Height on posets and graphs

A height function h is defined for any connected poset P with no infinite chains. A regularity condition which we call the minimal path condition is also defined. It is shown that the minimal path condition is equivalent to the gradability of P if h(x) = h(l) + h(u). It is shown that a poset P can be characterized by binary operations U, @? on the power set of P which generalize the usual inf and sup for lattices. A structure theorem for upper and lower semimodular posets is proved and some results are obtained concerning modular pairs in a poset. Finally, some applications of the above theory are given for digraphs.