Deductive systems and Galois connections
by
Ivan Chajda
Palacky University Olomouc, Dept.of Algebra and Geometry

In algebras which serve as an algebraic counterpart of propositional logic, a deductive system is a subset closed under production rule of Modus Ponens. Having an arbitrary algebra A and a subset H the set T₂(A) of binary term functions of A, one can generalize the concept of deductive system to be a subset of A closed under production rules derived by means of the term functions of H. Such a system is called an H-deductive system and the set Ded_HA of these system is an algebraic lattice. For H subset or equal H₁, Ded_H1A is a sublattice of Ded_HA thus we can recognize Galois connections between sublattices of Ded_HA and subsets of T₂(A). We will indicate whether H-deductive systems are congruence kernels and describe sets H subset or equal T₂(A) closed under the closer operator induced by the Galois connections.

Date received: January 3, 2001