nLab
semi-atom

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

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

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

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

Examples

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

Created on January 15, 2015 at 15:56:21. See the history of this page for a list of all contributions to it.