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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Regular semigroups satisfying certain conditions on idempotents and ideals

Author: Mario Petrich
Journal: Trans. Amer. Math. Soc. 170 (1972), 245-267
MSC: Primary 20M10
MathSciNet review: 0304522
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Abstract: The structure of regular semigroups is studied (1) whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idempotents in different $\mathcal {D}$-classes, or (2) all of whose ideals are categorical or satisfy a variation thereof. For this purpose the notions of $D$-majorization of idempotents, where $D$ is a $\mathcal {D}$-class, $\mathcal {D}$-majorization, $\mathcal {D}$-categorical ideals, and completely semisimple semigroups without contractions are introduced and several connections among them are established. Some theorems due to G. Lallement concerning subdirect products and completely regular semigroups and certain results of the author concerning completely semisimple inverse semigroups are either improved or generalized.

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Keywords: Completely regular semigroup, categorical ideal, completely semisimple semigroup, primitive regular semigroup, subdirect products, tree, normal band of groups, <!– MATH $\mathcal {D}$ –> <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathcal {D}$">-class, extensions, partial homomorphism
Article copyright: © Copyright 1972 American Mathematical Society