An incline is a commutative and idempotent semiring \((S,+,\cdot)\) satisfying the identities \(x+x=x\) and \(x+xy=x\), or equivalently, a semilattice-ordered commutative semigroup (S,\({\mathbb{O}},\leq)\) satisfying the identity x\(y\leq x\). The authors have studied this structure in a series of papers and they now present the whole theory in a systematic manner. The book comprises six chapters.
Chapter 1 deals with standard algebraic concepts. The main concern is to determine conditions under which an incline is a lattice or a distributive lattice and a kind of converse problem: when does a lattice have a nontrivial incline structure? A representation theorem is also proved.
The essentials of linear algebra over a set of n-dimensional incline- valued vectors are built up in Chapter 2. As might be expected, some ingeniousness has been necessary in order to obtain reasonably good analogues of the concept of basis and of its fundamental properties.
Chapter 3 deals with square matrices over an incline. Idempotent and regular matrices are studied in detail, as well as Moore-Penrose generalized inverses. The chapter also includes a study of non- commutative inclines and of anti-inclines, which differ from an incline by the lack of commutativity and by the axiom \(x\leq xy\), respectively.
Chapter 4 provides a characterization of inclines on [0,1], studies n- dimensional lattices and one-parameter semigroups and sketches the possibility of defining analogues of the basic concepts of analysis and of Hilbert space theory for inclines.
The behaviour of the sequence of powers of an incline matrix is studied in Chapter 5. The Schein rank and group-theoretic complexity are also investigated.
As Boolean algebras and fuzzy algebras are particular cases of inclines, the applications of the former can be also viewed as applications of the latter. Chapter 6 is a sketchy survey of possible applications of inclines, other than Boolean or fuzzy algebras, to linear systems, probabilistic reasoning, group choice theory, automata, graphs, clustering, programming and Boolean matrix theory.
Each chapter contains various examples and a list of open problems. The authors have adopted a very concise style. All of the examples and some theorems and propositions are given without proofs. The book is largely self-contained, including even such elementary definitions as e.g. those of partial order, lattices and Hausdorff topology; yet the reader is supposed to be familiar with some concepts from semigroup theory. The bibliography seems to be selective. Thus e.g. the authors mention the works of R. Cunningham-Green and U. Zimmermann on certain algebras resembling inclines and related to optimal path problems in graphs, but the literature on this topic is much more extensive. Also, several papers on asymptotic forms of Boolean matrices are not mentioned.