We derive author's discrete forms of the Beckman-Quarles theorem from the definability results of Raphael M. Robinson. The classical Beckman-Quarles theorem states that each map from R n to R n (n > 1) preserving all unit distances is an isometry, see [1]-[3]. In this note we derive author's discrete forms of this theorem ([6]) from the definability results of Raphael M. Robinson (see [4] and a...

Bisemigroups are algebras equipped with two independent associative operations. Labeled finite sp-biposets may serve as a possible representation of the elements of the free bisemigroups. For finite sp-biposets, an accepting device, called parenthesizing automaton, was introduced in [6], and it was proved that its expressive power is equivalent to both algebraic recognizability and monadic seco...

Regular synchronization languages can be used to define rational relations of finite words, and characterize subclasses relations, like automatic or recognizable relations. We provide a systematic study the decidability uniformization definability problems for defined in terms such languages. rephrase known results this setting complete picture by adding several new undecidability results.