1 research outputs found
On Birkhoff's common abstraction problem
In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his
readers to develop a “common abstraction” that includes Boolean algebras and latticeordered
groups as special cases. In this paper, after reviewing the past attempts to solve the
problem, we provide our own answer by selecting as common generalization of BA and LG
their join BA∨LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue
that such a solution is optimal under several respects and we give an explicit equational
basis for BA∨LG relative to FL. Finally, we prove a Holland-type representation theorem
for a variety of FL-algebras containing BA ∨ LG