Definition:Subset Product
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
- Definition:Direct Image Mapping
- Definition:Minkowski Sum: for when $S$ is a vector space and the operation is vector addition
- Results about subset products can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: Notation $3$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 27$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 41$: Multiplication of subsets of a group
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Definition $5.16$