Infinite Combinatorial Issues Raised by Lifting Problems in Universal Algebra

Abstract

The critical point between varieties $\mathcal{A}$ and $\mathcal{B}$ of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of $\mathcal{A}$ but of no member of $\mathcal{B}$ , if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (κ,r,λ)→m (for proving that a critical point is small). In the middle, we find λ-lifters of posets and the relation $(\kappa,{<}\lambda)\leadsto P$ , for infinite cardinals κ and λ and a poset P.