Hierarchy of families of theories and their rank characteristics

S. V. Sudoplatov

doi:10.26516/1997-7670.2020.33.80

Hierarchy of families of theories and their rank characteristics

Переведенное название: Иерархия семейств теорий и их ранговые характеристики

S. V. Sudoplatov

Кафедра алгебры и математической логики ММФ

Результат исследования: Научные публикации в периодических изданиях › обзорная статья › рецензирование

2 Цитирования (Scopus)

Аннотация

Изучение семейств элементарных теорий дает информацию о поведении и взаимосвязях теорий внутри семейств, возможности порождения и их сложности. Эта сложность выражается ранговыми характеристиками как для семейств, так и для их элементов внутри семейств. В работе вводится и описывается иерархия семейств теорий и их ранговые характеристики, включая динамику рангов. Рассматриваются регулярные семейства, базирующиеся на основе семейства праэлементов - теорий данной сигнатуры, и пошагового процесса, задающего искомую иерархию. Для отражения шагов этого процесса используется ординально-значный теоретико-множественный ранг. Вводится ранг RS и связанные с ним ранги для регулярных семейств относительно определимых предложениями подсемейств, обобщается известный RS-ранг для семейств праэлементов, а также их степень. На основе отделимости множеств праэлементов описываются связи и динамика для этих рангов и степеней. Вводятся и характеризуются графы и семейства окрестностей, свидетельствующие о рангах. Показано, что декомпозиции семейств окрестностей и ранговые связи, для дискретных разложений, задают аддитивность и возможность сведения меры сложности для семейств к более простым подсемействам

Переведенное название	Иерархия семейств теорий и их ранговые характеристики
Язык оригинала	английский
Страницы (с-по)	80-95
Число страниц	16
Журнал	Bulletin of Irkutsk State University, Series Mathematics
Том	33
DOI	https://doi.org/10.26516/1997-7670.2020.33.80
Состояние	Опубликовано - 2020

Ключевые слова

Closure
Decomposition
Family of theories
Hierarchy
Rank
Urelement

Доступ к документу

10.26516/1997-7670.2020.33.80

Другие файлы и ссылки

Link to publication in Scopus

Fingerprint

Подробные сведения о темах исследования «Иерархия семейств теорий и их ранговые характеристики». Вместе они формируют уникальный семантический отпечаток (fingerprint).

Цитировать

@article{36445e10e1994c9a98c42c304d5b8af4,

title = "Hierarchy of families of theories and their rank characteristics",

abstract = "Studying families of elementary theories produces an information on behavior and interactions of theories inside families, possibilities of generations and their complexity. The complexity is expressed by rank characteristics both for families and their elements inside families. We introduce and describe a hierarchy of families of theories and their rank characteristics including dynamics of ranks. We consider regular families which based on a family of urelements — theories in a given language, and on a step-by-step process producing the required hierarchy. An ordinal-valued set-theoretic rank is used to reflect steps of this process. We introduce the rank RS and related ranks for regular families, with respect to sentence-definable subfamilies and generalizing the known RS-rank for families of urelements, as well as their degrees. Links and dynamics for these ranks and degrees are described on a base of separability of sets of urelements. Graphs and families of neighbourhoods witnessing ranks are introduced and characterized. It is shown that decompositions of families of neighbourhoods and their rank links, for discrete partitions, produce the additivity and the possibility to reduce complexity measures for families into simpler subfamilies.",

keywords = "Closure, Decomposition, Family of theories, Hierarchy, Rank, Urelement, family of theories, closure, urelement, hierarchy, rank, decomposition, DEFINABLE FAMILIES",

author = "Sudoplatov, {S. V.}",

year = "2020",

doi = "10.26516/1997-7670.2020.33.80",

language = "English",

volume = "33",

pages = "80--95",

journal = "Bulletin of Irkutsk State University, Series Mathematics",

issn = "1997-7670",

publisher = "Irkutsk State University",

}

Hierarchy of families of theories and their rank characteristics. / Sudoplatov, S. V.

В: Bulletin of Irkutsk State University, Series Mathematics, Том 33, 2020, стр. 80-95.

Результат исследования: Научные публикации в периодических изданиях › обзорная статья › рецензирование

TY - JOUR

T1 - Hierarchy of families of theories and their rank characteristics

AU - Sudoplatov, S. V.

PY - 2020

Y1 - 2020

N2 - Studying families of elementary theories produces an information on behavior and interactions of theories inside families, possibilities of generations and their complexity. The complexity is expressed by rank characteristics both for families and their elements inside families. We introduce and describe a hierarchy of families of theories and their rank characteristics including dynamics of ranks. We consider regular families which based on a family of urelements — theories in a given language, and on a step-by-step process producing the required hierarchy. An ordinal-valued set-theoretic rank is used to reflect steps of this process. We introduce the rank RS and related ranks for regular families, with respect to sentence-definable subfamilies and generalizing the known RS-rank for families of urelements, as well as their degrees. Links and dynamics for these ranks and degrees are described on a base of separability of sets of urelements. Graphs and families of neighbourhoods witnessing ranks are introduced and characterized. It is shown that decompositions of families of neighbourhoods and their rank links, for discrete partitions, produce the additivity and the possibility to reduce complexity measures for families into simpler subfamilies.

AB - Studying families of elementary theories produces an information on behavior and interactions of theories inside families, possibilities of generations and their complexity. The complexity is expressed by rank characteristics both for families and their elements inside families. We introduce and describe a hierarchy of families of theories and their rank characteristics including dynamics of ranks. We consider regular families which based on a family of urelements — theories in a given language, and on a step-by-step process producing the required hierarchy. An ordinal-valued set-theoretic rank is used to reflect steps of this process. We introduce the rank RS and related ranks for regular families, with respect to sentence-definable subfamilies and generalizing the known RS-rank for families of urelements, as well as their degrees. Links and dynamics for these ranks and degrees are described on a base of separability of sets of urelements. Graphs and families of neighbourhoods witnessing ranks are introduced and characterized. It is shown that decompositions of families of neighbourhoods and their rank links, for discrete partitions, produce the additivity and the possibility to reduce complexity measures for families into simpler subfamilies.

KW - Closure

KW - Decomposition

KW - Family of theories

KW - Hierarchy

KW - Rank

KW - Urelement

KW - family of theories

KW - closure

KW - urelement

KW - hierarchy

KW - rank

KW - decomposition

KW - DEFINABLE FAMILIES

UR - http://www.scopus.com/inward/record.url?scp=85091927005&partnerID=8YFLogxK

U2 - 10.26516/1997-7670.2020.33.80

DO - 10.26516/1997-7670.2020.33.80

M3 - Review article

AN - SCOPUS:85091927005

VL - 33

SP - 80

EP - 95

JO - Bulletin of Irkutsk State University, Series Mathematics

JF - Bulletin of Irkutsk State University, Series Mathematics

SN - 1997-7670

ER -