1. Introduction. 2. Prerequisites on semigroup theory. 2.1 Semigroups. 2.2. Uniform semigroups. 2.3 Full linear semigroup. 2.4 Structure of linear semigroups. 2.5 Closure. 2.6 Semigroups over a field. 3. Prerequisites on ring theory. 3.1 Noetherian rings and rings satisfying a polynomial identity. 3.2 Prime ideals. 3.3 Group algebras of polycyclic-by-finite groups. 3.4 Graded rings. 3.5 Gelfand-Kirillov dimension. 3.6 Maximal orders. 3.7 Principal ideal rings. 4. Algebras of submonoids of polycylic-by-finite groups. 4.1 Ascending chain condition. 4.2 The unit group. 4.3 Almost nilpotent case. 4.4 Structure theorem. 4.5 Prime ideals of K[S]. 4.6 Comments and problems. 5. General Noetherian semigroup algebras. 5.1 Finite generation of the monoid. 5.2 Necessary conditions. 5.3 Monomial semigroups and sufficient conditions. 5.4 Gelfand-Kirillov dimension. 5.5 Comments and problems. 6. Principal ideal rings. 6.1 Group algebras. 6.2 Matrix embedding. 6.3 Finite dimensional case. 6.4 The general case. 6.5 Comments and problems. 7. Maximal orders and Noetherian semigroup algebras. 7.1 Maximal orders and monoids. 7.2 Algebras of submonoids of abelian-by-finite groups. 7.3 Comments and problems. 8. Monoids of I-type. 8.1 A characterization. 8.2 Structure of monoids of I-type. 8.3 Binomial monoids are of I-type. 8.4 Decomposable monoids of I-type. 8.5 Algebras of monoids of I-type. 8.6 Comments and problems. 9. Monoids of skew type. 9.1 Definition. 9.2 Monoids of skew type and the cyclic condition. 9.3 Non-degenerate monoids of skew type. 9.4 Algebras of non-degenerate monoids of skew type. 9.5 The cancellative congruence and the prime radical. 9.6 Comments and problems. 10. Examples. 10.1 Monoids of skew type and the Gelfand-Kirillov dimension. 10.2 Four generated monoids of skew type. 10.3 Examples of Gelfand-Kirillov dimension 2. 10.4 Non-degenerate monoids of skew type of Gelfand-Kirillov dimension one. 10.5 Examples of maximal orders. 10.6 Comments. Bibliography. Index. Notation.

Here is a comprehensive treatment of the main results and methods of the theory of Noetherian semigroup algebras. These results are applied and illustrated in the context of important classes of algebras that arise in a variety of areas and have recently been intensively studied. The focus is on the interplay between combinatorics and algebraic structure. Mathematical physicists will find this work interesting for its attention to applications of the Yang-Baxter equation.