duaLabRieg
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A rieg is a rig with exponentials: it has $0$, $1$, $+$, $\times$ and a binary operation $x^y$. Natural numbers, modular arithmetic, Heyting algebras and the isomorphism classes of a topos all give riegs.
1. Definition§
Definition 1.1 (Rieg)§
A rieg is a commutative rig $(R,0,1,+,\times)$ equipped with a rig homomorphism $R\to\operatorname{End}(R,1,\times)$, where $\operatorname{End}(R,1,\times)$ is the rig of monoid endomorphisms of $(R,1,\times)$. The image of $y$ is written $x\mapsto x^y$.
Equivalently, a rieg is a commutative rig satisfying $1^x=1$, $(yz)^x=y^xz^x$, $x^0=1$, $x^1=x$, $x^{y+z}=x^yx^z$ and $x^{yz}=(x^z)^y$. A rieg homomorphism preserves $0$, $1$, $+$, $\times$ and exponentiation. See Definitions 2.2.1, 2.2.4 and Proposition 2.2.2.
Definition 1.2 (Characteristic)§
The characteristic of a rieg $R$ is the least pair $(a,b)$, with $a\ge0$ and $b>0$, such that $a\cdot1=(a+b)\cdot1$ in $R$. If no such pair exists, it is $(\infty,0)$. See Definition 3.2.9.
2. Examples§
- The natural numbers $\mathbb N$ form the initial rieg, with $0^0=1$. A singleton is the terminal rieg (Examples 2.2.5–2.2.6).
- For $a\ge0$ and $b>0$, the quotient $\mathbb N_{a,b}$ identifies $n,m\ge a$ with $n\equiv m \bmod b$. It is a rieg exactly in the cases of the Burris–Lee theorem. For example, $\mathbb N_{1,2}$ is the parity rieg and $\mathbb N_{1,1}$ is the rieg of truth values (§3.2).
- The profinite natural numbers $\widehat{\mathbb N}=\mathbb N\sqcup\widehat{\mathbb Z}$ and the 10-adic natural numbers $\mathbb N\sqcup\mathbb Z_{10}$ are topological riegs. There is no 5-adic analogue (§3.3).
- For a small cartesian closed category with finite coproducts, the isomorphism classes of its objects form a rieg, its Burnside rieg. Finite sets give $\mathbb N$; sets of size below a strong limit cardinal give cardinal arithmetic (Proposition 4.1.2, Examples 4.1.6–4.1.7).
- The isomorphism classes of finite groups, with $+$ freely added, form a rieg. Its exponential $H^G$ is the sum of the stabilizers $\operatorname{Aut}(\varphi)$ over the conjugacy classes of homomorphisms $\varphi:G\to H$ (Theorem 4.3.7).
- Every Heyting algebra is a rieg with $x^y=(y\to x)$; this includes the truth values and the open sets of a space (§5.3).
- The interval $[0,1]$ with $x+y=\max(x,y)$, $xy=\min(x,y)$ and $x^y=\max(x,1-y)$ is a rieg; the exponent laws follow from the De Morgan laws for $y\mapsto1-y$. All operations are continuous, so it is a nontrivial connected compact Hausdorff topological rieg.
- The nonnegative reals $[0,\infty)$ form a rieg with $0^x=0$ for $x>0$. The max-plus algebra admits no rieg structure (Examples B.1.1 and 5.1.2).
- There are exactly 8 three-element riegs up to isomorphism (Proposition B.2.1).
3. Properties§
Proposition 3.1 (No rieg is a ring)§☆
A rieg that is a ring is the one-element rieg.
Proof§
See Proposition 2.3.2.
Theorem 3.2 (Burris–Lee)§☆
The rig $\mathbb N_{a,b}$ is a rieg if and only if, for every prime $p$ and every $e>0$, $p^e\mid b$ implies $e\le a$, and $p\mid b$ implies $(p-1)\mid b$. Equivalently, these are exactly the characteristics of riegs.
Proof§
See Burris–Lee, Theorem 3.12 and Theorem 3.2.2 and Corollary 3.2.10.
Corollary 3.3 (Cyclic models)§☆
The finite quotient riegs $\mathbb N_{1,b}$ are exactly $\mathbb N_{1,1}$, $\mathbb N_{1,2}$, $\mathbb N_{1,6}$, $\mathbb N_{1,42}$ and $\mathbb N_{1,1806}$.
Proof§
See Burris–Lee, Corollary 3.14, attributed there to D. Higgs, and Corollary 3.2.6.
Theorem 3.4 (Profinite natural numbers)§☆
The profinite completion $\widehat{\mathbb N}$ of the rieg $\mathbb N$ has underlying set $\mathbb N\sqcup\widehat{\mathbb Z}$. Its exponentiation is the unique continuous extension of that of $\mathbb N$, defined for all pairs of elements.
Proof§
Proposition 3.5 (Idempotent riegs)§☆
For a rieg $R$, the following are equivalent: $1+1=1$; there is a rieg homomorphism from the truth values to $R$; addition is idempotent; addition and multiplication are both idempotent. Moreover, the underlying rig is a bounded lattice if and only if $1+x=1$ for all $x$.
Proof§
4. Open questions§
Question 4.1 (Riegs and games)§
Objects of the free cartesian closed category on one object $x$ are built from $x$ by products and exponentials, and Dolan and Trimble read each of them as a two-player game (Baez; Trimble). On the other hand, Games carry sums, and game values such as the Grundy number turn them into monoid operations (Bouton monoid; Hora, §4). How are riegs related to game theory?
5. Related concepts§
6. References§
- Ryuya Hora, Notes on Rieg theory: semiring with exponentials in logic, profinite arithmetic, enumerative combinatorics, and category theory, prepreprint v0.1 (2026).
- Stanley Burris and Simon Lee, Small models of the high school identities, International Journal of Algebra and Computation 2 (1992), 139–178.
- John Baez, Classical vs Quantum Computation (Week 3), The n-Category Café (2006), on the guest lecture by James Dolan.
- Todd Trimble, Holodeck games and CCCs, joint work with James Dolan (2006).
- Ryuya Hora, Games as recursive coalgebras: A categorical view on the Nim-sum, arXiv:2510.22886v3, §4.