Regular article

algebra universalis

, Volume 48, Issue 3, pp 253-271

First online:

The finite embeddability property for residuated lattices, pocrims and BCK-algebras

  • W. J. BlokAffiliated withDepartment of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607, e-mail:
  • , C. J. Van AltenAffiliated withDepartment of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa, e-mail:

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A class of algebras has the finite embeddability property (FEP) if every finite partial subalgebra of an algebra in the class can be embedded into a finite algebra in the class. We investigate the relationship of the FEP with the finite model property (FMP) and strong finite model property (SFMP).¶ For quasivarieties the FEP and the SFMP are equivalent, and for quasivarieties with equationally definable principal relative congruences the three notions FEP, FMP and SFMP are equivalent. The variety of intuitionistic linear algebras –which is known to have the FMP–fails to have the FEP, and hence the SFMP as well. The variety of integral intuitionistic linear algebras (also known as the variety of residuated lattices) does possess the FEP, and hence also the SFMP. Similarly contrasting statements hold for various subreduct classes. In particular, the quasivarieties of pocrims and of BCK-algebras possess the FEP. As a consequence, the universal theories of the classes of residuated lattices, pocrims and BCK-algebras are decidable.

2000 Mathematics Subject Classification: 06F05, 03B47, 06F35, 08C15, 08A50.¶Key words and phrases: Finite embeddability property, finite model property, linear logic, intuitionistic linear algebra, residuated lattice, pocrim, BCK-algebra, well-quasi-order.