RESIDUATED ALMOST DISTRIBUTIVE LATTICES WITH MAXIMAL ELEMENTS SATISFYING ASCENDING CHAIN CONDITIONS

Abstract

The purpose of this project study is Residuated Almost Distributive Lattices with Maximal Elements satisfying Ascending Chain Conditions (A.C.C.). First, important preliminary conce pts, examples, lemmas and theorems were presented to make the concept of ADL with maximal element m to satisfying A.C.C., and we discuss some important properties of residuation ‘:’ and multiplication ‘.’ in a Residuated Almost Distributive Lattice (R-ADL) L. We study important results in a residuated ADL L. In this project; we discuss the concepts of residuation and multiplication in an Almost Distributive Lattice (ADL) L and also discuss a residuated ADL. We study those important relations between residuation and multiplication in an ADL L. Additionaly, discuss the concepts of the radical of an element and a p-primary element in a complete ADL L with a maximal element m is introduced and important properties of radical of an element in a complete residuated ADL are stuieded. And also, we show the idea of a simple element in a residuated ADL and the concept of Principal Residuated Almost Distributive Lattice (P-ADL). We discuss important results in a P-ADL. Moreover, we descriabe the concept of meet representation of an element in an ADL L with ascending chain conditions (a.c.c.). Further, we discuss decomposition theorems in a P-ADL, the concepts of normal primary decomposition and isolated component of an element a in a complete residuated ADL L. We also discuss the fundamental theorem on primary decompositions. Important properties of primary elements in a complete residuated ADL L and the uniqueness theorem in a complete complemented residuated ADL L are discussed