# w-independent quasi-equational basis of differential groupoids

## DOI:

https://doi.org/10.26577/jmmcs-2017-3-465## Keywords:

quasiidentity, quasivariety, quasi-equational bases, independent quasi-equational bases, 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.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*95*(3), 21–31. https://doi.org/10.26577/jmmcs-2017-3-465