Order polynomially complete lattices must be large

If L is an o.p.c. (order polynomially complete) lattice, then the cardinality ofL is a strongly inaccessible cardinal. In particular, the existence of o.p.c. lattices is not provable in ZFC, not even from ZFC + GCH. Received February 28, 1997; accepted in final form May 3, 1998.     

Markov approximations of chains of infinite order

We consider chains whose transition probabilities depend on the whole past, with summable continuity rates. We show that Ornstein's -distance between one such chain and its canonical Markov approximations of different orders is at worst proportional to the continuity rate of the chain. The result generalizes previous bounds obtained by X. Bressaud and ourselves, while relying on a similar coupling argument. Received: 9 April 2002     

Trivial, strongly minimal theories are model complete after naming constants

We prove that if is any model of a trivial, strongly minimal theory, then the elementary diagram is a model complete -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is .

     

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

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. Received February 16, 2001; accepted in final form November 2, 2001. RID="h1" ID="h1"The second author was supported by a postdoctoral research fellowship of the National Research Foundation of South Africa, hosted by the University of Illinois at Chicago.