Bicyclic Semigroup - Properties

Properties

The bicyclic semigroup has the property that the image of any morphism φ from B to another semigroup S is either cyclic, or it is an isomorphic copy of B. The elements φ(a), φ(b) and φ(e) of S will always satisfy the conditions above (because φ is a morphism) with the possible exception that φ(b) φ(a) might turn out to be φ(e). If this is not true, then φ(B) is isomorphic to B; otherwise, it is the cyclic semigroup generated by φ(a). In practice, this means that the bicyclic semigroup can be found in many different contexts.

The idempotents of B are all pairs (x, x), where x is any natural number (using the ordered pair characterisation of B). Since these commute, and B is regular (for every x there is a y such that x y x = x), the bicyclic semigroup is an inverse semigroup. (This means that each element x of B has a unique inverse y, in the "weak" semigroup sense that x y x = x and y x y = y.)

Every ideal of B is principal: the left and right principal ideals of (m, n) are

• (m, n) B = {(s, t) : s ≥ m} and
• B (m, n) = {(s, t) : t ≥ n}.

Each of these contains infinitely many others, so B does not have minimal left or right ideals.

In terms of Green's relations, B has only one D-class (it is bisimple), and hence has only one J-class (it is simple). The L and R relations are given by

• (a, b) R (c, d) if and only if a = c; and
• (a, b) L (c, d) if and only if b = d.

This implies that two elements are H-related if and only if they are identical. Consequently, the only subgroups of B are infinitely many copies of the trivial group, each corresponding to one of the idempotents.

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. The idempotents of B appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each L-class and each R-class must contain exactly one idempotent.

The bicyclic semigroup is the "simplest" example of a bisimple inverse semigroup with identity; there are many others. 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.