A SURVEY OF DIMENSION OF POSETS

In 1930, E. Szpilrajn proved that any order relation on a set $X$ can be extended to linear order on $X$. It also follows that order relation is the intersection of its linear extensions. In 1941 Dushnik and Miller introduced the concept of dimension of a poset $(X,P)$, denoted $Dim(X,P)$, as the smallest positive integer $n$ for which there exist linear extensions $L_1, L_2, ..., L_n$ of $P$ so that $P=L_1\cap L_2\cap ...\cap L_n$.In this paper, we do literature survey of the research articles on dimension theory of posets. This survey provides an overview of key concepts, results, and challenges in dimension theory of posets. The survey also examines advanced topics like fractional dimension, geometric representations, and topological considerations. Open problems and future research directions, emphasizing the interdisciplinary potential of dimension theory in mathematics and computer science.