Monoids, Acts and Categories: With Applications to Wreath Products and Graphs. a Handbook for Students and Researchers
Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev · 2000
Elementary properties of monoids, acts and categories: sets and relations groupoids, semigroups and monoids some classes of semigroups acts over monoids (monoid automata) decompositions and components categories functors. Constructions: products and coproducts pullbacks and pushouts free objects and generators cofree objects and generators tensor products wreath products of monoids and acts the wreath product of a monoid with a small category. Classes of acts: injective acts divisible acts principally weakly injective acts fg-weakly injective acts absolutely pure acts cogenerators and overview torsion free acts flatness of acts and related properties principally weakly flat acts weakly flat acts flat acts acts satisfying condition (P) acts satisfying condition (E) equalizer flat acts pullback flat acts and overview projective acts generators regular acts and overview. Homological classification of monoids: principal weak injectivity on fg-weak injectivity weak injectivity absolute purity injectivity and overview torsion freeness and principal weak flatness flatness condition (P) strong flatness projectivity projective generators freeness and overview regularity of acts. Equivalence and duality: adjoint functors categories equivalent to act - S Morita equivalence of monoids endomorphism monoids of generators on morita duality.