A generalized theory of ideals and algebraic substructures

In 2011, a topic containing the concepts of upper and lower periodic subsets of (basic) algebraic structures was introduced and studied. The concept of ``upper periodic subsets'' can be considered as a generalized topic of ideals and sub-structures (e.g., subgroups, sub-semigroups, sub-magmas, sub-rings, etc.). Hence, it can be improved to a theory in every algebraic structure which extends many basic concepts including the ideals. This paper follows the mentioned goals and studies related to algebraic and topological aspects. For this purpose, first, we state an improved fundamental theorem about the unique direct representation of left upper periodic subsets and then introduce some extensions and generalizations. As a result of the study, we classify sub-semigroups and subgroups of real numbers and introduce some related topics and concenterable upper periodic subsets. We end the study with many research projects and some future directions of the theory.