English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider classes of arbitrary (finite or infinite) graphs of boundedshrub-depth, specifically the class $\mathrm{TM}_{r, p}(d)$ of $p$-labeledarbitrary graphs whose underlying unlabeled graphs have tree models of height$d$ and $r$ labels. We show that this class satisfies an extension of theclassical L\"owenheim-Skolem property into the finite and for $\mathrm{MSO}$.This extension being a generalization of the small model property, we obtainthat the graphs of $\mathrm{TM}_{r, p}(d)$ are pseudo-finite. In addition, weobtain as consequences entirely new proofs of a number of known resultsconcerning bounded shrub-depth classes (of finite graphs) and $\mathrm{TM}_{r,p}(d)$. These include the small model property for $\mathrm{MSO}$ withelementary bounds, the classical compactness theorem from model theory over$\mathrm{TM}_{r, p}(d)$, and the equivalence of $\mathrm{MSO}$ and$\mathrm{FO}$ over $\mathrm{TM}_{r, p}(d)$ and hence over bounded shrub-depthclasses. The proof for the last of these is via an adaptation of the proof ofthe classical Lindstr\"om's theorem characterizing $\mathrm{FO}$ over arbitrarystructures.

Some classical model theoretic aspects of bounded shrub-depth classes