An online database of ordered algebraic structures
by
Peter Jipsen
Chapman University

Research in ordered algebraic structures has grown from the classical areas of ordered groups, rings, and fields to the study of a wide variety of classes of partially ordered algebras, where the order takes the form of (join- or meet-) semilattices, lattices, or Boolean algebras. In addition, subclasses determined by various completeness properties and residuation properties have received considerable attention, motivated by studies in algebraic logic, topology, and theoretical computer science. While researchers are familiar with the structures in their areas of expertise, it is no small task to be informed about the terminology, definitions, basic properties and relationships between the many classes of structures in this growing field.

This talk will present a preliminary version of an online database that aims to (eventually) survey most classes of ordered algebraic structures that have appeared in the literature. At minimum, a record for a class contains its name, (several) definition(s), some of the basic properties and results about the class, and the position of the class in this hierarchy relative to its ``nearest'' sub- and superclasses. Most of the classes are categories in a natural way, and concepts from category theory are used to express structure preserving relationships between different classes. Currently the database contains around a hundred classes from action algebras to weakly representable relation algebras. Although intended mainly as a reference for researchers, with citations to the literature for further detail, the database aims to store definitions in machine-readable form to allow additional processing with computational tools. It will also provide access to families of examples and (where feasible) implementations of algorithms for computable syntactic or semantic properties of a given class. For example, if a class has a decidable equational theory, an implementation of a decision procedure may be included in the entry for the class. Tools for visualizing (parts of) the partial order structure of specific examples and of the hierarchy of classes will be discussed, as well as recent significant online developments such as MathML and XML transformation tools that play a role in our approach.

Much work remains to be done to give this database the breadth and consistency that would ensure it is a useful research tool and is helpful to students entering this area. In fact, in a successful scenario, this project will not be completed at some stage, but will continue to grow in a collaborative style, thereby ensuring that the information in the database remains up-to-date.

Date received: December 31, 2002