Elementary extensions of almost o-minimal structures

This paper investigates almost o-minimal structures, a weakening of o-minimality introduced by Fujita to capture structures that lie outside the classical o-minimal framework. In contrast to o-minimality and local o-minimality, almost o-minimality is not preserved under elementary equivalence. This raises the natural question of whether every almost o-minimal structure admits a proper elementary extension that is again almost o-minimal. The main result of this paper provides an affirmative answer to this question.