Order Ideals on Lexicographic Direct Sum of Three Totally Ordered Abelian Groups

Order ideals play an important role in the study of abstract algebra, especially in the study of ordered groups. In this paper, we focus on the study of order ideals in lexicographic direct sums of totally ordered Abelian groups. We begin by examining the order ideals in the group of integers , and the group of real numbers  It is shown that there are no non-trivial order ideals in both groups. Next, we revisit the order ideals in the lexicographic direct sum of two totally ordered Abelian groups,  The only non-trivial order ideal of  is  Furthermore, our study extends to the lexicographic direct sum of three totally ordered Abelian groups:  and  We investigate the non-trivial order ideals in these structures. It is stated that the non-trivial order ideals of  are only  and  Furthermore, the non-trivial order ideals of  are only  and .Keywords: order ideal; lexicographic order; direct sum.