Algebras of Binary Formulas for Compositions of Theories
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Algebra and Logic, Vol. 59, No. 4, September, 2020 (Russian Original Vol. 59, No. 4, July-August, 2020)
ALGEBRAS OF BINARY FORMULAS FOR COMPOSITIONS OF THEORIES D. Yu. Emel’yanov,1∗ B. Sh. Kulpeshov,2∗ and S. V. Sudoplatov3∗
UDC 510.67
Keywords: algebra of binary formulas, composition of theories, e-definable composition, ℵ0 categorical theory, strongly minimal theory, stable theory, linear preorder, cyclic preorder. We consider algebras of binary formulas for compositions of theories both in the general case and as applied to ℵ0 -categorical, strongly minimal, and stable theories, linear preorders, cyclic preorders, and series of finite structures. It is shown that edefinable compositions preserve isomorphisms and elementary equivalence and have basicity formed by basic formulas of the initial theories. We find criteria for e-definable compositions to preserve ℵ0 -categoricity, strong minimality, and stability. It is stated that e-definable compositions of theories specify compositions of algebras of binary formulas. A description of forms of these algebras is given relative to compositions with linear orders, cyclic orders, and series of finite structures. Algebras of binary formulas are a tool for describing connections between realizations of types at a binary level relative to a superposition of binary definable sets. These algebras are characterized for the general case in [1-3] and for natural classes of theories in [4-11]. In the present paper, we consider specific features and describe properties and algebras for compositions of theories both in the general case and as applied to ℵ0 -categorical, strongly minimal, and stable theories, linear orders, cyclic orders, and series of finite structures. Our plan is as follows. In Sec. 1, we set out the notation and preliminary concepts. Namely, we define the following: an I-groupoid for an axiomatization of algebras of binary isolating formulas, a composition for graphs and monoids, and a composition of some natural monoids with groups ∗
Supported by RFBR (project No. 20-31-90004), by KN MON RK (grant No. AP08855544), and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2019-0002). 1
Novosibirsk State Technical University, Novosibirsk, Russia; [email protected]. 2 Kazakh-British Technical University. Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science RK, Alma-Ata, Kazakhstan; [email protected]. 3 Sobolev Institute of Mathematics. Novosibirsk State Technical University. Novosibirsk State University, Novosibirsk, Russia; [email protected]. Translated from Algebra i Logika, Vol. 59, No. 4, pp. 432-457, July-August, 2020. Russian DOI: 10.33048/ alglog.2020.59.402. Original article submitted April 9, 2019; accepted November 24, 2020. c 2020 Springer Science+Business Media, LLC 0002-5232/20/5904-0295
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generating algebras of binary isolating formulas for suitable theories (Thms. 1.4, 1.5). In Sec. 2, we introduce natural concepts of compositions for structures and theories that generalize the corresponding notion
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