Example

Let

From Proposition

Also, any

Any proper filter of

Assume that

The converse of the above proposition is not true, since in Example

It follows that any

A filter

Let

In Example

In Example

Let

Assume that

Now, by Proposition

A residuated lattice

One easily verifies that a residuated lattice

Let

From Propositions

Let

In [

Let

(i) It is easy to verify that

(ii) The class of

These two notions are weakenings of the corresponding ones for

In [

(i)

(ii)

Clearly,

Now, let us consider some statements about

Then we have the following implications.

Let

Now, it is easy to see that an

We may restate [

Let

If

If

Now, here are some consequences of

Let

This follows from Propositions

Let

If

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to thank the referees for their excellent suggestions that improved the presentation and the readability of the paper.