The Regular J-Class of the Power Semigroup of a Completely 0-Simple Semigroup

✍ Scribed by Takayuki Tamura

• Publisher: John Wiley and Sons
• Year: 1990
• Tongue: English
• Weight: 932 KB
• Volume: 149
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19901490102

✦ Synopsis

Next we consider when those idempotents are Ror &related. If idempotents ( A , B; 4) and (C, D; $) are R-(L)related, then it is necessary that A = C ( B = D).

lluinma 2.4. [lo]. Idempdents ( A , B ; 4) and ( A , D ; $J) of depth 1 are R-related i f and e r r l y i f A x (b, d ) is poportional in P /or 8ome b E B and d E D, in other words (A, b ; 4) I I ( A , d ; 9) i s an idempotent. J'roof. By Lemma 2.2 we have ( A , B ; 4) (A, D ; $) = (A, D; 9 ) if and only if (*) C i A P k I I l A = pz:ApfDIA for all iA, l A , kB, j D , h i ;)articular (**) N l t i c ~ BEPA and D&A by Lemma 2.2, we have (**) implies (*). h m m a 2.4'. Idempohts ( A , B ; 4) and (C, B ; $) of depth 1 are Erelated if and m l y 1' 1 (11, c) x B is poportiom1 in P for some a E A , c E C, in other words, (a, B ; 4) u (c, B ; 9 ) h (171 idempotent.

h n m a 2.6. Let (A, B ; 4) be an idempotent of depth 1 and let 1 E M , 1 E I with pGipblA = p&~dl, for all i,,, l A , for some b E B, d E D .