# Category:Power Structures

Jump to navigation
Jump to search

This category contains results about Power Structures.

Definitions specific to this category can be found in Definitions/Power Structures.

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\powerset S$ denote the power set of $S$.

Let $\circ_\PP$ denote the operation induced on $\powerset S$ by $\circ$ as follows:

- $\forall A, B \in \powerset S: A \circ_\PP B = \set {a \circ b: a \in A, b \in B}$

Then the resulting algebraic structure $\struct {\powerset S, \circ_\PP}$ is called the **power structure of $\struct {S, \circ}$**.

## Subcategories

This category has only the following subcategory.

## Pages in category "Power Structures"

The following 22 pages are in this category, out of 22 total.

### C

### P

- Power Structure of Group is Monoid
- Power Structure of Group is Semigroup
- Power Structure of Magma is Magma
- Power Structure of Monoid is Monoid
- Power Structure of Semigroup is Semigroup
- Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup
- Power Structure of Semigroup Ordered by Supersets is Ordered Semigroup
- Power Structure of Subset is Closed iff Subset is Closed
- Power Structure Operation on Set of Singleton Subsets is Closed
- Power Structure Operation on Set of Singleton Subsets preserves Associativity
- Power Structure Operation on Set of Singleton Subsets preserves Commutativity

### S

- Set of Closed Subsets of Power Structure of Entropic Structure is Closed
- Set of Finite Subsets under Induced Operation is Closed
- Set of Subgroups of Abelian Group form Subsemigroup of Power Structure
- Set of Subsemigroups of Commutative Semigroup form Subsemigroup of Power Structure
- Subset Relation is Compatible with Subset Product