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A non-deterministic hypersubstitution maps operation symbols to sets of terms of the corresponding arity. A non-deterministic hypersubstitution of type τ is said to be linear if it maps any operation symbol to a set of linear terms of the corresponding arity. We show that the extension of non-deterministic linear hypersubstitutions of type τ map sets of linear terms to sets of linear terms. As a consequence, the collection of all nondeterministic linear hypersubstitutions forms a monoid. Non-deterministic linear hypersubstitutions can be applied to identities and to algebras of type τ .

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Ideals play an essential part in studying ordered semigroups. There are several generalizations of ideals that are used to investigate ordered semigroups. It was known that (m, n)-ideals and n-interior ideals are an abstraction of bi-ideals and interior ideals, respectively. This paper introduces a generality of (m, n)-ideals and n-interior ideals, so-called (α, β)-fuzzy (m, n)-ideals and (α, β)-fuzzy n-interior ideals. Furthermore, we discuss our current notions with those that already exist. We examine connections between (m, n)- (resp., n-interior) ideals and (α, β)-fuzzy (m, n)- (resp., n-interior) ideals. A characterization of (α, β)-fuzzy (m, n)- (resp., n-interior) ideals, by a particular product, in ordered semigroups is provided. We demonstrate that our results generalize the known results through specific settings.