It is perhaps most easily understood as the syntactic monoid describing the Dyck language of balanced pairs of parentheses.
Thus, it finds common applications in combinatorics, such as describing binary trees and associative algebras.
Alfred H. Clifford and Gordon Preston claim that one of them, working with David Rees, discovered it independently (without publication) at some point before 1943.
It can then be shown that all elements of B in fact have the form qa pb, for some natural numbers a and b.
So B is the semigroup of pairs of natural numbers (including zero), with operation[1] This is sufficient to define B so that it is the same object as in the original construction.
A third way of deriving B uses two appropriately-chosen functions to yield the bicyclic semigroup as a monoid of transformations of the natural numbers.
The egg-box diagram for B is infinitely large; the upper left corner begins: Each entry represents a singleton H-class; the rows are the R-classes and the columns are L-classes.
Where the definition of B from ordered pairs used the class of natural numbers (which is not only an additive semigroup, but also a commutative lattice under min and max operations), another set with appropriate properties could appear instead, and the "+", "−" and "max" operations modified accordingly.