The articles [15], [18], [19], [21], [20], [7], [8], [10], [1], [2], [6], [14], [11], [16], [12], [17], [3], [4], [23], [9], [5], [22], and [13] provide the terminology and notation for this paper. Let L be a relational structure. We introduce L as a synonym of L`. We now state several propositions: (1) For every relational structure L and for all elements x, y of L holds x ¬ y iff xx ­ xy. (2)...

Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...