Classification theory for a $1$-ary function

Abstract

The classification problem, namely the problem of characterizing classifiable theories, was (almost) thoroughly solved by S. Shelah. It involves an obvious algebraic question, namely to find, given an elementary class K of 1st order structures, under which conditions a structure M∈K satisfies «Th(M) classifiable». This is the question we deal with for the class K of all structures M with a 1-ary (total) function