Universal algebra and lattice theory

The IRN project AP09058390 was carried out under a grant from the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan

Project Manager: S.M. Lutsak, PhD, e-mail sveta_lutsak@mail.rumailto:sveta_lutsak@mail.ru , ScopusAuthorID 57194636622, https://www.scopus.com/authid/detail.uri?authorId=57194636622 , ResearcherIDAAW-8985-2020, https://publons.com/researcher/3845860/svetlana-lutsak /

Project executors: Voronina O.A., Ph.D., Nurakhmetova G.N., Master's degree.

Terms of execution: 2021-2023.

Amount of funding: 50,648,800 tenge.

Relevance: The stated project relates to the field of universal algebra and lattice theory. This project is related to the fundamental Birkhoff-Maltsev problem on the description of lattices isomorphic to lattices (quasi)varieties. Despite the fact that the problem has been posed for a long time and the study of lattices of (quasi) manifolds has been intensively carried out for several decades in a number of countries (Kyrgyzstan, Poland, Russia, USA), effective approaches to its solution have not yet been developed. Moreover, the results obtained so far demonstrate the exceptional structural and algorithmic complexity of these lattices. Thus, progress in the study of the properties of such lattices is a very relevant issue. The results obtained within the framework of this project will be used for further study of lattices (quasi)varieties. The issues studied in the framework of the project are closely related to the issues of the basability of axiomatic classes, with the issues of algorithmic computability and can be applied in such related fields as model theory, computability theory, discrete mathematics. The results can also be used in computer science and biology.

The purpose of the project: Study of the complexity of the structure of derived lattices, such as lattices (relative) (quasi)manifolds, lattices of (relative) congruences, lattices of semigroups of semigroups of elementary theories.

Expected results: Theorem(s) on isomorphism of certain lattices and lattices of (relative) (quasi)manifolds, lattices of (relative) congruences, lattices of sub-semigroups of semigroups of elementary theories (in various special cases). Theorem(s) on nontrivial properties of the lattices under study (in certain special cases).

Achieved results:The Birkhoff-Maltsev problem on the description of lattices isomorphic to lattices (quasi) is investigatedmanifolds for Lukasevich algebras. A study of the complexity of the structure of the lattice of quasimanifolds of a manifold generated by the set of all finite Lukasevich algebras is performed. The existence of a finite B-class in this manifold is proved. The highest complexity of the lattice of quasimanifolds of a manifold generated by the set of all finite Lukasevich algebras is shown. It is proved that the lattice of quasimanifolds of a variety generated by the set of all finite Lukasevich algebras is Q–universal. Moreover, the variety generated by the set of all finite Lukasevich algebras contains a continuum of Q-universal quasimanifolds, a continuum of quasimanifolds that have no coverings in the lattice under study, a continuum of subclasses with property (N) (a subclass such that the set of isomorphism types of the class of all finite sublattices of its lattice (relative) (quasi)manifolds are uncountable), a continuum of subclasses with property (N), but nevertheless not Q-universal. These results are reflected in the article "On complexity of quasivariety lattices of Lukasiewicz algebra".

On the semigroup of complete theories of a first-order countable language with respect to the product of complete theories, the formula-definable subsemigroups and idempotently formula-definable subsemigroups of the semigroup of complete theories are investigated. The properties they satisfy are proved. It is shown that the idempotent elements of idempotently formula-definable subsemigroups form a complete lattice. These results are reflected in the article "Properties of formula-definable semigroups of complete theories".

Sufficient conditions of V.I. are investigated. In the case of which a locally finite quasi-manifold of lattices does not have a finite (independent) basis of quasi-identities. A finite lattice is found that does not have a finite basis of quasi-identities and does not satisfy one of the conditions of V.I. Tumanova. These results are reflected in the article "Onquasi-identitiesoffinitelattices".

