Applications of Mathematical Logic to Algebra

An algebraic structure consists of a base set A (e.g., the set of natural numbers, points in threedimensional space, or the 2-element set of truth values 1 true, false}), together with a sei F of "operations" an A (e.g., addition or multiplication in the first case, translations or convex combinations in the second case, negation or disjunction in the third case). The aim of this project is to investigate the family of all algebraic structures an a given infinite base sei A. If we identify any two such structures that have the same term functions, the set of all those structures becomes in a natural way a complete algebraic lattice (called Cl(A)), about whose structure little is known. Many deep results for the case of a FINITE base sei A are already known, and they are usually obtained by purely algebraic methods (combined with simple counting or other combinatorial arguments); as some results already indicate, the case of infinite sets is markedly different, in that methods from mathematical logic (in particular: from set theory) will be necessary. Preliminary results indicate that methods and notions from classical descriptive set theory, from infinite Ramsey theory, and from the calulus of forcing, can be employed to investigate CI(A). As in the finite case, a main application would be the establishment of a COMPLETENESS CRITERION: Given a set F of operations an A, how can we decide if F is complete, i.e., if every other operation an A be obtained by composing elements of F?