Congruences And Semi-direct Product On Some Generalized Regular Semigroup

Gomes described the R-unipotent congruence on a regular semigroup S by means of its kernal and hyper-trace. In this paper , we describe the r-semiprime R-unipotent congruence and the r-semiprime E-left normal congruence on a strong -regular semigroup. In the second part, we give a necessary and sufficient condition for relations to be a congruence on some special strong T-regular semigroups. At last, we discuss the congruence on orthodox semirings by Kernal normal system.Main Result:Defination 2.1. Let S be a strong 7r-regular semigroup, call its subsemi-group K is normal if K is full( E(S) C K), self-conjugate(for all a € RegS a 6 V(a),aKa C K} and satisfies:Defination 2.2. If semigroup S is strong T-regular semigroup, call congruencef on (E(S)) is normal if for VDefination 2.3. Let K is a normal subsemigroup of a strong 7r-regular semigroup, ?is a normal left-regular band congruence on (E(S)), then (, K) is called r-semiprime R-unimpotent congruence pair, if (, K) satifies: for Va, b 5, a' If is r-semiprime R-unipotent congruence pair as defined above, define relation P as follows:NOTES: Let RRCP(S) represent r-semiprime R-unipotent congruence pair and RRC(S) represent r-semiprime R-unipotent congruence, briefly noted P(,K) -P-Theory 2.6. Let S be a strong 7r-regular semigroup, If (,K) 6 RRCP(S), then p(S), andConversely, if p e RRC(S},then(htrp,Kerp) 6 RRCP(S)andp = P(Theory 3.6. Let S be a strong TT- regular semigoup and Ve, / 6 E(S), existsn € N, such as (ef)n = (e/)n+1, then the necessary and sufficient condition of to be a congruence are T = is a band of 7r-group and for Va, 6 6Defination 4.6. The set B = {B,- : i 6 /} is called regular kernal normal system of orthodox semiring S, if(Kl) every Bi is regular subsemigroup of (S, +).(K3) Every additive idempoent of S must be in some J(K4) For every a 6 5, a 6, exists j = (K5) For every I such as Bi + Bj + Bi C Bk.(K6) If a, a + 6, 6 + 6', 6' + 6 € Bi, then b B(K7) For every sucA as ,5, + Ej . where E, is additive idempotent set of Bi.(K8) For every Bi 5, Vc e 5, 3, Jb 6 7, such as Bfc C Bj, cBi C Bk.Let H = {} is regular kernal normal system of orthodox semiring S, defineTheory 4.8. Let S be a orthdox semiring, B is regular kernal normal system of S, then there is unique congruence p on S such as the regular kernal normal system of p on (S, +) is, and p = . Conversely, if p is a congruence on S, then the regular kernal normal system of p on (S, +) the regular kernal normal system of...