Linear Extensions of Orders Invariant under Abelian Group Actions
نویسنده
چکیده
Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a G-invariant linear preorder ≤ on the powerset PX such that if A is a proper subset of B, then A < B (i.e., A ≤ B but not B ≤ A).
منابع مشابه
On component extensions locally compact abelian groups
Let $pounds$ be the category of locally compact abelian groups and $A,Cin pounds$. In this paper, we define component extensions of $A$ by $C$ and show that the set of all component extensions of $A$ by $C$ forms a subgroup of $Ext(C,A)$ whenever $A$ is a connected group. We establish conditions under which the component extensions split and determine LCA groups which are component projective. ...
متن کاملN = 1 and N = 2 Supersymmetric Non - Abelian Born - Infeld Actions from Superspace 1
The Goldstone-Maxwell interpretation of the known abelian N=1 and N=2 su-persymmetric Born-Infeld actions in four dimensions is used to construct their new non-abelian generalizations in N=1 and N=2 superspace, respectively, to all orders in α ′. The proposed invariant actions are dictated by simple (manifestly supersymmetric and gauge-covariant) non-linear constraints.
متن کاملShift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups
We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...
متن کاملFrames and Homogeneous Spaces
Let be a locally compact non?abelian group and be a compact subgroup of also let be a ?invariant measure on the homogeneous space . In this article, we extend the linear operator as a bounded surjective linear operator for all ?spaces with . As an application of this extension, we show that each frame for determines a frame for and each frame for arises from a frame in via...
متن کاملComputation of invariants of finite abelian groups
We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the non-modular case. By diagonalization, such a group action can be described by integer matrices of orders and exponents. We make use of integer linear algebra to compute a minimal generating set of invariants along with the substitution needed to rewrite any invariant in t...
متن کامل