Rational closure of the group algebra of a linearly ordered group in the division ring of Mal'tsev series

1992 ◽  
Vol 32 (6) ◽  
pp. 939-948 ◽  
Author(s):  
N. I. Dubrovin
Keyword(s):  
Group Algebra ◽  
Division Ring ◽  
Ordered Group ◽  
Rational Closure ◽  
Linearly Ordered Group

Stable decomposability of matrices over the rational closure of a group algebra of an ordered group

2020 ◽  
Vol 211 (9) ◽  
pp. 1213-1232
Author(s):  
N. I. Dubrovin
Keyword(s):  
Group Algebra ◽  
Ordered Group ◽  
Rational Closure

scholarly journals Congruences on bicyclic extensions of a linearly ordered group

2020 ◽  
Vol 15 (2) ◽  
pp. 61-80
Author(s):  
Oleg Gutik ◽  
Dušan Pagon ◽  
Kateryna Pavlyk
Keyword(s):  
Positive Cone ◽  
Inverse Semigroups ◽  
Green’S Relations ◽  
Ordered Group ◽  
Green's Relations ◽  
Linearly Ordered Group

In the paper we study inverse semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G) which are generated by partial monotone injective translations of a positive cone of a linearly ordered group G. We describe Green’s relations on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G), their bands and show that they are simple, and moreover, the semigroups B(G) and B^+(G) are bisimple. We show that for a commutative linearly ordered group G all non-trivial congruences on the semigroup B(G) (and B^+(G)) are group congruences if and only if the group G is archimedean. Also we describe the structure of group congruences on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G).

scholarly journals A noncommutative ordinally simple linearly ordered group

1951 ◽  
Vol 2 (6) ◽  
pp. 902-902 ◽  
Author(s):  
A. H. Clifford
Keyword(s):  
Ordered Group ◽  
Linearly Ordered Group

Free pairs of symmetric and unitary units in normal subgroups of a division ring

2017 ◽  
Vol 16 (06) ◽  
pp. 1750108 ◽  
Author(s):  
Jairo Z. Goncalves
Keyword(s):  
Heisenberg Group ◽  
Free Group ◽  
Group Algebra ◽  
Division Ring ◽  
General Condition ◽  
Normal Subgroups ◽  
Unitary Subgroup

Let [Formula: see text] be the field of fractions of the group algebra [Formula: see text] of the Heisenberg group [Formula: see text], over the field [Formula: see text] of characteristic [Formula: see text]. We show that for some involutions of [Formula: see text] that are not induced from involutions of [Formula: see text], [Formula: see text] contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

scholarly journals Lexicographic factors of a linearly ordered group

1989 ◽  
Vol 39 (1) ◽  
pp. 111-119
Author(s):  
Ján Jakubík
Keyword(s):  
Ordered Group ◽  
Linearly Ordered Group

scholarly journals Lexicographic product decompositions of a linearly ordered group

1986 ◽  
Vol 36 (4) ◽  
pp. 553-563
Author(s):  
Ján Jakubík
Keyword(s):  
Lexicographic Product ◽  
Ordered Group ◽  
Linearly Ordered Group

Generalized Specker lattice-ordered groups and two types of distributivity

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Ján Jakubík ◽  
Judita Lihová
Keyword(s):  
Boolean Algebra ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Linearly Ordered Group ◽  
Specker Lattice Ordered Group ◽  
Group B ◽  
Generalized Boolean Algebra ◽  
Boolean Extension

AbstractLet A be a lattice-ordered group, B a generalized Boolean algebra. The Boolean extension A B of A has been investigated in the literature; we will refer to A B as a generalized Specker lattice-ordered group (namely, if A is the linearly ordered group of all integers, then A B is a Specker lattice-ordered group). The paper establishes that some distributivity laws extend from A B to both A and B, and (under certain circumstances) also conversely.

Undecidable wreath products and skew power series fields

1998 ◽  
Vol 63 (1) ◽  
pp. 237-246 ◽  
Author(s):  
Françoise Delon ◽  
Patrick Simonetta
Keyword(s):  
Power Series ◽  
Large Class ◽  
Division Ring ◽  
Wreath Products ◽  
Ordered Group ◽  
Division Rings ◽  
Skew Fields ◽  
Image Position ◽  
Rings Of Power Series

We prove the undecidability of a very large class of restricted and unrestricted wreath products (Theorem 1.2), and of some skew fields of power series (Section2). Both undecidabilities are obtained by interpreting some enrichments of twisted wreath products, which are themselves proved to be undecidable (Proposition 1.1).We consider division rings of power series in various languages:We show (Theorem 2.8) that every power series division ring k((B)), whose field of constants k is commutative and whose ordered group of exponents is noncommutative with a convex center, is undecidable in every extension of the language of rings where the valuation and the ordered group B are definable.For certain k and B we prove here the undecidability of the structurewhere X↾k((B))xB is the restriction of the multiplication to k((B)) Χ B,and γ is a given conjugation of k((B)). This shows that we cannot hope to improve our previous result, a sort of Ax-Kochen-Ershov principle for power series division rings, which ensures thatis decidable for every decidable solvable B.

A note on Lie nilpotent group rings

1992 ◽  
Vol 45 (3) ◽  
pp. 503-506 ◽  
Author(s):  
R.K. Sharma ◽  
Vikas Bist
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Group Rings ◽  
Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

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