On the complexity of posets

The purpose of this paper is to discuss several invariants each of which provides a measure of the intuitive notion of complexity for a finite partially ordered set. For a poset X the invariants discussed include cardinality, width, length, breadth, dimension, weak dimension, interval dimension and semiorder dimension denoted respectively X, W(X), L(X), B(X), dim(X). Wdim(X), Idim(X) and Sdim(X). Among these invariants the following inequalities hold. $\text{B(}\text{X}\text{)?Idim(}\text{X}\text{)?Sdim(}\text{X}\text{)?Wdim(}\text{X}\text{)?dim(}\text{X}\text{)?W(}\text{X}\text{)}$. We prove that every poset X with three of more points contains a partly with $\text{Idim}\text{(}\text{X}\text{)}\text{Idim}\text{(}\text{X}\text{) {x,v})}$. If M denotes the set of maximal elements and A an arbitrary anticham of X we show that $\text{Idim}\text{(}\text{X}\text{)?W(}\text{X-M}\text{)}$ and $\text{Idim}\text{(}\text{X}\text{)?2W(}\text{X-A}\text{)}$. We also show that there exist functions f(n,t) and (gt) such that $\text{I(}\text{X}\text{)?n}$ and $\text{Idim}\text{(}\text{X}\text{)?t}$simply $\text{dim}\text{(}\text{X}\text{)?f(n,t and Sdim(}\text{X}\text{)?t}$ implies $\text{dim}\text{(}\text{X}\text{)?g(t)}$.