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Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for researchers and graduate students. The appendix gives an introduction to the necessary set theory, in particular to the (non-)measurable cardinals, to help the reader make smooth progress through the text. A detailed index shows the numerous connections among the topics treated. Every chapter has a historical section to show the original sources for results and the subsequent development of ideas, and is rounded off with numerous exercises. More than 100 open problems and projects are presented, ready to inspire the keen graduate student or researcher. Many of the results are appearing in print for the first time, and many of the older results are presented in a new light.

Metric spaces --- Geometry, Algebraic --- Linear topological spaces --- Homomorphisms (Mathematics)

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Regular rings were originally introduced by John von Neumann to clarify aspects of operator algebras ([33], [34], [9]). A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not ?nite-dimensional ([8, page 155], [32, page V]). Von Neumann proved ([32, Theorem 14. 1, page 208], [8, page 162]): Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. The book of K. R. Goodearl ([14]) gives an extensive account of various types of regular rings and there exist several papers studying modules over regular rings ([27], [31], [15]). In abelian group theory the interest lay in determining those groups whose endomorphism rings were regular or had related properties ([11, Section 112], [29], [30], [12], [13], [24]). An interesting feature was introduced by Brown and McCoy ([4]) who showed that every ring contains a unique largest ideal, all of whose elements are regular elements of the ring. In all these studies it was clear that regularity was intimately related to direct sum decompositions. Ware and Zelmanowitz ([35], [37]) de?ned regularity in modules and studied the structure of regular modules. Nicholson ([26]) generalized the notion and theory of regular modules. In this purely algebraic monograph we study a generalization of regularity to the homomorphism group of two modules which was introduced by the ?rst author ([19]). Little background is needed and the text is accessible to students with an exposure to standard modern algebra. In the following, Risaringwith1,and A, M are right unital R-modules.

Endomorphism rings. --- Homomorphisms (Mathematics). --- Modules (Algebra). --- Rings (Algebra). --- Modules (Algebra) --- Homomorphisms (Mathematics) --- Rings (Algebra) --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Finite number systems --- Modular systems (Algebra) --- Mathematics. --- Algebra. --- Associative rings. --- Group theory. --- Associative Rings and Algebras. --- Group Theory and Generalizations. --- Algebraic fields --- Functions --- Finite groups --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematical analysis

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51 <082.1> --- Mathematics--Series --- Homomorphisms (Mathematics) --- C*-algebras. --- Homomorphismes (Mathématiques) --- C*-algèbres --- Homomorphismes (Mathématiques) --- C*-algèbres --- Algebraic topology --- Homotopy theory. --- C*-algebras --- Homotopy theory --- Deformations, Continuous --- Topology --- Functions --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Homotopie

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Homomorphic signature schemes are an important primitive for many applications and since their introduction numerous solutions have been presented. Thus, in this work we provide the first exhaustive, complete, and up-to-date survey about the state of the art of homomorphic signature schemes. First, the general framework where homomorphic signatures are defined is described and it is shown how the currently available types of homomorphic signatures can then be derived from such a framework. In addition, this work also presents a description of each of the schemes presented so far together with the properties it provides. Furthermore, three use cases, electronic voting, smart grids, and electronic health records, where homomorphic signature schemes can be employed are described. For each of these applications the requirements that a homomorphic signature scheme should fulfill are defined and the suitable schemes already available are listed. This also highlights the shortcomings of current solutions. Thus, this work concludes with several ideas for future research in the direction of homomorphic signature schemes.

Computer Science --- Engineering & Applied Sciences --- Data encryption (Computer science) --- Computer security. --- Homomorphisms (Mathematics) --- Computer privacy --- Computer system security --- Computer systems --- Computers --- Cyber security --- Cybersecurity --- Electronic digital computers --- Security of computer systems --- Data encoding (Computer science) --- Encryption of data (Computer science) --- Security measures --- Protection of computer systems --- Protection --- Functions --- Data protection --- Security systems --- Hacking --- Computer security --- Cryptography --- Data structures (Computer scienc. --- Data Structures and Information Theory. --- Discrete Mathematics. --- Data structures (Computer science) --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Data structures (Computer science). --- Discrete mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis