Congruences On E-bisimple Semigroups

In this paper, we first introduce the achivements of the research on Bruck-Reilly semigroups in these years, Then we study block-separating congruences lattices on E-bisimple semigroups. Let / be the set of natural numbers, and I the set of all nonnegative integers. Let T be a monoid with identity 1, and G the group of units in T. An E-bisimple semigroup is a bisimple semigroup with the set of idempotents being an w-chain of right zero semigroups and for ei Ei and fj Ej,i > j implies ei < fj. Let G be a group and 9 a endomorphism of G, and P, K be two sets. A product is defined on ((I x {0}) x (G x P)) ∪ ((I x I) x (G x K)) as follows,where 7 is a homomorphism from G into the symmetric group on K. Then ((I x {0}) x (G x P)) ∪ ((I x I) x (G x K)) is a semigroup under the aboveproduct, which is denoted as S(G, P, K,θ γ). Warne proved in[12] that any E-bisimple semigroup S has structure as S(G, P, K, θ,γ). and in [13] proved that a congruence on an E-bisimple semigroup is a group congruence or is block-separating congruence.In the second chapter,We study group congruences lattices on E-bisimple semigroups,and we determined completely the kernel of the group congruenecs, since we describe the group congruences on E-bisimple semigroup .The meet and join of two group congruences are also characterized.In the third chapter,We study block-separating congruences lattices on E-bisimple semigroups. Let S = S(G, P, K, θ,γ) be an E-bisimple semi-group, N a θ-adimissble normal subgroup of group G, A and 6 equivalences on P and K respectively. If for any and (k, r) implies then is called a admissible triple of S. The set of admissible triples of S will be denoted by AT(S).Let (N, @, ^) AT(S), define a relation p(N,@,^) as follows,Then is block-separating congruence on S and any block-separating congruence p can be obtained in this way as The set of all block-separating congruences on S is denoted by CBS(S). Define a relation W on CBS(S) below,we determine the greatest and smallest congruences in T-classes and W-classes .