Definition:Subset Product/Singleton

Definition

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

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

Then:

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

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

where $\set a \circ S$ and $S \circ \set a$ denote the subset product of $\set a$ with $S$.

That is:

$a \circ S = \set {a \circ s: s \in S}$

$S \circ a = \set {s \circ a: s \in S}$

Examples

Subsets of $\R$ under Multiplication

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

Let $S = \set 3$.

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

Then the subset product $S T$ is:

$ST = \set {-3, 6}$

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
• 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