Some Studies Of Congruences And The Translational Hulls On The G-generalized Regular Semigroups

A semigroup S is called an E-semiabundant,if S is L-abundant and R-abundant.A semigroup S is called an E-super semiabundant,if S is H-abundant,R is a left congruence and L is a right congruence of S.An E-super semi abundant semigroup S is called a normal E-super abundant semigroup,if the set of all idempotents of S forms a normal band.The E-semiabundants and the E-super semiabundants are respectively natural extensions of the regular semigroups and the completely regular semigroups.Recently,the research of E-semiabundants and their subclasses has attracted great attention from many scholars at home and abroad.In this paper,the theories of good congruences on the normal E-super abundant semigroup and the super r-wide semigroup are studied,and the theory of the translational hull of the strong r-semiadequate is studied.First of all,the concept of a(?)-good congruence pair is introduced,and it is proved that if(?,?_?)is a(?)-good congruence pair for a normal E-super abundant semigroup S=[Y;S?,??,?].then the relation ?(?,?_?)on S defined by:??(?,?_?)b if and only if ???,???,???_??b??,??is the unique(?)-good congruence on S induced by ? on Y and ?_? on Sa.Conversely,if ? is a(?)-good congruence on S.then(?VD/D,?|S?)is a(?)-good congruence pair on S,and ?(?VD/D,?|S?)=?.Some results about congruences on normal orthogroups on regular semigroups are generalized to E-semiabundant semigroups.Secondly,(*,?)-good congruences on the super r-wide semigroup are studies,some properties and equivalent characterizations of(*,?)-good congruences on the super r-wide semigroup are given.Finally,it is proved that the translational hull of the strong r-semiadequate satisfying R is a left congruence is still strong r-semiadequate.