w-independent quasi-equational basis of differential groupoids (Journal of Mathematics Mechanics and Computer Science, 2018)
The search for the solutions of the finite basis problem was and still is under the influence ofthe problem of Alfred Tarski (Tarski, 1966: 275–288), who asked in 1966: if there is exists analgorithm for deciding whether the quasi-equational theory of a finite set of finite algebras whichis assumed additionally to be equational, is finitely based. Tarski’s problem has been solved inthe negative by Ralph McKenzie (McKenzie, 1996: 49–104). The negative solution of Tarski’sproblem 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 betotally different. It would probably still exist, however, the main stream of the scientific effort wouldgo toward improving the known algorithms and classifying their complexity. However, the mainstream of the scientific effort would go toward improving the known algorithms and classifyingtheir complexity. We want to stress that the analogue of Tarski’s problem for the quasi-equationaltheories that are not equational is unsolved. There is, however, a common believe that it willalso have a negative answer. It has been established at the end of the 1990-ties by a number ofresearchers marvelous proof for the negative solution of Tarski’s problem cannot be easily modifiedfor quasi-equational theories. So today independent bases of quasiidentities have been found formany classes of algebras and models. We will note that recently in work (Kravchenko, 2017с) hasfound the general and sufficient condition for existence of a continuum of quasivarieties withoutindependent basis of quasiidentities, but with w - independent bases of quasiidentities. Howeverdifferential gruppoids aren’t sutisfied to these conditions and in this work we continued to studythe an independent quasiaquational basis of quasivarieties of differential gruppoids. The mainresult is exists a continuum of quasivarieties of differential gruppoid which have w - independentbasis of quasiidentities.
Keywords: quasiidentity; quasivariety; quasi-equational bases; independent quasi-equational bases; differential groupoids