The theory of subalgebra lattices for the class of arbitrary groupoids, that is, algebras with a binary operation, is investigated. It has earlier been proved that similar lattice theories for the classes of Abelian groups, of all groups, of monoids, and of semigroups admit an interpretation of elementary arithmetic. Thus, they are undecidable and have no recursive axiomatization. For the class of all groupoids, the question remained open, because the proof used the associativity of the binary operation. In the present paper, a new method which makes it possible to overcome this limitation is proposed; it applies to arbitrary groupoids. Thus, it is possible to interpret elementary arithmetic in the theory of subalgebra lattices for the class of all groupoids. Therefore, the theory of subalgebra lattices for the class of all groupoids is algorithmically undecidable and has no recursive axiomatization. Some results on the upper bound for the degree of undecidability of lattice theories for groupoids are also obtained.
S. M. Dudakov (Sun,) studied this question.