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Comparison between the universal theories of partially commutative metabelian groups. (English. Russian original) Zbl 1400.20021

Sib. Math. J. 58, No. 3, 382-391 (2017); translation from Sib. Mat. Zh. 58, No. 3, 497-509 (2017).
Summary: We find necessary and sufficient conditions for the coincidence of the universal theories of partially commutative groups of metabelian varieties defined by acyclic graphs.

MSC:

20F05 Generators, relations, and presentations of groups
20E10 Quasivarieties and varieties of groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F70 Algebraic geometry over groups; equations over groups
03B10 Classical first-order logic
03C60 Model-theoretic algebra
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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References:

[1] Gupta Ch. K. and Timoshenko E. I., “Partially commutative metabelian groups: centralizers and elementary equivalence,” Algebra and Logic, vol. 48, no. 3, 173-192 (2009). · Zbl 1245.20032 · doi:10.1007/s10469-009-9051-3
[2] Timoshenko E. I., “Universal equivalence of partially commutative metabelian groups,” Algebra and Logic, vol. 49, no. 2, 177-196 (2010). · Zbl 1220.20024 · doi:10.1007/s10469-010-9088-3
[3] Gupta Ch. K. and Timoshenko E. I., “Universal theories for partially commutative metabelian groups,” Algebra and Logic, vol. 50, no. 1, 1-16 (2011). · Zbl 1263.20032
[4] Romanovskiĭ N. S., “On Shmel’kin embeddings for abstract and profinite groups,” Algebra and Logic, vol. 38, no. 5, 326-334 (1999). · Zbl 0943.20020 · doi:10.1007/BF02671749
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