The results of the research were reported at international scientific conferences. To date, 6 papers have been published in the materials of international scientific conferences (3 – the Republic of Kazakhstan, 3 – the near abroad). Based on the results of the research, articles have been prepared for publication in peer-reviewed publications recommended by COXON MES RK, in a peer-reviewed scientific publication on the scientific direction of the project, included in the 3rd (third) quartile in the WebofScience database.

List of publications: 

1. Lutsak S., Voronina O. «Complexity of the quasivariety lattice for the variety of Lukasiewicz algebras». The traditional April International scientific conference in honor of Science Day, dedicated to the 75th anniversary of Academician of the National Academy of Sciences of the Republic of Kazakhstan T.Sh. Kalmenova. Abstracts of reports. Almaty, Institute of Mathematics and Mathematical Modeling of the KN MES RK, April 5-8, 2021, pp. 133-134.
http://www.math.kz/public/filemanager/userfiles/Abstracts_of_conference_2021.pdf

2. Lutsak S.M., Voronina O.A. «Quasivariety lattice of Lukasiewicz algebras». Materials of the international scientific and practical online conference "Youth and Science – 2021", Petropavlovsk, M.Kozybayev SKU, April 9, 2021, p. 206-208.
/files/conference/min2021/min2021_tom2.pdf

3. Lutsak S.M., Voronina O.A. Q is the universality of the lattice of quasi-manifolds of Lukasevich algebras. Collection of materials of the XVI International Scientific Conference "YLYLYM JÁNE BILIM – 2021", Nur-Sultan, L.N. Gumilyov ENU, April 12, 2021, pp. 1321-1324.
https://enu.kz/downloads/iyun-2021/nio-2021-sekciya-4.pdf

4. Lutsak S.M., Voronina O.A. «On the existence of a finite B-class in the variety of all Lukasiewicz algebra». Abstracts of the international scientific conference "Problems of modern Mathematics and its applications" dedicated to the 70th anniversary of Academician A.A. Borubaeva, Kyrgyzstan, Bishkek, on the basis of the NAS of the Kyrgyz Republic, June 16-19, 2021, p. 111.
http://math.aknet.kg/ru/conf.html

5. Lutsak S.M., Voronina O.A. «On complexity of quasivariety lattices of Lukasiewicz algebras». International Conference "Maltsev Readings 2021". Abstracts of reports. Novosibirsk (Russia). International Mathematical Center in Akademgorodok, NSU, S.L. Institute of Mathematics SobolevaSORAN, September 20-24, 2021, p. 167.
http://www.math.nsc.ru/conference/malmeet/21/maltsev21.pdf

6. Bekenov M.I., Kasatova A.M., Lutsak S.M. «On the properties of formula-definable semigroups of complete theories». International Conference "Maltsev Readings 2021". Abstracts of reports. Novosibirsk (Russia). International Mathematical Center in Akademgorodok, NSU, S.L. Institute of Mathematics Soboleva SB RAS, September 20-24, 2021, p. 160.
http://www.math.nsc.ru/conference/malmeet/21/maltsev21.pdf

Put into print:

1. Lutsak S.M., Voronina O.A., Kasatova A.M. «On complexity of quasivariety lattices of Lukasiewicz algebras». Bulletin of Karaganda University. Mathematics Series (indexing: Webof Science Core Collection). COXON MES RK.

2. Bekenov M.I., Kasatova A.M., Lutsak S.M. "Properties of formula-definable semigroups of complete theories". Siberian Mathematical Journal (published by the S.L. Institute of Mathematics Soboleva SB RAS, published in Russian, translated into English and printed in the USA, indexing: WebofScienceCoreCollection, Quartilein JCR Category: mathematics: Q3 (2020), Scopus, Quartilein SJR Category: mathematics (miscellaneous): Q1 (2020)).

3. Lutsak S.M., Voronina O.A., Nurakhmetova G.K. «Onquasi-identitiesoffinitelattices». Bulletin of the Treasury. Mathematics, Mechanics, Computer Science series (indexing: Webof Science). COXON MES RK.