Positively closed Sh(B)-valued models
Abstract
We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For $\mathbf{Set}$-valued models of coherent theories they coincide. We prove that if $\mathcal{E}=Sh(B,τ_{coh})$ for a complete Boolean algebra, then positively closed but not strongly positively closed $\mathcal{E}$-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed $\mathcal{E}$-valued models. A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment $L^g_{κκ}$ where $κ$ is weakly compact.
Fields of Study & Subjects
Constraint Satisfaction and Optimization
Publication details
- Journal / source
- Annals of Pure and Applied Logic
- Publisher
- Elsevier BV
- Volume
- 177
- Issue
- 6
- Pages
- 103726–103726
- Publication date
- 28 January 2026
- Language
- EN