definition. free algebra [ca-001D]

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FreeAlgebra FreeAlgebra.Pre FreeAlgebra.Rel

L∃∀N

Let \(X\) be an arbitrary set. An free \(R\)-algebra on \(X\) (or "generated by \(X\)"), named \(A\), is the ring quotient of the following inductively constructed set \(A_{\mathtt {pre}}\)

1. for all \(x\) in \(X\), there exists a map \(X \to A_{\mathtt {pre}}\).
2. for all \(r\) in \(R\), there exists a map \(R \to A_{\mathtt {pre}}\).
3. for all \(a, b\) in \(A_{\mathtt {pre}}\), \(a + b\) is in \(A_{\mathtt {pre}}\).
4. for all \(a, b\) in \(A_{\mathtt {pre}}\), \(a * b\) is in \(A_{\mathtt {pre}}\).

1. there exists a ring homomorphism from \(R\) to \(A_{\mathtt {pre}}\), denoted \(R \to _{+*} A_{\mathtt {pre}}\).
2. \(A\) is a commutative group under \(+\).
3. \(A\) is a monoid under \(*\).
4. left and right distributivity under \(*\) over \(+\).
5. \(0 * a \sim a * 0 \sim 0\).
6. for all \(a, b, c\) in \(A\), if \(a \sim b\), we have
   1. \(a + c \sim b + c\)
   2. \(c + a \sim c + b\)
   3. \(a * c \sim b * c\)
   4. \(c * a \sim c * b\)
   (compatibility with the ring operations \(+\) and \(*\))