This book offers a comprehensive survey of the area of monoids. It includes injective and weakly injective S-acts, the concept of projective S-acts, some fundamental and interesting results concerning strong flatness, condition(P) and flatness of S-acts, some results on homological classification of monoids, the study of sheaves for classes of monoids and S-acts, analogous to the sheaves for classes of rings and modules. This volume will be of interest to researchers, graduate students and scientists of mathematics.

Preface

Chapter 1 Fundamental Concepts

 1.1 Relations and Lattices

 1.2 Categories and Functors

 1.3 Basic Concepts in Semigroups

 1.4 S-acts and S-homomorphisms

 1.5 Prime and Semiprime S-acts over Monoids with Zero

Chapter 2 Injective and Weakly Injective S-Acts and Related Concepts

 2.1 Injective S-acts

 2.2 Weakly Injective S-acts

 2.3 Completely Right Injective Monoids

 2.4 Quasi-injective S-acts and Completely Quasi-injective Monoids

Chapter 3 Relatively Injective S-acts and Related Concepts

 3.1 P-injective and Divisible S-acts

 3.2 Characterization of Monoids by P-injective S-acts

 3.3 Weakly Regular Monoids and Normal S-acts

 3.4 Hereditary and Semihereditary Monoids

 3.5 Hereditary and Cohereditary S-acts

 3.6 Completely Right FSF-injective Monoids

Chapter 4 Projective S-Acts

 4.1 Projective S-acts

 4.2 Completely Right Projective Monoids

 4.3 Quasi-projective S-acts

 4.4 (x，y)-projective S-acts

 4.5 Products of Projective S-acts

 4.6 Projective Acts over Factorisable Semigroups

Chapter 5 Flat S-Acts

 5.1 The Functor

 5.2 Flatness and Condition (P)

 5 3 Strongly Flat S-acts

Chapter 6 Homological Classification of Monoids by Flatness

 6.1 Strong Flatness of Condition (P) Acts

 6.2 Condition (P) Property of Flat Acts

 6.3 Strong Flatness of Flat Acts

 6.4 Right Perfect Monoids

Chapter 7 Some Topics Relative to Flatness

 7.1 von Neumann Regular Monoids

 7.2 Regularity and Flatness of Acts

 7.3 Relative Flatness and Strongly Faithful Acts

 7.4 Purity of Acts

 7.5 Flatness in SPOS

Chapter 8 Sheaves for Classes of Monoids

 8.1 A Historical Introduction of Sheaf Theory in Algebra

 8.2 Sheaves of von Neumann Regular Monoids with Zero

 8.3 Pure Ideals of a Monoid with Zero

 8.4 Pure Spectrum of a Monoid with Zero

 8.5 S-acts

 8.6 First Representation Theorem

 8.7 Second Representation Theorem

Bibliography

Index