Categories of algebraic systems : vector and projective spaces, semigroups, rings, and lattices / Mario Petrich

From the author's preface: "The leitmotiv of this monograph is the unity of the modest part of mathematics encompassed by the algebraic and geometric structures mentioned in its title. Some of the relationships discussed here are very old, indeed `classical', some of them are quite recent, and still lack `status', but the categorical setting in which they are presented is considered here for the first time''.
The text is subdivided into an introduction and 8 chapters with the following headings: (I) Vector spaces; (II) Semigroups; (III) Rings; (IV) Lattices; (V) Projective spaces; (VI) The loop; (VII) Antiautomorphisms; (VIII) Special cases. Chapter I deals with categories of pairs of dual vector spaces, regular linear systems and weakly topologized vector spaces. In Chapter II the author considers the categories of completely 0-simple semigroups, their orthogonal sums and maximal dense extensions of both, morphisms being isomorphisms. A similar consideration is performed for certain categories of rings in Chapter III. Chapter IV is devoted to interconnections between certain categories of semisimple atomic rings and certain categories of subprojective lattices, and in Chapter V the category of pairs of dual projective spaces is introduced. Some of the categories considered in this book are found to be equivalent, in Chapter VI the resulting compositions of functors are proved to be naturally equivalent. In Chapter VII the objects of some of the categories previously considered are enriched by taking them together with an anti-automorphism in the case of semigroups, rings and lattices, scalar products in the case of vector spaces, and correlations in the case of projective spaces. A few special cases are considered in Chapter VIII. The last chapter is followed by a bibliography with 127 entries. The bibliography is followed by three diagrams, lists of categories, functors and special symbols, and an index.
An advanced graduate student or a research mathematician with a general interest in some of the subjects treated in the book should find reading the book useful. The principal gain will be the clear realization how essentially similar the underlying ideas in these different areas and subjects are, and what the mutual relationships are. (MathSciNet)