w-independent quasi-equational basis of differential groupoids

Abstract

The search for the solutions of the finite basis problem was and still is under the influence of
the problem of Alfred Tarski (Tarski, 1966: 275–288), who asked in 1966: if there is exists an
algorithm for deciding whether the quasi-equational theory of a finite set of finite algebras which
is assumed additionally to be equational, is finitely based. Tarski’s problem has been solved in
the negative by Ralph McKenzie (McKenzie, 1996: 49–104). The negative solution of Tarski’s
problem actually makes the finite basis problem more interesting and worthy of continued effort.
If Tarski’s problem had had a positive solution, the status of the finite basis problem would be
totally different. It would probably still exist, however, the main stream of the scientific effort would
go toward improving the known algorithms and classifying their complexity. However, the main
stream of the scientific effort would go toward improving the known algorithms and classifying
their complexity. We want to stress that the analogue of Tarski’s problem for the quasi-equational
theories that are not equational is unsolved. There is, however, a common believe that it will
also have a negative answer. It has been established at the end of the 1990-ties by a number of
researchers marvelous proof for the negative solution of Tarski’s problem cannot be easily modified
for quasi-equational theories. So today independent bases of quasiidentities have been found for
many classes of algebras and models. We will note that recently in work (Kravchenko, 2017с) has
found the general and sufficient condition for existence of a continuum of quasivarieties without
independent basis of quasiidentities, but with w - independent bases of quasiidentities. However
differential gruppoids aren’t sutisfied to these conditions and in this work we continued to study
the an independent quasiaquational basis of quasivarieties of differential gruppoids. The main
result is exists a continuum of quasivarieties of differential gruppoid which have w - independent
basis of quasiidentities.