Definition:Boolean Ring Axioms

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Definition

A Boolean ring is an algebraic structure $\left({R, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, satisfying the following conditions:

\((A0)\)   $:$   Closure under addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b \in R \)             
\((A1)\)   $:$   Associativity of addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right) \)             
\((A2)\)   $:$   Commutativity of addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b = b * a \)             
\((A3)\)   $:$   Identity element for addition: the zero      \(\displaystyle \exists 0_R \in R: \forall a \in R:\) \(\displaystyle a * 0_R = a = 0_R * a \)             
\((AC2)\)   $:$   Characteristic 2 for addition:      \(\displaystyle \forall a \in R:\) \(\displaystyle a * a = 0_R \)             
\((M0)\)   $:$   Closure under product      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a \circ b \in R \)             
\((M1)\)   $:$   Associativity of product      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right) \)             
\((M2)\)   $:$   Identity element for product: the unity      \(\displaystyle \exists 1_R \in R: \forall a \in R:\) \(\displaystyle 1_R \circ a = a = a \circ 1_R \)             
\((MI)\)   $:$   Idempotence of product      \(\displaystyle \forall a \in R:\) \(\displaystyle a \circ a = a \)             
\((D)\)   $:$   Product is distributive over addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right), \)             
\(\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right) \)             

These criteria are called the Boolean ring axioms.


Also see


Sources