nLab semi-atom

A semi-atom is a generalization of the notion of atom in a bottom bounded partial order.

An object $a$ is a semi-atom when the interval $[\bot, a ]$ is a chain?, i.e. any two objects $x$ and $y$ in that interval are comparable? ($x \le y$ or $y \leq x$).

All atoms are semi-atoms and usually the bottom is not considered one.

If $a$ and $b$ are semi-atoms then their meet exists and we have $a \wedge b \in \{\bot, a, b\}$

Examples

The divisor lattice? for some number $n$ contains prime numbers as atoms and may contain powers of primes as semi-atoms that are not atoms.