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Description

The contents of this talk can be considered as intractions of logic and general algebra, also can be considered as part of model theory in mathematical logic. The main idea is to find the finite algebraic computations in the algebraic structures. More than two decades ago, Stanley Burris ( Univ. waterloo ) and Joel Berman (UCI) invented a logic notion on algebraic structures called Definable Principal Congruences (DPC) . Unfortunately , Ralph Mckenzie (UC Berkeley) et al. had given examples to show that even in the varieties of lattice, this property does not hold. We invented the notion DPSC and prove that this logical property widely exist in many algebraic varieties. Also , we will present a result on finite groups: the class of subdirectly irreducible groups in the variety generated by any finite group is first-order language definable. And we will explain how those results will lead to a solution of a long-standing conjecture in the finite base study on algebraic structures.