Research Article | Open Access

# Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

**Academic Editor:**Jose A. Sanz

#### Abstract

In this paper, we introduce the concept of kernel fuzzy ideals and ⁎-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We observe that every fuzzy ideal cannot be a kernel of a ⁎-fuzzy congruence and we give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ⁎-fuzzy congruence. On the other hand, we show that every fuzzy filter is the cokernel of a ⁎-fuzzy congruence. Finally, we prove that the class of ⁎-fuzzy filters forms a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.

#### 1. Introduction

The theory of pseudocomplementation was introduced and extensively studied in semilattices and particularly in distributive lattices by O. Frink [1] and G. Birkhoff [2]. Later, pseudocomplement in Stone algebra has been studied by several authors like R. Balbes [3], G. Grtzer [4], etc. In 1973, W. H. Cornish [5] studied congruence on pseudocomplemented distributive lattices and identified those ideals and filters that are congruence kernels and cokernels, respectively. Later, T. S. Blyth [6] studied ideals and filters of pseudocomplemented semilattices.

On the other hand, the concept of fuzzy sets was firstly introduced by Zadeh [7]. Rosenfeld has developed the concept of fuzzy subgroups [8]. Since then, several authors have developed interesting results on fuzzy theory; see [8–19].

In this paper, we introduce the concept of kernel fuzzy ideals and -fuzzy filters of a pseudocomplemented semilattice. We studied a -fuzzy congruence on a pseudocomplemented semilattice. We observe that every fuzzy ideal cannot be a kernel of a -fuzzy congruence. We give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a -fuzzy congruence. We also show that the class of kernel fuzzy ideals can be made a complete distributive lattice. Moreover, we study the image and preimage of kernel fuzzy ideals under a -epimorphism mapping. In Section 4 we turn our attention to fuzzy filters. Here we show that every fuzzy filter is the cokernel of a -fuzzy congruence and investigate a certain type of fuzzy filter called a -fuzzy filter. We prove that these filters form a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.

#### 2. Preliminaries

In this section, we recall some definitions and basic results on lattices, meet semilattices, and fuzzy theory.

*Definition 1 (see [2]). *An algebra is said to be a lattice if it satisfies the following conditions: (1) and ,(2) and ,(3) and ,(4) and .

*Definition 2 (see [2]). *A lattice is a distributive lattice if and only if it satisfies the following identity:

A lattice is said to be bounded if there exist and in such that and for all .

If and are lattices, then is said to be a lattice morphism if for all .(1)(2).

If is onto, then is an epimorphism.

*Definition 3 (see [20]). *An algebra is a pseudocomplemented lattice if the following conditions hold: (1) is a bounded lattice,(2)for all . If is a bounded distributive lattice, then is a pseudocomplemented distributive lattice.

The lattice is said to be a complemented lattice if satisfies the following conditions for all there exists such that and .

*Definition 4 (see [3]). *A pseudocomplemented distributive lattice is called a Stone algebra if, for all , it satisfies the property:

*Definition 5 (see [1]). *An algebra is a pseudocomplemented semilattice if for all the following conditions are satisfied: (1),(2),(3),(4).

It is well known that is a partially ordered set relative to the order relation defined by and that relative to this order relation is the greatest lower bound of and . Hence the condition of Definition 5(4) is equivalent to

A nonempty subset of a semilattice is called an ideal of if, .

A nonempty subset of a semilattice is called a filter of if it satisfies the following:(1),(2).

Theorem 6 (see [1]). *For any two elements of a pseudocomplemented semilattice , we have the following: *(1)*,*(2)*,*(3)*,*(4)*,*(5)*,*(6)*,*(7)*,*(8)*,*(9)*.*

An element of a pseudocomplemented semilattice is called closed if .

*Definition 7 (see [7]). *Let be any nonempty set. A mapping is called a fuzzy subset of .

The unit interval together the operations min and max form a complete distributive lattice. We often write for minimum or infimum and for maximum or supremum. That is, for all we have and .

*Definition 8 (see [8]). *Let and be fuzzy subsets of a set . Define the fuzzy subsets and of as follows: for each , Then and are called the union and intersection of and , respectively.

For any collection, of fuzzy subsets of , where is a nonempty index set, the least upper bound , and the greatest lower bound of the ’s are given by for each , respectively.

For each the set is called the level subset of at [7].

The characteristics function of any set is defined as

*Definition 9 (see [8]). *Let be a function from into ; be a fuzzy subset of ; and be a fuzzy subset of . The image of under , denoted by , is a fuzzy subset of such that The preimage of under , symbolized by , is a fuzzy subset of and

*Definition 10 (see [18]). *A fuzzy subset of a bounded lattice is called a fuzzy ideal of if for all the following conditions are satisfied: (1),(2),(3).

*Definition 11 (see [18]). *A fuzzy subset of a bounded lattice is called a fuzzy filter of if for all the following conditions are satisfied: (1),(2),(3).

Theorem 12 (see [21]). *Let be a lattice, , and . Define a fuzzy subset of as is a fuzzy ideal of .*

*Remark 13 (see [21]). * is called the -level principal fuzzy ideal corresponding to .

