Algorithmic recognizability of finiteness property of finite-definite systems

The object of this work is a property of finiteness of finite-definite systems, specified with a number of quasi-identities. The purpose of this work is to show in what cases algorithm determining finiteness such systems exists. Using algorithmic methods in universal algebra theory, basing on previous results in adjacent regions (S. I. Adyan, M. O. Rabin, on algorithmic properties of finite-definite groups and semigroups) there was a success in following results: in case of signature consist of one unary operation and some constants — there is an unified algorithm determining finiteness of finite-definite systems, in case of several unary operations or one binary operation with at least one constant — there is no unified algorithm.
Results of this research are able to be applied in treat of correctness of abstract data type description with quasi-identities.