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 .

Anderson, D. F. and Chapman, S. T. "On the Elasticities of Krull Domains with Finite Cyclic Divisor Class Group." Comm.
Alg.28, 2543-2553, 2000.Chapman, S. T. "On the
Davenport Constant, the Cross Number, and Their Applications in Factorization Theory."
In Zero-Dimensional
Commutative Rings (Ed. D. F. Anderson and D. E. Dobbs).
New York: Dekker, pp. 167-190, 1997.