Closures and generating sets related to combinations of structures

• Main Author: S. Sudoplatov
• Format: Article
• Language: English
• Published: Irkutsk State University 2016-06-01
• Series: Известия Иркутского государственного университета: Серия "Математика"
• Subjects: E-combination; P-combination; closure operator; generating set
• Online Access: http://isu.ru/journal/downloadArticle?article=_59fb90a1a2eb4cb48f0b570354d5d0dd&lang=eng
• ISSN: 1997-7670; 2541-8785

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories including the negation of finite character and the exchange property. It is shown that closure operators for E-combinations correspond to the closures with respect to the ultraproduct operator forming Hausdorff topological spaces. It is also shown that closure operators for disjoint $P$-combinations form topological $T_0$-spaces, which can be not Hausdorff. Thus topologies for $E$-combinations and $P$-combinations are rather different. We prove, for E-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set: it is shown that elements of the least generating set are isolated and dense in its $E$-closure. Related properties for P-combinations are considered: it is proved that again the existence of a minimal generating set of theories is equivalent to the existence of the least generating set but it is not equivalent to the isolation of elements in the generating set. It is shown that $P$-closures with the least generating sets are connected with families which are not $\omega$-reconstructible, as well as with families having finite $e$-spectra. Two questions on the least generating sets for E-combinations and P-combinations are formulated and partial answers are suggested.