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Keywords:
Boolean ring; commutative ring; lattice-like structure; difference
Boolean ring; commutative ring; lattice-like structure; difference
Summary:
It is well known that to every Boolean ring $\mathcal {R}$ can be assigned a Boolean algebra $\mathcal {B}$ whose operations are term operations of $\mathcal {R}$. Then a symmetric difference of $\mathcal {B}$ together with the meet operation recover the original ring operations of $\mathcal {R}$. The aim of this paper is to show for what a ring $\mathcal {R}$ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached interesting results if the characteristic of the ring $\mathcal {R}$ is either an odd natural number or a power of 2.
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