Arithmetic for octonionic-closed balls

With inspiration given by circular complex interval arithmetic, an algebraic framework for octonionic-closed balls – here called arithmetic for octonionic-closed balls – is proposed. More concretely, an addition and two multiplications are defined on octonionic-closed balls and analyzed. Diverse algebraic properties, such as (anti-)commutativity, (power-)associativity, existence of neutral element and reciprocal of each element, and its square root(s), are studied. Furthermore, inclusion monotonicity – the basis for several applications of interval arithmetic – and the (sub)distributivity of each multiplication relative to the addition, are explored. Throughout the manuscript, the underlying structure of composition algebra of the involved octonion algebra plays a key role in the results. In particular, the multiplications on octonionic-closed balls under consideration are both related to the multiplication of this octonion algebra. The results highlight how interval-like objects behave in the non-associative setting of octonions, giving rise to rich algebraic structures with potential applications in generalized interval analysis.