kingroups

Kingroups are a proposed algebraic structure in abstract algebra that generalize the notion of a group by equipping a set with a family of binary operations instead of a single multiplication. A kingroup consists of a nonempty set G together with a collection {⊗_k | k ∈ K} of binary operations G × G → G, indexed by a nonempty index set K (often the natural numbers). A distinguished element e ∈ G, called the king, serves as a common identity: e ⊗_k g = g ⊗_k e = g for all g ∈ G and all k ∈ K. Each operation ⊗_k is required to make G into a group on its own, with its own inverse function, but the family is linked by coherence laws.

One common coherence requirement is that there exists a function φ: K × K → K such that for all g,h,i ∈ G and m,n ∈ K, g ⊗_m (h ⊗_n i) = (g ⊗_φ(m,n) h) ⊗_n i, providing a way to pass between different operation indices during associative regrouping. Substructures consisting of subsets closed under every ⊗_k are called kingroupsubgroups or simply subkingroups.

Examples and variants: If all ⊗_k are identical, a kingroup reduces to an ordinary group. If only a finite subset of operations is used, one obtains a partial kingroup. Variants include commutative kingroups where every pair of operations commute, and left/right kingroups with direction-specific associativity. Applications: Kingroups serve as a framework for modeling systems with multiple kinds of composition, such as multi-relational networks, layered symmetry, or clone-theoretic algebraic theories. The concept is primarily of theoretical interest and can connect with studies in universal algebra and category theory.