**My Research**

**
***J.D.
Phillips*

*Problems worthy *

*of attack*

*prove their worth *

*by fighting back*

*—*Piet
Hein

For a list of my publications, please
see my curriculum vitae. Most of my recent papers can be
downloaded at the Front
for the Mathematics ArXiv. Here are the Math Reviews of my papers on MathSciNet
(note: a subscription is needed to access this page). For a link to some
supplementary items (mostly Prover9 and Mace4 files) to a few of my papers,
click here.

Most of my research is in quasigroup and loop theory. Quasigroups are
"nonassociative groups." More precisely, a *quasigroup* is a set together with
three binary operations *, \, and /satisfying the following identities: *x \ (x * y) = x *
(x \ y) = y, *and* (x / y) * y = (x * y) / y = x*. A *loop* is a quasigroup with a 2-sided neutral element: *x * 1 = 1 * x = x*.

The theories, then, of quasigroups and loops are generalizations of the theory
of groups. This generalization takes two main forms. First, since all groups
can also be viewed as quasigroups, the theory of quasigroups includes the
theory of groups as an appealing subtheory. This forces the theory into a leaner
elegance and greatly simplifies certain notions from group theory. Applying
techniques from quasigroup theory to groups viewed as quasigroups allows the
mathematician to address questions whose answers are usually hidden under the
additional assumptions of group theory. Second, the techniques involved in
studying quasigroups and loops closely resemble the techniques of group theory.
In fact, the theory of quasigroups is intimately related to the theory of
certain groups associated with quasigroups. And so a knowledge of traditional
group theory is required to study quasigroups and loops. This explains, for
instance, the rich history of involvement by some of the world's most eminent
group theorists in developing the theory of quasigroups and loops.

From another perspective, quasigroups and loops can be viewed as universal
algebras, as above. As such, quasigroups lend themselves naturally to the full
spectrum of techniques available to the universal algebraist, especially the
powerful tools and techniques of computational mathematics. In fact, these
techniques have proven to be much more effective in dealing with quasigroups
than in dealing with groups, particularly in the rich area of representation
theory. And so by virtue of their intimate connection with groups, quasigroups
are natural vehicles for the group theorist wishing to use universal algebra
and computational mathematics in his or her investigations.

My work exploits the powerful computational tools—for instance, automated
theorem provers like Prover9,
as well as finite model builders like Mace4—that are now
widely available to algebraists and that are transforming some areas of
algebra, especially those involving inquiries about structures that can be
defined equationally, for instance, quasigroups and loops, as above.
Ultimately, I enjoy working on problems involving any algebras that can be
defined equationally, e.g., medial groupoids, digroups, alternative rings, etc.

One of the things I enjoy most about mathematics research is collaborating with
interesting people. Here are links to the websites of some of the
mathematicians I've been lucky enough to work with: