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MODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS: NECESSARY AND SUFFICIENT CONDITIONS

Part of: Model theory Distributive lattices Ordered structures Boolean algebras (Boolean rings) General logic

Published online by Cambridge University Press:  10 January 2022

GEORGE METCALFE Open the ORCID record for GEORGE METCALFE [Opens in a new window]  and
LUCA REGGIO Open the ORCID record for LUCA REGGIO [Opens in a new window]
GEORGE METCALFE*
Affiliation:
MATHEMATICAL INSTITUTE UNIVERSITY OF BERN SIDLERSTRASSE 5 3012 BERN, SWITZERLAND
LUCA REGGIO
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF OXFORD 15 PARKS ROAD OXFORD OX1 3QD, UK E-mail: luca.reggio@cs.ox.ac.uk
*
E-mail: george.metcalfe@math.unibe.ch
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Abstract

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion.

Keywords

model completionuniversal classequationally definable principal congruenceslattice-ordered abelian groupMV-algebraresiduated lattice

MSC classification

Primary: 03C05: Equational classes, universal algebra
Secondary: 03C10: Quantifier elimination, model completeness and related topics 03C40: Interpolation, preservation, definability 03C64: Model theory of ordered structures; o-minimality 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 06D35: MV-algebras 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 1 , March 2023 , pp. 381 - 417
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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