We compare the monadic second-order theory of an arbitrary linear ordering L with the theory of the family of subsets of L endowed with the operation on subsets obtained by lifting the max operation on L. We show that the two theories define the same relations. The same result holds when lifting the min operation or both max and min operations.
Choffrut, C., & Grigorieff, S. (2015). Monadic theory of a linear order versus the theory of its subsets with the lifted min/max operations. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9300, pp. 109–128). Springer Verlag. https://doi.org/10.1007/978-3-319-23534-9_6