nLab multiplicatively idempotent semiring

This is about idempotent semirings whose multiplication is idempotent. For idempotent semirings whose addition is idempotent, see additively idempotent semiring.

This article is about Boolean semirings as defined by Toby Bartels. For other notions of “Boolean semiring” or “Boolean rig”, see Boolean semiring.

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Idea

Recall that a semiring is a set $R$ equipped with two binary operations, denoted $+$, and $\cdot$ and called addition and multiplication, satisfying the ring (or rng) axioms except that there may or may not be either be a zero nor a negative nor an inverse, for which reason do check.

Definition

A semiring or rig is (multiplicatively) idempotent or Boolean if and only if multiplication is idempotent; that is, $x^2 = x$ holds for all $x \in S$.

Terminology

From now on we will assume that semirings and rigs are synonyms of each other; i.e. the semiring, $S$, has a neutral element $\varepsilon$ for + and one $e$ for $\cdot$. Moreover we assume that for all $s\in S$, $s\cdot \varepsilon =\varepsilon \cdot s = \varepsilon$, making $S$ into a rig.

Multiplicatively idempotent semirings are sometimes called idempotent rigs, see e.g. Baez 2022. However, the term idempotent rig and idempotent semiring usually refers to additively idempotent semirings in the literature, and now redirects to the disambiguation page idempotent semiring on the nLab.

Multiplicatively idempotent semirings are called Boolean rigs or Boolean semirings by Toby Bartels, see e.g. Bartels 2020. The name originated from the fact that a Boolean ring is defined in the literature as a ring for which multiplication is idempotent (Bartels 2020, Rogers 2024). However, the terms “Boolean rig” and “Boolean semiring” have multiple meanings in the literature representing some generalization of Boolean rings from rings to rigs and semirings, and now redirects to the disambiguation page Boolean semiring on the nLab.

Properties

Let CMon be the concrete monoidal category of abelian groups, and let $U: CMon \to Set$ be the lax monoidal underlying-set functor.

Theorem

Every multiplicatively idempotent rig $R$ satisfies the equation

$$1_{U(R)} = \left(U(R) \stackrel{\delta}{\to} U(R) \times U(R) \stackrel{\lambda}{\to} U(R \otimes R) \stackrel{U(mult)}{\to} U(R) \right)$$

where $\lambda$ is a lax monoidal constraint.

Examples

The main examples are probably distributive lattices.

Of course, a Boolean ring is a multiplicatively idempotent semiring, since any ring is a semiring. However, since it's also a distributive lattice, a Boolean ring is actually a multiplicatively idempotent semiring in two different ways.

Similarly, a Boolean semiring as defined by Fernando Guzmán is a multiplicatively idempotent semiring by definition.

Smallest multiplicatively idempotent semirings

Let’s see what are the smallest multiplicatively idempotent semirings.

• The only multiplicatively idempotent semiring of cardinality $1$ is the zero ring $0$.

There are exactly two multiplicatively idempotent semirings of cardinality $2$. In such a multiplicatively idempotent semiring, we necessarily have $0 \neq 1$, because $0=1$ would imply that $x=1 \cdot x=0 \cdot x=0$ for every $x$ and then the multiplicatively idempotent semiring would be the zero ring $0$. The two elements of the multiplicatively idempotent semirings are thus $0$ and $1$. From the axioms of a semiring, we have $0+0=0$, $1+0=0+1=1$, $0 \cdot 0=0$, $1 \cdot 0=0 \cdot 1=0$ and $1 \cdot 1=1$. We then have two possibilities for $1+1$, either $0$ or $1$. The two possibilities give a multiplicatively idempotent semiring.

• One of the two multiplicatively idempotent semirings of cardinality $2$ is $\mathbb{Z}/2\mathbb{Z}$ ($0$ can be interpreted as “False”, $1$ as “True”, $+$ as the exclusive disjunction and $\cdot$ as the conjunction).
• One of the two multiplicatively idempotent semirings of cardinality $2$ is $\mathbb{B}=\{0,1\}$ where $1+1=1$ ($0$ can be interpreted as “False”, $1$ as “True”, $+$ as the disjunction and $\cdot$ as the conjunction).

In either case, the multiplication is commutative and distributive over addition.

• The three-element distributive lattice $\{0, u, 1\}$ with $0 \leq u \leq 1$ is a multiplicatively idempotent semiring with cardinality $3$.

• The four-element distributive lattice $\{0, u, v, 1\}$ with $0 \leq u \leq v \leq 1$ is a multiplicatively idempotent semiring with cardinality $4$.

• The four-element distributive lattice $\{0, u, v, 1\}$ with $0 \leq u \leq 1$ and $0 \leq v \leq 1$ but $u$ and $v$ incomparable (i.e. $\neg (u \leq v) \wedge \neg (v \leq u)$) is a multiplicatively idempotent semiring with cardinality $4$.

• The previous four-element distributive lattice is also a Boolean algebra, and so one can define a Boolean ring structure, resulting in yet another multiplicatively idempotent semiring with cardinality $4$.

• The initial multiplicatively idempotent semiring, i.e. the free multiplicatively idempotent semiring on the empty set, has cardinality $4$ (Rogers 2024), and is defined as the quotient of the natural numbers by the relation $n + 4 \sim n + 2$. This semiring is commutative.

One can construct a non-commutative multiplicatively idempotent semiring of cardinal $9$ (Rogers 2024). It is currently unknown if there are any smaller non-commutative multiplicatively idempotent semirings.

 Related concepts

• semiring, rig
• idempotent monoid
• Boolean semiring (Guzmán)
• additively idempotent semiring
• distributive lattice
• Boolean ring
• 01-bounded semilattice

References

The term (multiplcatively) idempotent appears in:

• Morgan Rogers, From free idempotent monoids to free multiplicatively idempotent rigs [arXiv:2408.17440]

The term idempotent rig appears in:

• John Baez, Free Idempotent Rigs and Monoids, Azimuth Blog, 21 December 2022. [web]

The term Boolean rig was used by Toby Bartels in this current article on the nLab, before it was renamed to multiplcatively idempotent rig:

• Toby Bartels, Boolean rig, 5th revision of the nLab article “multiplicatively idempotent rig”, August 15, 2020. [nLab]

The term Boolean rig was also used in this Category Theory Zulip discussion:

• J-B Vienney, Are Boolean rigs commutative?, Category theory Zulip, (web)