nLab Boolean semiring (Chajda & Kotrle)

This page is about Boolean semirings as defined by Ivan Chajda and Miroslav Kotrle. For other notions of “Boolean semirings”, see Boolean semiring.

This page is about the notion of skeleton of a semiring as defined by Ivan Chajda and Miroslav Kotrle. For other notions of skeleton in mathematics, see at skeleton.

Definition

The authors Chadja & Kotrle 1994 assume that semirings are non-unital in that they do not have either the multiplicatively absorptive additive unit $0$ or the multiplicative unit $1$ that define rigs. Instead, their semirings only have an element $0$ such that $0 \cdot a = 0$.

Chadja & Kotrle 1994 defined the skeleton $S(A)$ of a semiring $A$ to be the subset of all elements $c \in A$ such that $c$ is the sum of two elements $a \in A$ and $b \in A$.

A semiring $A$ is a Boolean semiring if:

• the semiring $A$ is skeletal: the skeleton $S(A)$ of the semiring is a subrng of $A$.

• for all $a$ and $b$, $a \cdot a = a + 0$ and $a \cdot b + 0 = a \cdot b$. By definition, $0 = 0 \cdot (a + a) = 0 \cdot a + 0 \cdot a = 0 + 0$ is an element of $S(A)$.

• there exists an element $1$ such that for all $a$ and $b$, $(a \cdot b) \cdot 1 = a \cdot b$ and $1 \cdot (a \cdot b) = a \cdot b$

Properties

Every boolean semiring is commutative.

Related concepts

• semiring

References

• Ivan Chajda, Miroslav Kotrle, Boolean semirings, Czechoslovak Mathematical Journal, Volume 44, Issue 4, pp. 763-767, 1994. [doi:10.21136/CMJ.1994.128495, pdf]