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The notions of union-soft semigroups, union-soft  l -ideals, and union-soft  r -ideals are introduced, and related properties are investigated. Characterizations of a union-soft semigroup, a union-soft  l -ideal, and a union-soft  r -ideal are provided. The concepts of union-soft products and union-soft semiprime soft sets are introduced, and their properties related to union-soft  l -ideals and union-soft  r -ideals are investigated. Using the notions of union-soft  l -ideals and union-soft  r -ideals, conditions for an ordered semigroup to be regular are considered. The concepts of concave soft sets and critical soft points are introduced, and their properties are discussed.

Concave Soft Sets, Critical Soft Points, and Union-Soft Ideals of Ordered Semigroups