Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties.
In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set.
In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult.
In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.
For example, the definition xab = xba should be read as: The third column states whether this set of semigroups forms a variety.