Monogenic semi-group
cyclic semi-group
A semi-group generated by one element. The monogenic semi-group generated by an element is usually denoted by
(sometimes by
) and consists of all powers
with natural exponents. If all these powers are distinct, then
is isomorphic to the additive semi-group of natural numbers. Otherwise
is finite, and then the number of elements in it is called the order of the semi-group
, and also the order of the element
. If
is infinite, then
is said to have infinite order. For a finite monogenic semi-group
there is a smallest number
with the property
, for some
;
is called the index of the element
(and also the index of the semi-group
). In this connection, if
is the smallest number with the property
, then
is called the period of
(of
). The pair
is called the type of
(of
). For any natural numbers
and
there is a monogenic semi-group of type
; two finite monogenic semi-groups are isomorphic if and only if their types coincide. If
is the type of a monogenic semi-group
, then
are distinct elements and, consequently, the order of
is
; the set
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is the largest subgroup and smallest ideal in ; the identity
of the group
is the unique idempotent in
, where
for any
such that
;
is a cyclic group, a generator being, for example,
. An idempotent of a monogenic semi-group is a unit (zero) in it if and only if its index (respectively, period) is equal to 1; this is equivalent to the given monogenic semi-group being a group (respectively, a nilpotent semi-group). Every sub-semi-group of the infinite monogenic semi-group is finitely generated.
References
[1] | A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961) |
[2] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |
Monogenic semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monogenic_semi-group&oldid=33128