Definition:Subset Product

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Definition

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


We can define an operation on the power set $\powerset S$ as follows:

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


This is called the operation induced on $\powerset S$ by $\circ$, and $A \circ_\PP B$ is called the subset product of $A$ and $B$.


It is usual to write $A \circ B$ for $A \circ_\PP B$.


Subset Product with Singleton

When one of the subsets in a subset product is a singleton, we can (and often do) dispose of the set braces. Thus:

Let $A \subseteq S$ be a subset of $S$.

Then:

$(1): \quad a \circ S := \set a \circ S$
$(2): \quad S \circ a := S \circ \set a$


Also defined as

When the operation $\circ$ is in fact addition ($+$), the induced operation $+$ is then also called setwise addition.


When used, it is best to state explicitly that $+$ means setwise addition.

This is because some sources use $A + B$ also to denote set union and disjoint union.


Also known as

Also known as a complex.

As there are a number of conflicting definitions for the word complex in the context of group theory, it is highly recommended that the word not be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in this context.


Examples

Let $G$ be a group.

Example 1

Let $a \in G$ be an element of $G$.

Let:

\(\ds X\) \(=\) \(\ds \set {e, a^2}\)
\(\ds Y\) \(=\) \(\ds \set {e, a, a^3}\)

Let $\order a = 4$.

Then:

$\card {X Y} = 4$

where $\card {\, \cdot \,}$ denotes cardinality.


Example 2

Let $a \in G$ be an element of $G$.

Let:

\(\ds X\) \(=\) \(\ds \set {e, a^2}\)
\(\ds Y\) \(=\) \(\ds \set {e, a, a^3}\)

Let $\order a = 6$.

Then:

$\card {X Y} = 5$

where $\card {\, \cdot \,}$ denotes cardinality.


Example 3

Let the order of $G$ be $n \in \Z_{>0}$.

Let $X \subseteq G$ be a subset of $G$.

Let $\card X > \dfrac n 2$.

Then:

$X X = G$

where $X X$ denotes subset product.


Example 4

Let $S$ be the initial segment of the natural numbers $\N_{<3}$:

$\N_{<3} = \set {0, 1, 2}$

Let $\circ$ be the operation defined on $S$ by the Cayley table:

$\begin {array} {c|cccc}

\circ & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ \end {array}$

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Then every non-empty subset of $S$ which does not contain $0$ is invertible in $\struct {\powerset S, \circ_\PP}$.


Subset Product with Empty Set

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

Let $A, B \in \powerset S$.


If $A = \O$ or $B = \O$, then $A \circ B = \O$.


Subsets of $\R$ under Multiplication

Let $\struct {\R, \times}$ be the multiplicative group of (non-zero) real numbers.

Let $S = \set {-1, 2}$.

Let $T = \set {1, 2, 3}$.

Then the subset product $S T$ is:

$ST = \set {-1, -2, -3, 2, 4, 6}$


Also see

  • Results about subset products can be found here.


Sources

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