Quotient space theory is one of the most important models of granular computing.The structure in the existing quotient space model is usually a topology.If the domain structure is an algebraic structure

the equivalence relation in topological structure is no longer applicable.So the congruence relation in use is stricter than the equivalence relation.When a given relation is a non congruent equivalence relation

it introduces the concept of congruence closure and congruence interior to approximate the relation.The method

the properties and the conclusions are systematically demonstrated in the paper.On this basis

the algebraic quotient space model is systematically studied.The paper provides theoretical basis for solving complicated problems with the algebraic structure using quotient space theory.