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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 763428, 9 pages
Nilpotent Elements of Residuated Lattices
1Department of Mathematics, Bam Higher Education Complexes, Kerman, Iran
2Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran
Received 28 March 2012; Revised 13 June 2012; Accepted 27 June 2012
Academic Editor: Siegfried Gottwald
Copyright © 2012 Shokoofeh Ghorbani and Lida Torkzadeh. 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.
Some properties of the nilpotent elements of a residuated lattice are studied. The concept of cyclic residuated lattices is introduced, and some related results are obtained. The relation between finite cyclic residuated lattices and simple MV-algebras is obtained. Finally, the notion of nilpotent elements is used to define the radical of a residuated lattice.
Ward and Dilworth  introduced the concept of residuated lattices as generalization of ideal lattices of rings. The residuated lattice plays the role of semantics for a multiple-valued logic called residuated logic. Residuated logic is a generalization of intuitionistic logic. Therefore it is weaker than classical logic. Important examples of residuated lattices related to logic are Boolean algebras corresponding to basic logic, BL-algebras corresponding to Hajek’s basic logic, and MV-algebras corresponding to Lukasiewicz many valued logic. The residuated lattices have been widely studied (see [2–8]).
In this paper, we study the properties of nilpotent elements of residuated lattices. In Section 2, we recall some definitions and theorems which will be needed in this paper. In Section 3, we study the nilpotent elements of a residuated lattice and study its properties. In Section 4, we define the notion of cyclic residuated lattice and we obtain some related results. In particular, we will prove that a finite residuated lattice is cyclic if and only if it is a simple MV-algebra. In Section 5, we investigate the relation between nilpotent elements and the radical of a residuated lattice.
In this section, we review some basic concepts and results which are needed in the later sections.
A residuated lattice is an algebraic structure such that(1) is a bounded lattice with the least element 0 and the greatest element 1,(2) is a commutative monoid where is a unit element,(3) if and only if , for all . We denote the residuated lattice by . We use the notation for the bounded lattice .
An MV-algebra is an algebra with one binary operation , one unary operation , and one constant 0 such that is a commutative monoid and, for all , , , . If is an MV-algebra, then the binary operations , , , and the constant 1 are defined by the following relations: for all , , , , , .
Remark 2.2. A residuated lattice is an MV-algebra if it satisfies the additional condition: , for any .
Definition 2.3. A nonempty subset of is called a filter of if and only if it satisfies the following conditions:(i)for all and all , if then ,(ii)for all , .
Let be a filter of . For all , we denote and say that and are congruent if and only if and . is a congruence relation on . The quotient residuated lattice with respect to the congruence relation is denoted by and its elements are denoted by , for .
For all elements of a residuated lattice , define and for all . The order of , in symbols , is the smallest positive integer such that . If such does not exist, we say has infinite order.
Definition 2.4. The residuated lattice is called simple if the only filters of are and .
Proposition 2.5 (see ). A residuated lattice is simple if and only if , for every such that .
A filter of is called a maximal filter if and only if it is a maximal element of the set of all proper filters of . The set of all maximal filters of is called the maximal spectrum of and is denoted by . For any , we will denote . For any , will be denoted by .
Proposition 2.6 (see ). Let be a residuated lattice and a proper filter of . Then the followings are equivalent:(i) is a simple residuated lattice,(ii) is a maximal filter, (iii)for any , if and only if , for some .
Definition 2.7. A residuated lattice is said to be local if and only if it has exactly one maximal filter.
Proposition 2.8 (see ). Any simple residuated lattice is local.
We denote by the Boolean center of , that is the set of all complemented elements of the lattice . The complements of the elements in the Boolean center of a residuated lattice are unique.
Theorem 2.9 (see ). If is a local residuated lattice, then .
3. Nilpotent Elements of Residuated Lattices
We recall that an element is called nilpotent if and only if is finite. We denote by the set of the nilpotent elements of .
Example 3.1 (see ). Consider the residuated lattice with the universe . Lattice ordering is such that , , and elements from and are pairwise incomparable. The operations of and are given by the following: Then , , , and . Hence .
Remark 3.2. Let be a residuated lattice. Then has order if and only if , for some .
Example 3.3. Let , for and for all . Also, we have and . Define Then becomes a residuated lattice. Let . Then there exists such that . We get that , because . Hence is a simple residuated lattice.
Theorem 3.4. is a lattice ideal of the residuated lattice .
Proof. It is clear that . Suppose that and . There exists such that . We have . Therefore .
Suppose that . Then there exists such that . By Proposition 2.1(4), we have Hence , and then is a lattice ideal of .
Remark 3.5. An element of a residuated lattice is nilpotent if and only if there is no proper filter of such that .
Theorem 3.6. Let be a residuated lattice and a family of residuated lattices. Then(1),(2).
Proof. (1) Let . Since , then . Hence we get that for all . Also, we have . So there exists such that . Therefore, we obtain that .
(2) Suppose that . Then where is nilpotent in . Thus, for each , there exists such that . Put . Then , that is . The proof of reverse inclusion is straightforward.
For a nonempty subset , the smallest filter of which contains is said to be the filter of generated by and will be denoted by . If with , we denote by the filter generated by . Also, we have for some .
