On Birkhoff’s Common Abstraction Problem

Studia Logica 100 (6):1079-1105 (2012)

Abstract

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a "common abstraction" that includes Boolean algebras and lattice-ordered 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 and their join ∨ in the lattice of subvarieties of ℒ (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for ∨ relative to ℒ. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing ∨

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Francesco Paoli
Universita di Cagliari

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