Partially commutative metabelian groups: centralizers and elementary equivalence

Abstract

Keywords: metabelian group, centralizer, annihilator, elementary equivalence.For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsionfree nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory.

Algebraic geometry over algebraic structures. II. Foundations

“…In the last few years considerable progress has been made towards understanding algebraic geometry over partially commutative groups. Here we would like to mention the following papers [6, 7,8,35,49,78].…”

“…In the last few years considerable progress has been made towards understanding algebraic geometry over partially commutative groups. Here we would like to mention the following papers [6, 7,8,35,49,78].Algebraic geometry over algebraic structures is also being developed for algebraic structures other than groups. Nice results were obtained in algebraic geometry over commutative monoids with cancellation [50,76,77].…”

Universal equivalence of partially commutative metabelian Lie algebras

“…We can assume without loss of generality that s = j and t = k. So, g i+1 = γx j + δx k . By (6) we obtain [[g i−1 , g i+1 ], g i ] = 0. Thus,…”

“…Partially commutative groups are studied very heavily nowadays (see [12,2,13,14,5,6,15,16,7]). In some papers (for example, in [6,15,7]), universal theories of partially commutative metabelian groups were studied.…”

Bases for partially commutative Lie algebras

“…Partially commutative groups (the term "graph groups" is also used) are studied very heavily nowadays (see [11,7,12,13,9,5,17], for example). Although, there are some results obtained for other partially commutative structures (see [10,4,6]).…”