We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform word problem for Q is solvable in (...) polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)

We determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.

In an earlier paper we proved that a universal Horn class generated by finitely many finite structures has a model companion. If the language has only finitely many fundamental operations then the theory of the model companion admits a primitive recursive elimination of quantifiers and is primitive recursive. The theory of the model companion is ℵ 0 -categorical iff it is complete iff the universal Horn class has the joint embedding property iff the universal Horn class is generated by a (...) single finite structure. In the last section we look at structure theorems for the model companions of universal Horn classes generated by functionally complete algebras, in particular for the cases of rings and groups. (shrink)

We present two conditions which are equivalent to having an almost χ0-categorical model companion.

This paper augments Hailperin's substantial efforts to place Boole's algebra of logic on a solid footing. Namely Horn sentences are used to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic—we call this principle The Rule of 0 and 1.