On the inexpressibility a linear order by means of non-deterministic programs in groupoids of a special type

In this paper we consider a special class of groupoids, which was presented in the article by M.A.Taitslin "Пример полиномиального запроса не распознаваемого в недетерминированной логарифмической памяти". In this paper we study a question of unexpressibility of linear order on the elements of such groupoid using non-determined programs. It is shown, that the question of linear order unexpressibility can be reduced to the question of unsolvability of some equations system in four-element quasi-group.