ON THE DIMENSION OF AMALGAMATED ORDERED SETS

The dimension problem has been one of central themes in the theory of ordered sets. In this paper we focus on amalgamated ordered sets. Although some results can be obviously applied to infinite cases, we assume throughout that all ordered set are finite. If A and B are ordered sets whose orders agree on A.cap.B, then the amalgam of A and B is defined to the the set A.cup.B in which the order is the transitive closure of the union of the two orders, i.e., the smallest order containing the two orders, and is denoted by A .or. B .leq. dim A + dim B for any ordered sets A and B. But it is quite surprising that the dimension of the amalgam of certain 2-dimensional ordered sets can be arbitrarily large.