A note on the convexity of lattices generated by the set of nonnegative integers.
The notion of convexity of lattices is due to \textit{E.~Fried} (Krems, 1988); he defined it as a nonempty class of lattices which is closed with respect to homomorphisms, direct products and convex sublattices. \textit{J.~Lihová} [Publ. Math. 72, No. 1-2, 35-43 (2008; Zbl 1164.06001)] investigated relations between convexities generated by chains and particularly, she proved that convexities generated by nonisomorphic finite chains are different.NEWLINENEWLINE The present paper shows that an analogous assertion fails to hold for convexities generated by different infinite ordinals. Namely, it is proved that all infinite ordinals generate the convexity which can be generated by \(\omega_0\); this is a consequence of the fundamental result being as follows: The convexity generated by nonnegative integers contains all ordinal numbers. The used model-theoretic approach throughout the paper appears to be appropriate and effective.
scientific article; zbMATH DE number 6333341