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We consider a category of all finite partial orderings with quotient maps asarrows and construct a Fra\"iss\'e sequence in this category. Then we usecommonly known relations between partial orders and lattices to construct asequence of lattices associated with it. Each of these two sequences has alimit 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 aFra\"iss\'e sequence in category of finite posets with quotient maps andproperties of inverse limit of this sequence. In the third chapter weinvestigate connections between posets and order ideals corresponding to them,getting an inductive sequence made of these ideals; then we study properties ofthe inverse limit of this sequence.

An inverse Fra\"iss\'e limit for finite posets and duality for posets and lattices