Tame group
Encyclopedia
In mathematical
group theory
, a tame group is a certain kind of group
defined in model theory
.
Formally, we define a bad field as a structure of the form (K, T), where K is an algebraically closed field
and T is an infinite, proper, distinguished subgroup
of K, such that (K, T) is of finite Morley rank
in its full language. A group G is then called a tame group if no bad field is interpretable
in G.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a tame group is a certain kind of group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
defined in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
.
Formally, we define a bad field as a structure of the form (K, T), where K is an algebraically closed field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and T is an infinite, proper, distinguished subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of K, such that (K, T) is of finite Morley rank
Morley rank
In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.-Definition:Fix a theory T with a model M...
in its full language. A group G is then called a tame group if no bad field is interpretable
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.-Informal definition:Assume T and S are formal theories...
in G.