Bicyclic semi-group

The semi-group with unit element $\mathbf{1}$ and two generators $a,b$ subject to the single generating relation $ab = \mathbf{1}$. One of the realizations of the bicyclic semi-group is the Cartesian square $\mathbf{N} \times \mathbf{N}$, where $\mathbf{N}$ is the set of non-negative integers, with respect to the operation $$ (k,l) \cdot (m,n) = (k+m-\min(l,m), l+n - \min(l,m)) \ . $$

The bicyclic semi-group is an inversion semi-group and as such is monogenic, i.e. is generated by a single element. The idempotents of the bicyclic semi-group form a chain, which is ordered with respect to the type of the positive numbers. The bicyclic semi-group is bisimple (cf. Simple semi-group).

The bicyclic semi-group often occurs in theoretical investigations concerning semi-groups, not only as a representative of certain important classes of semi-groups, but also as a "block" , which defines the structure of individual semi-groups. Thus, for any idempotent $e$ of a $0$-simple, but not completely $0$-simple semi-group $S$ there exists a bicyclic sub-semi-group in $S$ containing $e$ as the unit element (cf. [1], Para. 2.7). The elements $a$ and $b$ of the bicyclic semi-group $B$ defined as above, are, respectively, its left and right multiplying elements (i.e. there exist proper subsets $X$ and $Y$ in $B$ such that $aX = B$, $Yb= B$). Moreover, in a semi-group $S$ with unit element $e$ the element $c$ will be a left multiplier if and only if $S$ contains the bicyclic semi-group whose unit element is identical with $c$; a similar theorem is also valid for right multiplying elements, so that $S$ has left multiplying elements if and only if it also has right multiplying elements.

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) Zbl 0111.03403 Zbl 0178.01203
[2]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) Zbl 0303.20039