On structures in hypergraphs of models of a theory

We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types of models of a theory, are given

Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.Comment: 16 page

ON ORDERED MINIMAL STRUCTURES

We investigate minimal rst-order structures and consider interpretability and denability of orderings on them. We also prove the minimality of their canonical substructures

On structures in hypergraphs of models of a theory

Hypergraphs of models of a theory are derived objects allowing toobtain an essential structural information about both giventheories and related semantic objects including graph ones. In the present paper we define and study structural properties of hypergraphs of modelsof a theory including lattice ones. Characterizations for thelattice properties of hypergraphs of models of a theory, as wellas for structures on sets of isomorphism types of models of atheory, are given