Ordinal sum and its application in BE-algebras

This paper aims to develop a simultaneous study of ordinal sums in BE-algebras and to apply this construction to several central notions in the theory. We first introduce the ordinal sum of a totally ordered family of BE-algebras and establish that it yields a well-defined BE-algebra in which each component embeds naturally. A detailed analysis of the resulting structure is provided, including explicit descriptions of its filters, subalgebras, and congruence relations. Building on these foundations, we characterize the behavior of dense BE-algebras, multipliers, stabilizers, and closure operators in ordinal sums, thereby refining and extending several earlier results in the literature. We provide a representation for dense BE-algebras as the ordinal sum of [Formula: see text], where [Formula: see text] is the unique two-element BE-algebra and [Formula: see text] is an arbitrary BE-algebra. This representation leads us to a categorically equivalent relation between dense BE-algebras and BE-algebras. The ordinal sum construction thus serves as a unifying framework for understanding the internal organization of BE-algebras. It offers new tools for creating various examples and transferring structural properties between their components.