## Arithmetical completely simple semigroups |

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### Contents

Introduction | 1 |

An Arithmetic for Arithmetical Completely Simple Semigroups | 41 |

Lattices Semilattices and Rooted Trees | 61 |

5 other sections not shown

### Common terms and phrases

0-right a e E(0 A-admissible a.c.s. semigroup M(G;I,I;P anti-atom Assume columns completely simple semigroup coproduct COROLLARY cyclic structure group defined DEFINITION descending steps equal equivalent exists finite lengths group G i,k e ideals in Q identity element implies indecomposable factors index set integral idempotents integral subsemigroup ir-adic numbers isomorphic it-adic join-semilattice lattice ordered group left order Lemma mapping maximal orders maximal Q-unit minimal left ideals modular modular lattice multiplication noetherian a.c.s. semigroup normal ideals number of descending nxn nxn Observe order in Q ordered minimal left partially ordered semigroup partially ordered set path permutation matrix prime numbers principal ideal ring product of indecomposable PROPOSITION proves THEOREM quasi-uniserial semigroup Rees matrix semigroup right ideal rooted tree I,V satisfies semilattice I,V Similarly simple semigroup subdirect product Theorem Theorem 1.9 two-sided ideal uefi uniquely determined upper bound valid zero