Date of Award

Spring 2020

Document Type

Open Access Dissertation



First Advisor

George F. McNulty


Study of general algebraic systems has long been concerned with finite basis results that prove finite axiomatisability of certain classes of general algebras. In the 1970’s, Bjarni Jónsson speculated that a variety generated by a finite algebra might be finitely based provided the variety has a finite residual bound (that is, a finite bound on the cardinality of subdirectly irreducible algebras in the variety). As such, most finite basis results since then have had the hypothesis of a finite residual bound. However, Jónsson also speculated that it might be sufficient to replace the finite residual bound with the weaker hypothesis that the subdirectly irreducible algebras themselves be finitely axiomatisable.

In this dissertation, we give an overview of the concepts and history involving this topic. We also prove that two types of varieties that are already known to be finitely based have the property that their subdirectly irreducible members are finitely axiomatisable.

• Varieties generated by finite nilpotent groups have this property.

• Congruence permutable varieties generated by finite nilpotent algebras of finite signature that are the product of algebras of prime power order have this property.


© 2020, Joshua Thomas Grice

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Mathematics Commons