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
