Nilpotent Elements of Residuated Lattices

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. 1. Introduction Ward and Dilworth [1] 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. 2. Preliminaries 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 . Proposition 2.1 (see [5, 9]). Let be a residuated lattice. Then one has the following properties: for all ,(1) if and only if ,(2) , ,(3) , ,(4) ,(5) ,(6) ,(7)if , then and . 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