Discussion and research on some properties of congruence relations and compatible relations 同余关系和相容关系的一些性质讨论和研究
Pointed out a family of congruence relations induced by a fuzzy filter of a lattice implication algebra and proved that this family of congruence relations is a complete chain , and then discussed the relations between the quotient lattice implication algebras induced by this family of congruence relations . 3 . discussed the property of fuzzy li - ideals , pointed out a family of congruence relations induced by a fuzzy li - ideal of a lattice implication algebra and proved that this family of congruence relations is a complete chain , and then discussed the relations between the quotient lattice implication algebras induced by this family of congruence relations 对模糊li -理想的性质作了进一步的研究，证明了利用格蕴涵代数上的一个模糊li -理想可以得到格蕴涵代数上的一簇同余关系，证明了这簇同余关系构成一个完备链，在此基础上讨论了这簇同余关系诱导的格蕴涵商代数之间的关系，并用一个例子说明了这簇同余关系的确可以包含更多的同余关系。
Moreover , in a group , the concepts of fuzzy rough subgroups and fuzzy rough normai subgroups are first given . under the condition of the congruence relation determined by a given normal sungroup in a group , fuzzy rough sets of a fuzzy subsgroup is proved to be its fuzzy subgroup , while that of a fuzzy normal one is also proved to be its fuzzy normal subgroup 接着，在群中提出了模糊粗糙子群和模糊粗糙正规子群的概念，证明了在群中一固定的正规子群所决定的同余关系下，模糊子群的模糊粗糙集是模糊子群，模糊正规子群的模糊粗糙集是模糊正规子群。
Moreover , rough sets in a ring is studied and the concepts of rough subrings , rough ideals are been first introduced . under the condition of the congruence relation determined by a given ideal in a ring , rough sets of a subring is proved to be its subring , while that of a ideal is also proved to be its ideal 进一步，研究了环中的粗糙集，引入了粗子环和粗理想的概念，证明了在环中一固定的理想所决定的同余关系下，子环的粗糙集是子环，左(右，双)理想的粗糙集是左(右，双)理想。
Rough ideals in a semigroup have been studied and the concepts of rough subsemigroups , rough ideals been first introduced by kuroki n . under the condition of the congruence relation , rough sets of a subsemigroup was proved to be its subsemigroup , while that of a left ( right , bisides ) ideal was also proved to be its left ( right , bisides ) ideal . next , rough sets in a group have been studied and the concepts of rough subgroups , rough nornal subgroups been first introduced Kurokin研究了半群中的粗理想，首次提出了粗子半群和粗理想的概念，证明了在同余关系下，半群的粗糙集是半群，左(右，双)理想的粗糙集是左(右，双)理想。接着，他又研究了群中的粗糙集，首次提出了粗子群和粗正规子群的概念，证明了在群中一固定的正规子群所决定的同余关系下，子群的粗糙集是子群，正规子群的粗糙集是正规子群。
Fonally , in a ring , the corresponding results of a semigroup and a group are generalized . fuzzy rough sets in a ring is studied and the concepts of r fuzzy rough subrings , fuzzy rough ideals are been first introduced . under the condition of the congruence relation determined by a given ideal in a ring , fuzzy rough sets of a fuzzy subring is proved to be its fuzzy subring , while that of a fuzzy ideal is also proved to be its fuzzy ideal 最后，把半群和群中的相应结果推广到来了环中，引入了模糊粗糙子环和模糊粗糙理想的概念，得出在环中一固定的理想所决定的同余关系下，模糊子环的模糊粗糙集是模糊子环，模糊左(右，双)理想的模糊粗糙集是模糊左(右，双)理想。
