scholarly journals Nilpotency and strong nilpotency for finite semigroups

2018 ◽  
Vol 70 (2) ◽  
pp. 619-648
Author(s):  
J Almeida ◽  
M H Shahzamanian ◽  
M Kufleitner
Keyword(s):  
Finite Rank ◽  
Nilpotent Groups ◽  
Finite Semigroups ◽  
Semigroup Identities ◽  
Mal'cev Product ◽  
Mal’Cev Product

AbstractNilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

Modules over nilpotent groups of finite rank

1985 ◽  
Vol 24 (6) ◽  
pp. 412-436 ◽  
Author(s):  
D. I. Zaitsev ◽  
L. A. Kurdachenko ◽  
A. V. Tushev
Keyword(s):  
Finite Rank ◽  
Nilpotent Groups

On the derived subgroups of the free nilpotent groups of finite rank

2008 ◽  
Author(s):  
Russell D. Blyth ◽  
Primož Moravec ◽  
Robert Fitzgerald Morse
Keyword(s):  
Finite Rank ◽  
Nilpotent Groups

Noetherian modules over nilpotent groups of finite rank

1991 ◽  
Vol 56 (5) ◽  
pp. 433-436
Author(s):  
L. A. Kurdachenko ◽  
A. V. Tushev ◽  
D. I. Zaitsev
Keyword(s):  
Finite Rank ◽  
Nilpotent Groups ◽  
Noetherian Modules

scholarly journals Mal'cev products of varieties of completely regular semigroups

1987 ◽  
Vol 42 (2) ◽  
pp. 227-246 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Mal'cev Product ◽  
Mal’Cev Product

AbstractWhilst the Mal'cev product of completely regular varieties need not again be a variety, it is shown that in many important instances a variety is in fact obtained. However, unlike the product of group varieties this product is nonassociative.Two important operators introduced by Reilly are studied in the context of Mal'cev products. These operators are shown to generate from any given variety one of the networks discovered by Pastijn and Trotter, enabling identities to be provided for the varieties in the network. In particular the join O V BG of the varieties of orthogroups and of bands of groups is determined, answering a question of Petrich.

scholarly journals Hyperdecidability of pseudovarieties of orthogroups

2001 ◽  
Vol 43 (1) ◽  
pp. 67-83 ◽  
Author(s):  
Jorge Almeida ◽  
Peter G. Trotter
Keyword(s):  
Word Problem ◽  
Weak Inversion ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Mal'cev Product ◽  
Mal’Cev Product

Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product of the pseudovariety of bands with a pseudovariety V of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in the case in which terms involving only multiplication and weak inversion are considered. It is also shown that, if V is a hyperdecidable (respectively canonically reducible) pseudovariety of groups, then so is W.

COMMUTATOR SUBGROUPS OF FREE NILPOTENT GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1021-1031
Author(s):  
N. GUPTA ◽  
I. B. S. PASSI
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Finite Chain ◽  
Free Nilpotent Group ◽  
Infinite Rank ◽  
Derived Subgroup ◽  
Finite Set

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γn(F) of the free group F of rank m with respect to a certain finite chain F = F(0) > F(1) > ⋯ > F(l-1) > F(l) = {1} of free groups in which F(k) is of finite rank m(k) and is contained in the kth derived subgroup δk(F) of F. The derived subgroups δk(F/γn(F)) of the free nilpotent group F/γn(F) are isomorphic to the quotients F(k)/(F(k) ∩ γn(F)) and admit presentations of the form 〈xk,1,…,xk,m(k): γ(n)(F(k))〉, where γ(n)(F(k)), contained in γn(F), is a certain partial lower central subgroup of F(k). We give a complete description of γn(F) as a staggered product Π1 ≤ k ≤ l-1(γ〈n〉(F(k))*γ[n](F(k)))F(k+1), where γ〈n〉(F(k)) is a free factor of the derived subgroup [F(k),F(k)] of F(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ[n](F(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

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