By Kar Ping Shum, Zhe-Xian Wan, Jiping Zhang

On Normalized desk Algebras Generated via a devoted Non-Real section of measure three (Z Arad & G Chen); Graph Semigroups (V Dlab & T Pospichal); Moor-Penrose Generalized Inverses of Matrices Over department earrings (Z-X Wan); M-Solid Pseudovarieties and Galois Connections (K Denecke & B Pibaljommee); Indecomposable Decompositions of CS-Modules (J L Gomez Pardo & P A Guil Asensio); Hereditary jewelry, QF2 jewelry and earrings of Finite illustration style (C R Hajarnavis); good Burst errors Detecting Cyclic Codes (S Jain); at the Homology Bifunctors Over Semimodules (X T Nguyen); a few difficulties and Conjectures in Modular Representations (J-P Zhang); and different papers

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A = m y a z , for some z, y , z E S. Then y a z = y ( z a y a z ) z = ( y z ) a ( y a z 2 )= ( y z ) ( z a y a z ) ( y a z 2 ) = ( y z 2 a ) ( y a z ) 2 zE S ( y a z ) 2 S , so we have that yaz is an intra-regular element of S. The converse is clear. e. a = z a y a , for some z, y E 5’. Then Ya = Y ( W a ) = ( y z ) a ( y a ) = ( y z ) ( z a y a ) y a ) = (Yz2a)(Ya)2E s(Y42, so yu is a left regular element of S. The converse is evident. The assertions (b) and (c) can be proved similarly. 0 It is well-known that an element a of a semigroup S is regular if and only if the principal left ideal L ( a ) (or the principal right ideal R ( a ) ) has an idempotent generator.

Mat. Univ. Padoua, 36:129-157, 1966. 18. V. I. Mysovskikh. Investigation of subgroup embeddings by the computer algebra package GAP. In Computer algebra in scientific computing-CASC’99 (Munich), pages 309-315, Berlin, 1999. Springer. 19. A. Ballester-Bolinches and R. Esteban-Romero. On finite 7-groups. To appear in J. Austral. Math. SOC. 20. A. Ballester-Bolinches and R. Esteban-Romero. On finite soluble groups in which Sylow permutability is a transitive relation. Preprint. 21. M. Asaad and A.

If S satisfies anyone of the regularity conditions from Figure 3, then S, satisfies the same condition, for every 0 E Y . Figure 3. Proof. We will prove only the assertion concerning the condition (( c = zcycz)). The remaining cases can be proved similarly. Let S satisfies the condition (( c = zcycz)) and let a E Y be an arbitrary element. For any a E S, there exists z, y , z E S such that a = xayaz, and we have that z E Sp, y E S, and z E 5'6, for some p , y , b E Y . By a E S, and a = xayaz it follows that ap = a y = a6 = a , so a = z a y a z = z ( z a y a z ) y ( z a y a z ) z= = (z'ay)a( zyz)a(yaz') = (z'ay) (zayaz)( z y z ) a ( y a z ' ) = = (z%yz)a( yaz' yz) a(yaz') E s,as,as,.