On direct decompositions of elements and on Goldie numbers in balanced lattices
Let \(L\) be a non-trivial lattice with 0. A subset \(X\subseteq L\backslash\{0\}\) is said to be \(g\)-independent if for all \(x\in X\), \(x\land (\lor(X_ 1\backslash\{0\}))= 0\), where \(X_ 1\) is an arbitrary finite subset of \(X\). An element \(x\in L\backslash\{0\}\) is called \(d\)- indecomposable if the conditions \(x= x_ 1\lor x_ 2\), \(x_ 1\land x_ 2=0\) imply \(x_ 1= 0\), \(x_ 2= x\) or \(x_ 1= x\), \(x_ 2= 0\). The decomposition of \(x\in L\backslash\{0\}\) in the form \(x= x_ 1\lor\cdots\lor x_ n\) is a direct one if \(\{x_ 1,\dots,x_ n\}\) is \(g\)-independent. This is denoted by \(x= x_ 1\dot\lor\cdots \dot\lor x_ n\). If \(x= x_ 1\lor\cdots\lor x_ n\) is a direct decomposition and \(x_ 1,\dots,x_ n\) are \(d\)-indecomposable, then this representation of \(x\) is called a finite \(d\)-decomposition. The following theorems form our starting point: Theorem A. If \(L\) is a modular lattice of finite length, then any non- zero element \(x\in L\) possesses a finite \(d\)-decomposition. Theorem B. (Ore) Let \(L\) be a modular lattice of finite length. If an element \(x\in L\) possesses two \(d\)-decompositions \(x= x_ 1\dot\lor\cdots \dot\lor x_ n\) and \(x= y_ 1\dot\lor\cdots \dot\lor y_ m\), then: 1) \(n= m\); 2) for any element \(x_ i\) there is an element \(y_ j\) such that \(x= x_ 1\dot\lor\cdots \dot\lor x_{i-1}\dot\lor y_ j \dot\lor x_{i+1}\dot\lor\cdots \dot\lor x_ n\). Our aim is to extend Theorems A and B to a wider class of lattices. Such a class is the class of balanced lattices possessing a finite Goldie number. We say that a lattice is balanced (from below) if for any elements \(x,y,z\in L\), the conditions \(y\land z=0\), \(x\land(y\lor z)= 0\) imply \(y\land (x\lor z)= 0\).
scientific article; zbMATH DE number 640177