Unification in Primal Algebras, Their Powers and Their Varieties

Tobias Nipkow

This paper examines the unification problem in the class of primal algebras and the varieties they generate. An algebra is called primal if every function on its carrier can be expressed just in terms of the basic operations of the algebra. The two-element Boolean algebra is the simplest non-trivial example: every truth-function can be realized in terms of the basic connectives, for example negation and conjunction.

It is shown that unification in primal algebras is unitary, i.e. if an equation has a solution, it has a single most general one. Two unification algorithms, based on equation solving techniques for Boolean algebras due to Boole and Löwenheim, are studied in detail. Applications include certain finite Post algebras and matrix rings over finite fields. The former are algebraic models for many-valued logics, the latter cover in particular modular arithmetic.

Then unification is extended from primal algebras to their direct powers which leads to unitary unification algorithms covering finite Post algebras, finite, semisimple Artinian rings, and finite, semisimple non-abelian groups.

Finally we use the fact that the variety generated by a primal algebra coincides with the class of its subdirect powers. This yields unitary unification algorithms for the equational theories of Post algebras and p-rings.


@article{Nipkow-JACM, author="Tobias Nipkow", title="Unification in Primal Algebras, Their Powers and Their Varieties", journal=JACM,year=1990,volume=37,pages="742--776"}