Lattices and Ordered Algebraic Structures

On pp. 95-96 of T. S. Blyth’s Lattices and Ordered Algebraic Structures we encounter proofs of the following three results: in a distributive lattice every maximal ideal is prime and every proper ideal is the intersection of prime ideals; in a complemented lattice every prime ideal is maximal; and every Boolean algebra is isomorphic to the algebra of clopen subsets of a compact, totally disconnected Hausdorff space.  The first two results conspire to show that in a Boolean algebra prime ideals and maximal ideals coincide (a marvelous algebraic result in its own right) even as it is also featured in the derivation of the third fact, a topological representation theorem due to none other than M. H. Stone.

Thus, a raison d’être for the study of lattices and ordered algebraic structures is the facility this subject imparts to demonstrating a certain class of results from algebra and topology.  But aside from such contextual connections the subject is of course autonomous and imbued with its own unique characteristics.  Consider, for example, the very last result in the book, a structure theorem (due to Janowitz in 1991) with a flavor all its own: the ordered monoid of residuated mappings on a bounded distributive lattice of finite length is regular if and only if the lattice is a vertical sum of lattices of the form M(k) where k≤2 and, by definition, M(k) is “the lattice formed by adding top and bottom elements to the discretely ordered set {1,2,…,k}.”  For further explication of the jargon the reader is referred to the book.  Suffice it to say that the preceding example is an indication of much of the thrust of this book.

More precisely, the focus of Lattices and Ordered Algebraic Structures falls on such things as Stone and Heyting algebras, ordered groups and ordered semigroups, with the notion of residuated mapping taking central stage much of the time.  Indeed Blyth introduces these last-mentioned players already in the Introduction where he likens them to continuous functions in analysis.  To wit, just as the inverse image of an open set under a continuous function is open, the inverse image of a principal down-set under a residuated mapping is again a principal down-set (all situated in the respective ordered sets).  Again, for further explanation of the jargon, see the book.

Lattices and Ordered Algebraic Structures is extensive and scholarly, dense but accessible.  There are a decent number of exercises and a great deal of interesting (if occasionally arcane) material is covered, well beyond what little I have indicated above.

Recommended!