  We consider a category of all finite partial orderings with quotient maps as
arrows and construct a Fra\"iss\'e sequence in this category. Then we use
commonly known relations between partial orders and lattices to construct a
sequence of lattices associated with it. Each of these two sequences has a
limit object -- an inverse limit, which is an object of our interest as well.
  In the first chapter there are some preliminaries considering partial orders,
lattices, topology, inverse limits, category theory and Fra\"iss\'e theory,
which are used later. In the second chapter there are our results considering a
Fra\"iss\'e sequence in category of finite posets with quotient maps and
properties of inverse limit of this sequence. In the third chapter we
investigate connections between posets and order ideals corresponding to them,
getting an inductive sequence made of these ideals; then we study properties of
the inverse limit of this sequence.