Theorem 3.7. Let be an element of a residuated lattice . Then is nilpotent if and only if .
Proof. Let be a finite order. Then there exists integer such that , that is . Therefore .
Conversely, if , then . So there exists an integer such that . Hence has finite order.
Theorem 3.8. Let be an element of order of a residuated lattice . Then the elements , , , of are pairwise distinct.
Proof. Suppose that , for some . Then . But which is a contradiction with the order of . Hence .
4. Cyclic Residuated Lattices
The order of a residuated lattice is the cardinality of and denoted by .
Definition 4.1. Let be a finite residuated lattice. is called cyclic, if there exists such that . is called a generator of .
Theorem 4.2. Let be a cyclic residuated lattice of order . Then there exists an element of order such that where , .
Proof. Since is cyclic of order , then there exists an element of order . By Theorem 3.8, the elements , and , of are pairwise distinct. Hence the cardinality of is . Since and , we get that .
An element of a residuated lattice is called a coatom if it is maximal among elements in .
Theorem 4.3. Let be a cyclic residuated lattice of order . Then is linearly ordered. Moreover the generator of is a unique coatom.
Proof. Since is a cyclic residuated lattice of order , then there exists an element such that by Theorem 4.2. If , then . By Proposition 2.1(7),. Hence is linearly ordered.
It is clear that is a coatom of . We will show that . Since , then there exists such that . We have Therefore . By definition order of , we get that . Hence . Therefore .
In the following example, we will show that the converse of the above theorem may not be true in general.
Example 4.4. Consider the residuated lattice with the universe . Lattice ordering is such that . The operations of and are given by the following: is a finite linearly residuated lattice of order but it is not cyclic because we have , , , and .
In the following theorems, we characterize cyclic residuated lattices.
Theorem 4.5. Let be a cyclic residuated lattice of order . Then is isomorphic to the simple residuated lattice of Example 3.3.
Proof. By Theorems 4.2 and 4.3, there exists an element such that , where . We denote , , and for . By Theorem 4.3, is the generator of and it is a coatom. We will show that
It is clear that , if . We will prove that , if . Suppose that . Then
Hence . Since is a coatom, then or . If , then . Since , then which is a contradiction by Theorem 3.8. Hence .
Now, suppose that where . Since , then there exists such that . We have Therefore . We get that . Thus, . On the other hand, since and for all , then . Therefore . We obtain that .
Now, we will show that Suppose that . Then . Since , we get that . Thus .
Suppose that . Since , we have . Therefore . We get that . Let . Then . Consider the following cases.(1)If , then which is a contradiction.(2)If , then which is a contradiction.(3), then . We get that . Therefore which is a contradiction.We obtain that .
Hence is an isomorphism between and . Since is a simple residuated lattice, then is a simple residuated lattice.
Remark 4.6. Consider the residuated lattice in Example 3.3. Since for all , , then is an MV-algebra.
Corollary 4.7. Every cyclic residuated lattice is a finite simple MV-algebra.
Corollary 4.8. Every cyclic residuated lattice is local and .
Theorem 4.9. Every finite simple MV-algebra is a cyclic residuated lattice.
Proof. Any simple MV-algebra is isomorphic to a subalgebra of , and also , () is the only subalgebra of with elements . Since , then . Therefore it is cyclic.
Corollary 4.10. A finite residuated lattice is cyclic if and only if it is a simple MV-algebra.
Theorem 4.11. Every nonzero subalgebra of a cyclic residuated lattice is cyclic.
Proof. Suppose that is a nonzero subalgebra of a cyclic residuated lattice . Then is a simple MV-algebra and is isomorphic to a subalgebra of . Moreover , () is the only subalgebra of with elements. Hence is a simple MV-algebra. By Theorem 4.9, is a cyclic MV-algebra.
Theorem 4.12. Every finite MV-algebra is a direct product of cyclic residuated lattices.
5. Semisimple Residuated Lattices
The intersection of the maximal filters of residuated lattice is called the radical of and will be denoted by .
Theorem 5.1. Let be a residuated lattice. Then for all .
Proof. See .
Theorem 5.2. Let be a residuated lattice. Then(1),(2) and are homomorphic topological spaces.
Proof. (1) It is easily seen that , thus
(2) Define , for all , . This function is well defined and surjective. For any and any , we have if and only if . We get that is injective.
Now, we will prove that is continuous and open. Let . By using the above, we get Thus is open. Since is injective and open, then . So is continuous.
Definition 5.3. A residuated lattice is called semisimple if the intersection of all congruences of is the congruence (where, for all , if and only if ).
Remark 5.4 (see ). A residuated lattice is semisimple if and only if .
Lemma 5.5. Let be a residuated lattice, , and . Then is a filter of .
Proof. (1) Suppose that . Then . We have
(2) Let , where and . We have . Since , then . Therefore and .
Hence is a filter of .
Theorem 5.6. Let be a residuated lattice such that for all there exists such that . Then is semisimple.
Proof. We will show that . Suppose that . By Theorem 5.1, we have that is nilpotent. By assumption, there exists such that . We have . Since is a filter of by Lemma 5.5 and is nilpotent, we get that . Thus . We obtain that and then is semisimple by Remark 5.4.
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