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Advances in Fuzzy Systems
Volume 2019, Article ID 4263923, 13 pages
https://doi.org/10.1155/2019/4263923
Research Article

Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

1Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia
2Department of Mathematics, University of Gondar, Gondar, Ethiopia

Correspondence should be addressed to Wondwosen Zemene Norahun; moc.liamg@6791eidnow

Received 26 March 2019; Revised 13 May 2019; Accepted 17 June 2019; Published 8 July 2019

Academic Editor: Jose A. Sanz

Copyright © 2019 Berhanu Assaye Alaba and Wondwosen Zemene Norahun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [819].

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.

Figure 1

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 . Thus . Based on this fact we have Thus . So is distributive.
Now we show the completeness. Since is a poset and is greatest element of , it is enough to show that every subfamily of has infimum. Let be a subfamily of . Then is a fuzzy ideal of . For any , Thus is a kernel fuzzy ideal of . So is a complete distributive lattice.

Corollary 39. If is a pseudocomplemented lattice, then forms a distributive lattice with respect to inclusion ordering of fuzzy sets.

Proof. Clearly is a partially ordered set. For , define Obviously . Let and . Then If and , then , which implies