nLab atomic Boolean algebra


Given an element aa of a Boolean algebra (or other poset) AA, recall that aa is an atom in AA if aa is minimal among non-trivial (non-bottom) elements of AA. That is, given any bAb \in A such that bab \leq a, either b=0b = 0 or b=ab = a.

AA is atomic if we have b= Ia ib = \bigvee_I a_i for every bAb \in A, where {a i} I\{a_i\}_I is some set of atoms in AA.

If AA is complete we can write it: if for every bAb \in A, we have b=𝒜(b)b = \bigvee \mathcal{A}(b) where 𝒜(b)\mathcal{A}(b) is the set of all the atoms aa in AA such that aba \leq b. Or: for every bAb \in A, we have b𝒜(b)b \le \bigvee \mathcal{A}(b).


Last revised on December 29, 2023 at 22:30:55. See the history of this page for a list of all contributions to it.