We show that every function f: A × B → A × B, where |A| ≤ 3 and |B| < ω, can be represented as a composition f₁ ∘ f₂ ∘ f₃ ∘ f₄ of four axial functions, where f₁ is a vertical function. We also prove that for every finite set A of cardinality at least 3, there exist a finite set B and a function f: A × B → A × B such that f ≠ f₁ ∘ f₂ ∘ f₃ ∘ f₄ for any axial functions f₁, f₂, f₃, f₄, whenever f₁ is a horizontal function.
In this paper, we introduce six basic types of composition of ternary relations, four of which are associative. These compositions are based on two types of composition of a ternary relation with a binary relation recently introduced by Zedam et al. We study the properties of these compositions, in particular the link with the usual composition of binary relations through the use of the operations of projection and cylindrical extension.
We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1 decidable computably categorical structure is relatively Δ⁰₂ categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable...
We use ideas and machinery of effective algebra to investigate computable structures on the space C[0,1] of continuous functions on the unit interval. We show that (C[0,1],sup) has infinitely many computable structures non-equivalent up to a computable isometry. We also investigate if the usual operations on C[0,1] are necessarily computable in every computable structure on C[0,1]. Among other results, we show that there is a computable structure on C[0,1] which computes + and the scalar multiplication,...
The Rabin index of a rational language of infinite words given by a parity automaton with n states is computable in time O(n2c) where c is the cardinality of the alphabet. The number of values used by a parity acceptance condition is always greater than the Rabin index and conversely, the acceptance condition of a parity automaton can always be replaced by an equivalent acceptance condition whose number of used values is exactly the Rabin index. This new acceptance condition can also be...