On the Decision Problem of the Congruence Lattices of Pseudocomplemented Semilattices( )

Abstract

Publisher Summary This chapter focuses on the decision problem of the congruence lattices of pseudocomplemented semilattices. The essential undecidability of the theories of closure algebras, Brouwerian algebras, the algebras of bodies, the algebras of convexity, and the semi-projective algebra are discussed in the chapter. The study of decision problems for lattices of subgroups shows that the theory of subgroup lattices, and also of the congruence lattices, of Abelian torsion-free reduced groups is undecidable. Boolean algebras are the only non-trivial proper subvariety of the variety of pseudocomplemented semilattices, and the class of Boolean algebras is a subclass of the congruence-distributive pseudocomplemented semilattices. The theory of congruence lattices of countable Boolean algebras is decidable. The theory of Heyting lattices and filter lattices of pseudocomplemented semilattices are recursively inseparable. The chapter also discusses various lemmas, theorems, and elementary theory of congruence lattices.