Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory.

In the article we describe algebras of binary formulas for the theories of Archimedean solids. For the obtained algebras, Cayley tables are given. It is shown that these algebras are exhausted by described algebras for a truncated cube, truncated octahedron, rhombocuboctahedron, icosododecahedron, truncated tetrahedron, cubooctahedron, flatnosed cube, flat-nosed dodecahedron, truncated cubooctahedron, rhomboicosododecahedron, truncated icosahedron, truncated dodecahedron, rhombo-truncated icosododecahedron.

Dmitry Emel’yanov, Postgraduate Student, Novosibirsk State Technical University, 20, K. Marx Ave., 630073, Novosibirsk, Russian Federation; tel.: (383)3461166, e-mail: dima-pavlyk@mail.ru

Emel’yanov D.Yu. Algebras of Distributions of Binary Formulas for Theories of Archimedean Solids. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 36-52. (in Russian) https://doi.org/10.26516/1997-7670.2019.28.36

1. Ashkinuze V.G. On the number of semi-regular polyhedra. Matem. prosv., 1957, vol. 1, pp. 107-118.
2. Baikalova K.A., Emelyanov D.Yu., Kulpeshov B.Sh., Palyutin E.A., Sudoplatov S.V. On algebras of distributions of binary isolating formulas for theories of abelian groups and their ordered expansions. Russian Math., 2018, no. 4, pp. 1-13. https://doi.org/10.3103/s1066369x18040011
3. Baizhanov B.S. Orthogonality of one types in weakly o-minimal theories. Algebra and Model Theory 2, Collection of papers, Pinus A.G., Ponomaryov K.N. (eds.). Novosibirsk, NSTU Publisher, 1999, pp. 5-28.
4. Baizhanov B.S., Sudoplatov S.V., Verbovskiy V.V. Conditions for non-symmetric relations of semi-isolation. Siberian Electronic Mathematical Reports, 2012, vol. 9, pp. 161-184.
5. Emel’yanov D.Yu. On algebras of distributions of binary formulas for theories of unars. The Bulletin of Irkutsk State University. Series Mathematics, 2016, vol. 17, pp. 23-36. (in Russian)
6. Emel’yanov D.Yu., Kulpeshov B.Sh., Sudoplatov S.V. Algebras of distributions for binary formulas in countably categorical weakly o-minimal structures. Algebra and Logic, 2017, vol. 56, no. 1, pp. 13-36. https://doi.org/10.17377/alglog.2017.56.102
7. Emel’yanov D.Yu., Kulpeshov B.Sh., Sudoplatov S.V. Algebras of distributions of binary isolating formulas for quite o-minimal theories. Algebra and Logic, 2018, vol. 57, no. 6, pp. 429-444. https://doi.org/10.1007/s10469-019-09515-5
8. Emel’yanov D.Yu., Sudoplatov S.V. On deterministic and absorbing algebras of binary formulas of polygonometrical theories. The Bulletin of Irkutsk State University. Series Mathematics, 2017, vol. 20, pp. 32-44. (in Russian) https://doi.org/10.26516/1997-7670.2017.20.32
9. Emel’yanov D.Yu. Algebras of binary isolating formulas for simplex theories. Algebra and Model Theory 11. Collection of papers, Novosibirsk, NSTU Publisher, 2017, pp. 66–74.
10. Gurin A.M. To history of studying of convex polyhedra with regular faces. Siberian Electronic Mathematical Reports, 2010, vol. 7, pp. A.5–A.23.
11. Pillay A. Countable models of stable theories. Proc. Amer. Math. Soc., 1983, vol. 89, no. 4, pp. 666-672.
12. Shulepov I.V., Sudoplatov S.V. Algebras of distributions for isolating formulas of a complete theory. Siberian Electronic Mathematical Reports, 2014, vol. 11, pp. 380-407.
13. Sudoplatov S.V. Classification of countable models of complete theories. Part 1. Novosibirsk, NSTU Publisher, 2018, 376 p.
14. Sudoplatov S.V. Hypergraphs of prime models and distributions of countable models of small theories. J. Math. Sciences, 2010, vol. 169, no. 5, pp. 680-695. https://doi.org/10.1007/s10958-010-0069-9
15. Sudoplatov S.V. Algebras of distributions for semi-isolating formulas of a complete theory. Siberian Electronic Mathematical Reports, 2014, vol. 11, pp. 408-433.
16. Sudoplatov S.V. Algebras of distributions for binary semi-isolating formulas for families of isolated types and for countably categorical theories. International Mathematical Forum, 2014, vol. 9, no. 21, pp. 1029-1033.