TY - JOUR
T1 - Infinite lexicographic products
AU - Meir, Nadav
N1 - Funding Information:
The author was partially supported by ISF Grant 181/16 and the Hillel Gauchman scholarship. The author was partially supported by Leverhulme Trust grant number RPG-2017-179. The author is supported by the Narodowe Centrum Nauki grant no. 2016/22/E/ST1/00450.
Funding Information:
The work in this paper is part of the author's Ph.D. studies at the Department of Mathematics, Ben-Gurion University of the Negev under the supervision of Assaf Hasson. The author is grateful to Assaf Hasson for presenting the question which motivated this paper, as well as for the fruitful discussions and the warm support along the way. The author thanks Itay Kaplan for remarks on a previous version that improved the quality of this paper. The author is grateful to Julia Gauchman and the Gauchman family for the financial support and encouragement.
Publisher Copyright:
© 2021
PY - 2022/1/1
Y1 - 2022/1/1
N2 - We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.
AB - We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.
KW - Elementary indivisibility
KW - Homogeneous structures
KW - Indivisibility
KW - Infinitary logic
KW - Lexicographic product
KW - Quantifier elimination
UR - http://www.scopus.com/inward/record.url?scp=85112787151&partnerID=8YFLogxK
U2 - 10.1016/j.apal.2021.102991
DO - 10.1016/j.apal.2021.102991
M3 - Article
AN - SCOPUS:85112787151
VL - 173
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
SN - 0168-0072
IS - 1
M1 - 102991
ER -