Some Studies About Congruences On Eventually Regular Semigroups

This thesis is mainly devoted to study the R-unipotent congruence and orthodox congruence on an eventually regular semigroup,regular congruence on E-inversive semigroups and congruences and operators on the lattice of subvar eties of periodic semigroups varieties.Details are as follows:First,we generalized the kernel-trace congruence pairs approach about regular semigroup to eventually regular semigroup by using weak inversive,introduced the definitions of R-unipotent congruence pairs and orthodox congruence pairs for eventually regular semigroup S and characterized the R-unipotent congruence(orthodox congruence)on such a semigroup by means of R-unipotent congruence pair(orthodox congruence pair).We proved that the R-unipotent congruence(orthodox congruence) on an eventually regular semigroup S is uniquely determined by its R-unipotent congruence pair(orthodox congruence pair)for such a semigroup and established an order preserving one-one correspondence between the set of all R-unipoten(?)congruence(orthodox congruence)on S to the set of all R-unipotent congruence pairs(orthodox congruence pairs)for S.These results generalized congruence theory on a regular semigroup of Gomes[21]and[22].Next.we are concerned with the regular congruence on an E-inversive semigroup, generalized the Green relation on a semigroup by using of the weak inversive. Depended heavily on these relations we studied the relationships between the regular congruence and its kernel and trace on such a semigroup,and introduced kernel-trace congruence pairs for E-inversive semigroup.We obtained that the regular congruence on E-inversive semigroup is uniquely determined by its regular congruence pair,and given out a abstract and distinct description of regular congruence by means of regular congruence pairs on E-inversive semigroup.These results generalized the correspond- ing results of congruence on a regular semigroup.Finally,parallel to the results of finite semigroup and pseudovarieties of semigroups we introduced the concepts of preimage-class and radical congruence system on a periodic semigroup,established the complete congruences and theirs corresponding operators,and obtained the characterization and description of the complete congruences and theirs corresponding operators on periodic semigroup.