TY - JOUR
T1 - On products of elementarily indivisible structures
AU - Meir, Nadav
N1 - Funding Information:
The work in this paper is part of the author’s M.Sc. thesis, prepared under the supervision of Assaf Hasson. The author would like to gratefully acknowledge him for presenting the questions discussed in the paper, as well as for fruitful discussions and the great help and support along the way. The author was partially supported by an Israel Science Foundation grant number 1156/10.
Publisher Copyright:
© 2016, Association for Symbolic Logic.
PY - 2016/9/1
Y1 - 2016/9/1
N2 - We say a structure M in a first-order language L is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure M′ ⊆ M such that M ≅ M. Additionally, we say thatMis symmetrically indivisible if M′ can be chosen to be symmetrically embedded in M (that is, every automorphism of M′ can be extended to an automorphism of M). Similarly, we say that M is elementarily indivisible if M′ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.
AB - We say a structure M in a first-order language L is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure M′ ⊆ M such that M ≅ M. Additionally, we say thatMis symmetrically indivisible if M′ can be chosen to be symmetrically embedded in M (that is, every automorphism of M′ can be extended to an automorphism of M). Similarly, we say that M is elementarily indivisible if M′ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.
KW - Coloring
KW - Elementary indivisibility
KW - Indivisibility
KW - Quantifier elimination
UR - http://www.scopus.com/inward/record.url?scp=84987829356&partnerID=8YFLogxK
U2 - 10.1017/jsl.2015.74
DO - 10.1017/jsl.2015.74
M3 - Article
AN - SCOPUS:84987829356
SN - 0022-4812
VL - 81
SP - 951
EP - 971
JO - Journal of Symbolic Logic
JF - Journal of Symbolic Logic
IS - 3
ER -