This entry is about scales in algebra and linear logic. For scales in geometry and physics, see length scale.
symmetric monoidal (∞,1)-category of spectra
The idea of a scale comes from Peter Freyd, in his attempt to give an algebraic (in the sense of universal algebra) description of the real interval, which he believed to be more fundamental than the real numbers themselves in analysis, and of concepts from analysis such as Lipschitz continuity, limits, differentiation and differential equations. Freyd also discovered that scales are models of multiplicative-additive linear logic with a midpoint operation.
A scale is a minor scale $M$ satisfying the scale identities:
for all $a$ and $b$ in $M$, $a \otimes b = (a^\vee \otimes b^\wedge) \vert (a^\wedge \otimes b^\vee)$
for all $a$ and $b$ in $M$, $a \oplus b = (a^\wedge \oplus b^\vee) \vert (a^\vee \oplus b^\wedge)$
Every scale with $\bot = \top$ is trivial.
As a scale is a closed midpoint algebra, a scale has a partial order $\leq$. For all $a$ and $b$ in $M$, $a \leq b$ if and only if $a \multimap b = \top$.
A subset $\mathcal{I}$ of a scale $M$ is an ideal if $\bot \in \mathcal{I}$, and $x \vee y \in \mathcal{I}$ if and only if $x \in \mathcal{I}$ and $y \in \mathcal{I}$. An ideal is a zoom-invariant ideal if it is closed under $\bot$-zooming.
A subset $\mathcal{I}$ of a scale $M$ is a $\bot$-face if $\bot \in \mathcal{I}$, and $x \vert y \in \mathcal{I}$ if and only if $x \in \mathcal{I}$ and $y \in \mathcal{I}$.
Every $\bot$-face is a zoom-invariant ideal.
Given an element $a$ in scale $M$, a principal $\bot$-face $((a))$ is the subset of all $b$ in $M$ such that $(\bot\vert)^n b \leq a$ for all large $n$ in $\mathbb{N}$.
A Jacobson radical of a scale $M$ is the set $J(M)$ of all $a$ in $M$ such that for all $n$ in $\mathbb{N}$, $x \leq (\bot\vert)^n \top$. $M$ is semi-simple if $J(M)$ is trivial.
The proof of the Linear Representation Theorem in section 8 of Algebraic Real Analysis by Peter Freyd requires the use of excluded middle through its implicit definition of the quasiorder $\lt$ from the algebraically defined partial order $\leq$. In particular, that every scale is a *-autonomous category and thus a model for linear logic and that every equational axiom added to the theory of minor scales is either a consequence of the scale identity for scales or is inconsistent with the theory of minor scales are classical results, as certain lemmas used in the proofs have only been derived from the scale identities through the Linear Representation Theorem. The same is true of the definition of simple scales in section 10, of the algebraic construction of the standard interval $I$ from simple scales in section 11, and various results involving absolute retracts in section 25. Since quasiorders can be constructed from partial orders in any inequality space, these results hold if the scales have a tight apartness relation, but it is unknown if these results still hold for general scales in constructive mathematics.
The unit interval with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = \frac{1}{2}$, $a^\bullet = 1 - a$, $\bot = 0$, $\top = 1$, $a^\wedge = max(2a-1,0)$, and $a^\vee = min(2a,1)$ is an example of a scale.
The set of truth values in Girard’s linear logic is a scale.