The set of all zero-systems of a group is denoted and is called the block monoid of since it forms a commutative monoid under the operation of zero-system addition defined by
The monoid has identity , the zero-system consisting of no elements.
For a nonempty subset of , is defined as the set of all zero-systems of containing only elements from . For any subset of , is a commutative submonoid of . If context makes it obvious, is often omitted and is written.
A zero-system is minimal if it contains no proper zero-systems. The set is defined as the set of all minimal zero-systems of .