Modal Characterization of the Classes of Finite and Infinite Quasi-Ordered Sets

To formulate the main aim of this paper we will begin with some definitions and notations. A system $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{U} $$=(U,≤), where U≠Ø and ≤ is a binary relation in U, is called quasi-ordered set — qoset for short, — if the relation ≤ is reflexive and transitive one. By QO, QO_fin and QO_inf we will denote respectively the class of all qosets, all finite qosets and all infinite qosets. If U is a qoset and in addition the relation ≤ is an antisymmetric one /x≤y & y≤x → x=y/ then U is called partially ordered set — poset for short. By PO, PO_fin and PO_infwe denote respectively the class of all posets, all finite posets, and all infinite posets. It is a well known fact that the modal logic S4 is complete in the classes QO and QO_fin. So QO and QO_fin cannot be separated by S4. However, the situation is different when we consider posets: S4 is complete in the class PO_inf but not in the class PO_fin and S4Grz / the so called Grzegorczik system/ is complete in PO_fin but not in PO_inf /see [l]/. Roughly speaking, we will say in this case, that S4 is a modal characterization of PO_inf and S4Grz is a modal characterization of PO_fin.