Upward Saturated Hyperclones

Hyperoperations are mappings that assign a non-void subset of a set to each n-tuple of elements from the set. They present a fundamental notion for modeling non deterministic processes. Even though for each non deterministic automaton there is a language equivalent deterministic automaton, they do not have the same behavior. As behavior of concurrent processes took place in emerging fields of theoretical computer science, the necessity for development of algebraic theory of hyperstructures significantly increased. In [9] and [10] Rosenberg introduced the notion of a hyperclone as a subuniverse of the algebra of hyperoperations (HA, ◦, ζ, τ, , π2 1 ) whose fundamental operations are generalization of Mal’tsev-type operations and a projection. It can equivalently be defined as a composition closed set of hyperoperations containing all projections. The set of all hyperclones ordered by set inclusion forms an algebraic lattice. The least element and the greatest element of the lattice are the set of all projections and the set of all hyperoperations, respectively. The lattice has continuum cardinality even on a twoelement set (see [4]).

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