Abstract
We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart AlgS of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in AlgS are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.
| Original language | English |
|---|---|
| Pages (from-to) | 1502-1519 |
| Number of pages | 18 |
| Journal | Annals of Pure and Applied Logic |
| Volume | 161 |
| Issue number | 12 |
| DOIs | |
| Publication status | Published - 1 Sept 2010 |
| Externally published | Yes |
Funding
I The research of the second author has been partially supported by SGR2005-00083 research grant of the research funding agency AGAUR of the Generalitat de Catalunya and by the MTM2008-01139 research grant of the Spanish Ministry of Education and Science. II The research of the third author has been supported by the VENI grant 639.031.726 of the Netherlands Organization for Scientific Research (NWO).
Keywords
- Abstract Algebraic Logic
- Canonical extensions
- Deduction theorem
- Hilbert algebras
- Tarski algebras