Research

I am interested in questions pertaining to the finite axiomatizability of equational theories of finite algebras. My current research includes applications of the Shift Automorphism Method (of Baker, McNulty, Werner). I am particularly interested in resolving the following open problems:

The Problem of Park and Jónsson (1976):
Is every finite algebra of finite signature that generates a variety with a finite residual bound finitely based?
The Problem of Eilenberg and Schützenberger (1976):
If V is a variety generated by a finite algebra, W is a finitely based variety, and V and W share the same finite algebras, must V be finitely based?
The Problem of Ježek and Quackenbush (1990), Modified:
Is there a finite commutative directoid that is inherently nonfinitely based?

Regarding the Problem of Park and Jónsson, in my dissertation and in a recent paper I was able to show that every shift automorphism variety contains a countably infinite subdirectly irreducible member. In other words, if a finite algebra can be shown to be inherently nonfinitely based by way of the Shift Automorphism Theorem (of Baker, McNulty, and Werner in 1989) then it cannot be a counter example to the Problem of Park and Jónsson.

My work on the modified problem of Ježek and Quackenbush led to the construction of a variety of commutative directoids that is inherently nonfinitely based in the finite sense. If this variety turns out to be generated by a finite algebra, this would be a resolution to this problem. I have also shown that if this variety is generated by a finite commutative directoid, then its finite generator will not be a counter example to either the Problem of Park and Jónsson or the Problem of Eilenberg and Schützenberger. I am also interested in resolving the finite basis status of Hajilarov’s six-element commutative directoid (1996).

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