semihypergroups

Semihypergroups are algebraic structures in which the product of two elements is not a single element but a nonempty subset of the underlying set. They generalize semigroups by replacing the binary operation with a hyperoperation that yields a set of possible outcomes.

Formally, let A be a nonempty set and define a hyperoperation ◦: A × A → P(A)\{∅}, where P(A) is the power set of A. The pair (A, ◦) is a semihypergroup if for all a, b, c in A the associativity condition holds in the hyper sense: a ∘ (b ∘ c) = (a ∘ b) ∘ c, where a ∘ (b ∘ c) is defined as ⋃_{x ∈ b ∘ c} a ∘ x and (a ∘ b) ∘ c as ⋃_{y ∈ a ∘ b} y ∘ c. The products are interpreted as unions of outcomes.

Relation to other structures: If every a ∘ b is a singleton {ab}, then (A, ◦) is a semigroup. Semihypergroups are a type of hyperstructure; hypergroups extend this idea with additional axioms such as reversibility and identity-like structures and often involve more elaborate convolution-like operations.

Examples: 1) Any semigroup (A, ⋅) can be viewed as a semihypergroup by setting a ∘ b = {a b}. 2) Let A = {0,1} and define a ∘ b = {a, b} for all a,b ∈ A. This operation is associative in the hyper sense, making (A, ∘) a semihypergroup.

Notes: Research on semihypergroups focuses on substructures, homomorphisms, and representations, with connections to algebraic combinatorics and multivalued algebra. See also hypergroup, semigroup, and hyperoperation.