Bicyclic semi-group
The semi-group with unit element and two generators subject to the single generating relation
. One of the realizations of the bicyclic semi-group is the Cartesian square
, where
is the set of non-negative integers, with respect to the operation
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The bicyclic semi-group is an inversion semi-group and as such is monogenic, i.e. is generated by a single element. The idempotents (cf. Idempotent) 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 of a
-simple, but not completely
-simple semi-group
there exists a bicyclic sub-semi-group in
containing
as the unit element (cf. [1], Para. 2.7). The elements
and
of the bicyclic semi-group
defined as above, are, respectively, its left and right multiplying elements (i.e. there exist proper subsets
and
in
such that
,
). Moreover, in a semi-group
with unit element
the element
will be a left multiplier if and only if
contains the bicyclic semi-group whose unit element is identical with
; a similar theorem is also valid for right multiplying elements, so that
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) |
[2] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |
Bicyclic semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicyclic_semi-group&oldid=42741