Ring of Sets vs Ring in Universal Algebra

Question

I actually want to continue this post. I believe the naming convention in Mathematics is consistent, such that there are no clearly distinguish objects have the same name. However, I'm not sure how to relate the Ring of Sets and the Ring in Universal Algebra.

The definition of Ring in Universal Algebra is

1. $R$ is an abelian group under addition, meaning that:
   • $(a + b) + c = a + (b + c)$ for all $a, b, c \in R$ ($+$ is associative).
   • $a + b = b + a$ for all $a, b \in R$ ($+$ is commutative).
   • There is an element $0 \in R$ such that $a + 0 = a$ for all $a \in R$ ($0$ is the additive identity).
   • For each $a \in R$ there exists $−a \in R$ such that $a + (−a) = 0$ ($−a$ is the additive inverse of $a$).
2. $R$ is a monoid under multiplication, meaning that:
   • $(a · b) · c = a · (b · c)$ for all $a, b, c \in R$ ($·$ is associative).
   • There is an element $1 \in R$ such that $a · 1 = a$ and $1 · a = a$ for all $a \in R$ ($1$ is the multiplicative identity).
3. Multiplication is distributive with respect to addition:
   • $a ⋅ (b + c) = (a · b) + (a · c)$ for all $a, b, c \in R$ (left distributivity).
   • $(b + c) · a = (b · a) + (c · a)$ for all $a, b, c \in R$ (right distributivity).

Next, the definition of Ring of Sets is

1. $A,B\in\mathcal{R}$ implies $A\cap B\in \mathcal {R}$ and
2. $A,B\in \mathcal {R}$ implies $A\cup B\in \mathcal {R}$.

Then, I tried to relate between this two definition.

1. Let assume that $\cap$ is the additive operator. Then, $\mathcal {R}$ must contain the additive identity and inverse for all $A \in \mathcal {R}$.

   • The additive identity is the whole element set $S$, such that for all $A \in \mathcal {R}, A \subseteq S$. It can be easily seen that $A \cap S = A$, for all $A \in \mathcal {R}$.
   • Nevertheless, pick $A \in \mathcal{R}$, such that $A \subset S$. Then, $A$ doesn't have the inverse, i.e. no $B \in \mathcal{R}$, such that $A \cap B = S$.

2. Let assume that $\cup$ is the additive operator. The argument is similar to (1).

I might be wrong, but what I'm trying to say is the structure of Ring of Sets isn't consistent with the structure of general Ring. Kindly need your explanation. Thank you.

Answer 1

Ring of Sets by corresponding union to addition and intersection to multiplication ($\varnothing$ to $0$ and $E$ (universe of set) to $1$ as well) satisfy most ring axioms. However the fundamental difference is that it doesn't have an additive inverse because $X\cup Y=\varnothing$ if and only if $X=\varnothing$ and $Y=\varnothing$.

Thus your definition of Ring of Sets is not a ring rigorously.

Another way to define Ring of Sets is known as Boolean Ring, i.e by corresponding symmetric difference of sets to addition and intersection to multiplication ($\varnothing$ to $0$ and $E$ (universe of set) to $1$ as well). It satisfies all ring axioms with an additive inverse.

Please read this post for more detail.

Answer 2

A "ring of sets" should really be called a "distributive lattice of sets." Does that answer your question?