Advances in Fuzzy Systems

Volume 2019, Article ID 4263923, 13 pages

https://doi.org/10.1155/2019/4263923

## Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

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 [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.