Multiplication and reciprocals of Dedekind cuts

Multiplication of Dedekind cuts (Dedekind cut, The real numbers $\mathbb{R}$ as Dedekind cuts) is defined first for nonnegative cuts, then extended to all cuts by their signs (Order on the Dedekind reals) via the absolute value.

Positive case. For cuts $A, B > 0^{*}$ (strictly positive),

$A \cdot B := \{\, q \in \mathbb{Q} : q \le 0 \,\} \;\cup\; \{\, a b : a \in A,\ b \in B,\ a > 0,\ b > 0 \,\}.$

Absolute value. $|A| := A$ if $A \ge 0^{*}$, and $|A| := -A$ otherwise (Addition, negation, and subtraction of Dedekind cuts for $-A$); thus $|A| \ge 0^{*}$ always, and $|0^{*}| = 0^{*}$.

Sign extension. For arbitrary cuts $A, B$,

• $A \cdot B := 0^{*}$ if $A = 0^{*}$ or $B = 0^{*}$;
• $A \cdot B := |A| \cdot |B|$ if $A, B$ are both $> 0^{*}$ or both $< 0^{*}$;
• $A \cdot B := -\bigl(|A| \cdot |B|\bigr)$ if $A, B$ have opposite signs.

Identity. The multiplicative identity is $1^{*} = \{\, r \in \mathbb{Q} : r < 1 \,\}$.

Reciprocal. For $A > 0^{*}$,

$A^{-1} := \{\, p \in \mathbb{Q} : p \le 0 \,\} \;\cup\; \{\, p > 0 : \exists\, s \in \mathbb{Q},\ s > 0,\ s \notin A,\ p < 1/s \,\}.$

Equivalently, a positive rational $p$ lies in $A^{-1}$ iff $1/p$ is an upper rational bound of $A$ that is not the least one (Rudin's construction). For $A < 0^{*}$, set $A^{-1} := -\bigl((-A)^{-1}\bigr)$. Division is $A / B := A \cdot B^{-1}$ for $B \ne 0^{*}$.

• The product formula is stated only for strictly positive cuts $A, B > 0^{*}$: then there exist positive $a \in A$, $b \in B$, so the positive products $ab$ are nonempty and, because $A, B$ have no greatest element (axiom (C3)), $A \cdot B$ has none either; together with downward closure this makes $A \cdot B$ a genuine cut, the clause $\{q \le 0\}$ being absorbed below those positive products. The formula is deliberately not applied at the boundary $A = 0^{*}$ or $B = 0^{*}$, where $\{q \le 0\}$ would leave $0$ as a greatest element and so fail to be a cut; products with a zero factor are supplied instead by the first sign rule, $A \cdot B = 0^{*}$, so the operation is well posed on all cuts.
• On the rational embedding $q \mapsto q^{*}$ the operation agrees with $\mathbb{Q}$: $(pq)^{*} = p^{*} \cdot q^{*}$ (The rational cuts embed densely in $\mathbb{R}$, preserving sums, products, $0$, $1$ and the order); with $q^{*} \cdot (1/q)^{*} = 1^{*}$ that gives $(q^{*})^{-1} = (1/q)^{*}$ for $q > 0$, the reciprocal being the one supplied by For a positive cut $A$, the reciprocal $A^{-1}$ satisfies $A \cdot A^{-1} = 1^{*}$.
• That these operations send cuts to cuts and satisfy the field axioms ( commutativity, associativity, distributivity over addition, identity $1^{*}$, and $A \cdot A^{-1} = 1^{*}$ for every $A \ne 0^{*}$) is The Dedekind reals form a field.