Similarly, a fuzzy subset of defined as is the -level principal fuzzy filter corresponding to .

*Definition 14 (see [18]). *A fuzzy subset of is said to be a fuzzy congruence on if and only if, for any , the following hold: (1),(2),(3),(4).

#### 3. Kernel Fuzzy Ideals and -Fuzzy Ideals

In this section, we introduce the concept of kernel fuzzy ideals and -fuzzy ideals of a pseudocomplemented semilattice. We give necessary and sufficient conditions for fuzzy ideals to be a kernel of a -fuzzy congruence.

Throughout the rest of this paper, stands for a pseudocomplemented semilattice and a partially ordered set unless it is specified.

Now we define a fuzzy ideal of a semilattice.

*Definition 15. *A fuzzy subset of is called a fuzzy ideal of if for all the following conditions are satisfied: (1),(2).

Theorem 16. *A fuzzy subset of is a fuzzy ideal if and only if each level subset of is an ideal of . (In particular, a nonempty subset of is an ideal of if and only if is a fuzzy ideal.)*

*Proof. *Suppose is a fuzzy ideal of . Let and . Then and . Since is a fuzzy ideal, we get that . Thus . So is an ideal of .

Conversely, suppose that every level subset of is a fuzzy ideal of . Then is a fuzzy ideal of . Thus . Now we proceed to show that ; let such that and . Then either or . Without loss of generality, . Since is an ideal of and , then . Thus . So is a fuzzy ideal of .

*Definition 17. *A fuzzy subset of is said to be a fuzzy congruence on if and only if, for any , the following hold: (1),(2),(3),(4).

*Definition 18. *A fuzzy congruence relation on is called a -fuzzy congruence if for all .

It is clear that a fuzzy congruence is a -fuzzy congruence on if and only if each level subset of is a -congruence on . Also, let be a congruence on . Then is a -congruence on if and only if its characteristic function is a -fuzzy congruence on .

Theorem 19. *A fuzzy congruence on is a -fuzzy congruence if and only if *

*Proof. *If is a -fuzzy congruence on , then by Definition 18. Suppose conversely that the condition holds. Let . Since is a fuzzy congruence on , . By the assumption, we have . Thus . Similarly, . Since is a fuzzy congruence, then . Thus . So is a -fuzzy congruence on .

If is a fuzzy congruence on and , then the fuzzy subset of defined by is called a fuzzy congruence class of determined by and . We thus have the following theorem.

Theorem 20. *If is a fuzzy congruence on , then the fuzzy congruence class of determined by and is a fuzzy ideal of .*

*Proof. *Suppose is a fuzzy congruence on . Then . Thus . Again for any , . Since is a fuzzy congruence on , we have . Similarly, . Thus . So is a fuzzy ideal of .

Now we define a kernel fuzzy ideal of a semilattice.

*Definition 21. *A fuzzy ideal of is called a kernel fuzzy ideal if , where is a -fuzzy congruence on .

*Example 22. *Consider the semilattice whose Hasse diagram is given in Figure 1.

Then is pseudocomplemented; we have for . The fuzzy ideal of defined as , and for is not the kernel fuzzy ideal of a -fuzzy congruence. For, suppose that were a -fuzzy congruence with kernel ; then . Since , we have . This is a contradiction.

In Example 22 we observe that every fuzzy ideal of a pseudocomplemented semilattice is not a kernel fuzzy ideal. In the following theorem we give the necessary and sufficient conditions for a fuzzy ideal of a pseudocomplemented semilattice to be a kernel of a -fuzzy congruence.

Theorem 23. *A fuzzy ideal of is a kernel fuzzy ideal if and only if for all .*

*Proof. *Let be a kernel fuzzy ideal of a -fuzzy congruence on . Then . Suppose . Then by the assumption and . Thus . Since is a fuzzy congruence on , we have Thus .

Conversely, suppose that the condition holds. Consider the fuzzy relation defined on by Clearly, is both reflexive and symmetric. It is also transitive; if , then If , then . By the assumption, we have So that is a fuzzy equivalence relation on . By the similar procedure it can be easily verified that is a fuzzy congruence on .

To show is a -fuzzy congruence on , let . Then . Since is a pseudocomplemented semilattice, . Thus . So thatIt follows by Theorem 19 that is a -fuzzy congruence on .

Now taking in the condition we obtain . Now we proceed to show that the kernel of is . For any ,Let satisfying . Then . This implies that is an upper bound of . This shows that . Thus . So is a kernel of .

Corollary 24. * is a kernel fuzzy ideal if and only if *(1)*,*(2)*.*

*Proof. *Let be a kernel fuzzy ideal of . Then for all .

(1) Since and is a fuzzy ideal, we get . Again if , then . Thus for all .

(2) Let such that . Then . By (1) and by the assumption, we have . This implies . On the other hand, since and , we have . Thus .

Conversely, if (1) and (2) hold, for any there exists such that , then . Thus by (1), we get that . So is a kernel fuzzy ideal of .

