364 research outputs found
Singly generated quasivarieties and residuated structures
A quasivariety K of algebras has the joint embedding property (JEP) iff it is
generated by a single algebra A. It is structurally complete iff the free
countably generated algebra in K can serve as A. A consequence of this demand,
called "passive structural completeness" (PSC), is that the nontrivial members
of K all satisfy the same existential positive sentences. We prove that if K is
PSC then it still has the JEP, and if it has the JEP and its nontrivial members
lack trivial subalgebras, then its relatively simple members all belong to the
universal class generated by one of them. Under these conditions, if K is
relatively semisimple then it is generated by one K-simple algebra. It is a
minimal quasivariety if, moreover, it is PSC but fails to unify some finite set
of equations. We also prove that a quasivariety of finite type, with a finite
nontrivial member, is PSC iff its nontrivial members have a common retract. The
theory is then applied to the variety of De Morgan monoids, where we isolate
the sub(quasi)varieties that are PSC and those that have the JEP, while
throwing fresh light on those that are structurally complete. The results
illuminate the extension lattices of intuitionistic and relevance logics
Categoricity and multidimensional diagrams
We study multidimensional diagrams in independent amalgamation in the
framework of abstract elementary classes (AECs). We use them to prove the
eventual categoricity conjecture for AECs, assuming a large cardinal axiom.
More precisely, we show assuming the existence of a proper class of strongly
compact cardinals that an AEC which has a single model of some high-enough
cardinality will have a single model in any high-enough cardinal. Assuming a
weak version of the generalized continuum hypothesis, we also establish the
eventual categoricity conjecture for AECs with amalgamation.Comment: 63 page
Quantifier elimination and other model-theoretic properties of BL-algebras
This work presents a model-theoretic approach to the study of firstorder theories of classes of BL-chains. Among other facts, we present several classes of BL-algebras, generating the whole variety of BL-algebras whose firstorder theory has quantifier elimination. Model-completeness and decision problems are also investigated. Then we investigate classes of BL-algebras having (or not having) the amalgamation property or the joint embedding property and we relate the above properties to the existence of ultrahomogeneous models. © 2011 by University of Notre Dame.Peer Reviewe
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