The first section of this chapter is a history of the notions and previous theorems upon which this monograph is based and inspired by. Notable contributors begin with Boole, Noether, Birkhoff, Leray, Foster, and include many more from the last half century.The second section is a survey of the principal results presented in this book. The first part of the second section covers the background. The next part summarizes complexes, sheaves, and the representation of algebras by these; it also tells about Boolean algebras of factor congruences.The third part applies this theory to shells, which is a common generalization of rings and bounded lattices; these have factor ideals and elements that capture products. The last part extends a number of classical theorems from ring and semigroup theory.