Structure and order: A concise overview of ordered semi(hyper)groups

This paper explores a hierarchy of algebraic structures beginning with ordered semigroups and extending through their generalizations and fuzzifications. An ordered semigroup is a semigroup equipped with a partial order that is compatible with its binary operation. Building on this foundation, we investigate fuzzy ideals within ordered semigroups, where fuzzy set theory introduces gradation and uncertainty into ideal theory. We then extend the framework to ordered semihypergroups, where the binary operation is replaced by a hyperoperation mapping element pair to subsets, allowing the modeling of systems with multi-valued outputs under an order-preserving structure. From there, we examine methods for constructing ordered semigroups derived from ordered semihypergroups, using representative elements and congruences to recover associativity in a classical sense. Finally, we introduce the fuzzification of ordered semihypergroups, integrating fuzzy logic into hyperoperations to develop fuzzy hyperideals, hyperfilters and explore their properties. This comprehensive study highlights the interplay between order, hyperstructure, and fuzziness, providing a generalized framework for both theoretical advancement and practical applications in areas involving uncertainty and complex relationships.