id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-1240	Guo, Xiaojiang; Shum, K.P.	Naturally Ordered Abundant Semigroups for which each Idempotent has a Greatest Inverse	2011	11	.pdf	application/pdf	5243	339	84	By Lemma 3, s♭ = s◦◦ and by Lemma 2, we obtain the following equalities: s◦♭ = (s◦s◦◦)◦(s◦s◦◦)s◦(s◦◦s◦)(s◦◦s◦)◦ = (s◦◦ ◦ s◦◦)(s◦s◦◦)s◦(s◦◦s◦)(s◦◦s◦◦ ◦ ) = s◦s◦◦s◦s◦◦s◦s◦◦s◦ = s◦s◦◦s◦ = s◦, Hence, s♭s◦♭s♭ = s◦◦s◦s◦◦ = s◦◦ = s♭ and s◦♭s♭s◦♭ = s◦s◦◦s◦ = s◦ = s◦♭, whence s◦♭ is an inverse of s♭. Note that s[u, s◦s]s◦[ss◦, x]s = ss◦ss◦ss◦s = ss◦s = s, we have, (x , s,u)(s◦s, s◦, ss◦)(x , s,u) = � x(s[u, s◦s]s◦[ss◦, x]s)† , s[u, s◦s]s◦[ss◦, x]s, (s[u, s◦s]s◦[ss◦, x]s)∗u � = (xs†, s, s∗u) = (x , s,u) and similarly, (s◦s, s◦, ss◦)(x , s,u)(s◦s, s◦, ss◦)	cache/ejpam-1240.pdf	txt/ejpam-1240.txt