Definition:Set Product/Family of Sets
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Definition
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
Let $P$ be a set.
Let $\family {\phi_i}_{i \mathop \in I}$ be an indexed family of mappings $\phi_i: P \to S_i$ for all $i \in I$ such that:
- For all sets $X$ and all indexed families $\family {f_i}_{i \mathop \in I}$ of mappings $f_i: X \to S_i$ there exists a unique mapping $h: X \to P$ such that:
- $\forall i \in I: \phi_i \circ h = f_i$
- that is, such that for all $i \in I$:
$\quad \quad \begin {xy} \xymatrix@+1em@L+3px { X \ar@{-->}[d]_*+{h} \ar[dr]^*+{f_i} \\ P \ar[r]_*{\phi_i} & S_i } \end {xy}$
- is a commutative diagram.
Then $P$, together with the family of mappings $\family {\phi_i}_{i \mathop \in I}$, is called a product of (the family) $\family {S_i}_{i \mathop \in I}$.
This product of $\family {S_i}_{i \mathop \in I}$ can be denoted $\struct {P, \family {\phi_i}_{i \mathop \in I} }$.
Also see
- Results about set products can be found here.
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations