Atoms in a Boolean algebra

Question

I am trying to understand the concept of an atom in a Boolean algebra.

To fix the ideas, let $X=\{a,b,c\}$ be a set, and $\mathcal{A}=\{\emptyset,\{a\},\{b,c\},X\}$ be one of the five possible algebras of subsets of $X$. Of course, $\mathcal{A}$ being an algebra of set, it is also a Boolean algebra.

Now, an element $x\in\mathcal{A}$ in is a atom of $\mathcal{A}$ if, for every $y\in\mathcal{A}$, either $x\wedge y=x$ or $x\wedge y=0$. Am I correct to assert that $\{a\}$ and $\{b,c\}$ are the atoms of $\mathcal{A}$, and that, consequently, atoms in a Boolean algebra are not necessarily singleton sets (though every singleton set is an atom)?

score 2 · Accepted Answer · answered Feb 25 '15 at 18:25

2

In any partially ordered set with a minimum element, an atom is defined to be an element that covers the minimum element. The elements you refer to are atoms in this sense. This definition coincides with the one you give when the set is a finite lattice, as is a finite Boolean algebra.

answered Feb 25 '15 at 18:25

Matt Samuel

58,164

Thanks for your reply. Can I ask you what would be the definition of an atom when a Boolean algebra is denumerable, yet still atomic? – Feb 25 '15 at 18:51
An atomic lattice is locally finite, so it still works. The same definition applies in general I believe, I was just being careful really. – Matt Samuel Feb 25 '15 at 18:54
OK. Thank you for your reply. – Feb 25 '15 at 19:34
@student you're welcome. – Matt Samuel Feb 25 '15 at 19:37

Atoms in a Boolean algebra

1 Answers1