The set is called the skeleton of . The elements of are called skeletal.

Corollary 25. *Let . is a kernel fuzzy ideal if and only if is a skeletal element of .*

*Proof. *Let be a kernel fuzzy ideal. Then by Corollary 24(1) , which implies . Since for all , we get that . Thus is a skeletal element.

Conversely, Suppose that is a skeletal element of . To show is a kernel fuzzy ideal, let such that and . Then and . Thus . So . This shows that . If and , then trivially holds. Hence is a kernel fuzzy ideal.

Corollary 26. *The following conditions on are equivalent: *(1)*Every fuzzy ideal of is a kernel fuzzy ideal.*(2)*Every level principal fuzzy ideal is a kernel fuzzy ideal.*(3)* is a Boolean algebra.*

Theorem 27. *A fuzzy subset of is a kernel fuzzy ideal if and only if each level subset of is kernel ideal of .*

*Proof. *Let be a kernel fuzzy ideal of . Then by Theorem 16 is an ideal of . To show is a kernel ideal, let . Then by the assumption, . Thus .

Conversely, suppose that every level subset of is a kernel ideal of . Then by Theorem 16 is a fuzzy ideal of . To show is a kernel fuzzy ideal, let such that and . Then either or . Without loss of generality, . Then and by the assumption, . This shows that . Thus . So is a kernel fuzzy ideal of .

Corollary 28. *A nonempty subset of is a kernel ideal of if and only if is a kernel fuzzy ideal.*

*Definition 29. *A fuzzy ideal of is called a -fuzzy ideal if for all .

Corollary 30. *Every kernel fuzzy ideal is a -fuzzy ideal.*

Corollary 31. *If is a pseudocomplemented distributive lattice, then a fuzzy ideal of is a kernel fuzzy ideal if and only if it is a -fuzzy ideal.*

*Proof. *Suppose is a kernel fuzzy ideal of . Then by Corollary 24(1) is a -fuzzy ideal of .

Conversely, suppose is a -fuzzy ideal. Since is a pseudocomplemented distributive lattice, for any we have . Now . Since is a -fuzzy ideal, we get . Thus is a kernel fuzzy ideal of .

Theorem 32. *A -fuzzy ideal is a kernel fuzzy ideal if and only if for all .*

*Proof. *Let . Define . Since for all , we have . Thus . Since is a kernel fuzzy ideal, .

Theorem 33. *A fuzzy subset of is a -fuzzy ideal if and only if each level subset of is a -ideal of .*

*Proof. *Let be a -fuzzy ideal of . Then for each and by Theorem 16 is an ideal of . To show is a -ideal, let . Then and . Thus each level subset of is a -fuzzy ideal.

Conversely, suppose every level fuzzy subset of is a -ideal of . Then by Theorem 16 is a fuzzy ideal of . Since and is a fuzzy ideal, we have . Let . Then . Thus . So is a -fuzzy ideal of .

Corollary 34. *A nonempty subset of is a -ideal of if and only if is a -ideal.*

Theorem 35. *Let be a kernel fuzzy ideal of . Then the smallest -fuzzy congruence on with kernel is given by*

*Proof. *It is shown in the proof of Theorem 23 that is a -fuzzy congruence with kernel . Now we proceed to show that is the smallest -fuzzy congruence with kernel .

Let be a -fuzzy congruence with kernel . For ,Since is a -fuzzy congruence, we have . Similarly, . If , then This shows that is an upper bound of . Thus . So is the smallest -fuzzy congruence with kernel .

Lemma 36. *If and are -fuzzy ideals of , then so is .*

*Proof. *Suppose and are -fuzzy ideals of . Clearly . Let . Then . Since and are fuzzy ideals, we have . Thus is a fuzzy ideal of . Now . Since and are -fuzzy ideals, we get that for all . Thus is a -fuzzy ideal of .

The class of all -fuzzy ideals of is denoted by . It is clear that the set of -fuzzy ideals of , ordered by fuzzy set inclusion, is a complete distributive lattice in which the lattice operations are fuzzy set-theoretic.

The class of all kernel fuzzy ideals of is denoted by . We now prove that is a complete distributive lattice.

Theorem 37. *If , the supremum of and is given by *

*Proof. *Let . First, we need to show that is a kernel fuzzy ideal of . Since , we have . For any , This shows that . Thus is a fuzzy ideal of .

We now show that is a kernel fuzzy ideal. For any , If and , then and . Thus . Since and are kernel fuzzy ideals, we get that and . Based on this we have Thus is a kernel fuzzy ideal of .

To show is the smallest kernel fuzzy ideal of , let be a kernel fuzzy ideal of containing and . Then for any , If , then . This implies is an upper bound of . Thus . Hence is the smallest kernel fuzzy ideal containing and .

Theorem 38. *The set forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.*

*Proof. *Clearly is a partially ordered set. For , clearly . So is a lattice.

For , clearly . Now for any , If , then . Since is a pseudocomplemented semilattice, we have for all . Now we proceed to show that .