**Research Description**

My primary research interest is in Near Vector Spaces, with the aim of developing a theory which parallels that of vector spaces.

As a secondary research interest, in collaboration with Professor Carl Maxson (Texas A & M University, USA), we investigate algebras of functions associated with a given algebraic structure. The question is when the set of endomorphisms of the universal algebra A will be a maximal structure in the set of all maps of A?

The investigation begin with finite monoids and our goal is to extend this to an investigation of endomorphism algebras of finite universal algebras.

For an idea of the research involved, please refer to:

- J. Andre, Algebra uber Fastkorpern, Math.Z.136 (1974), 295-313.
- K-T Howell, Contributions to the Theory of Near Vector Spaces, PhD thesis (2008).
- A. Kreuzer and C.J. Maxson, E-locally Cyclic Abelian Groups and Maximal Near-rings of Mappings, Forum Math. 18 (2006), no.1, 107-114.
- J.D.P. Meldrum and M. Samman, On Endomorphisms of Semilattices of Groups, Algebra Colloquim 12:1 (2005) 93-111.
- A.P.J. van der Walt, Near-Linear Transformations of Near-Vector Spaces, Proceedings of a Conference held at Oberwolfach, 5-11 November 1989 (ed. G.Betsch, G.Pilz).
- Near Rings : http://www.algebra.uni-linz.ac.at/Nearrings/.

### Research Rating

In September 2014 I was awarded a Y2 rating from the South African National Research Foundation (NRF).

### Professional Societies

I am a member of the South African Mathematical Society and the American Mathematical Society. As of November 2014 I am a reviewer for the American Mathematical society.

#### Students

Foundations Course Projects

- M. Chimes – supervised a third-year project, “Near-vector spaces” for the Foundations Course (2013)
- R. Ferguson - supervised a second-year Foundations Course project on Near-rings and one on Rings (2015)
- P. Thompson – supervised a second-year Foundations Course project on Rings (2015)

AIMS Projects

- S.P. Sagnon – supervised a project “An introduction to Nearrings” (2016), completed with distinction
- P. Djagba – supervised a project “Nearfields” (2017), completed with distinction
- K. Tefoetsile – supervised a project “Near-vector spaces constructed over Z_p for p a prime” (2018)
- J. Kalunga – supervised a project “The Decomposition Theorem for Near-vector spaces” (2018)

MSc

- S.P. Sagnon (2016-2017) – graduated cum laude.

PhD

- P. Djagba (supervised with Dr G. Boxall) (2017-)

“All wish to possess knowledge, but few, comparatively speaking, are willing to pay the price.”

Juvenal