## Some results on Arf numerical semigroups with multiplicity 8

#### Citation

Süer, M., İlhan, S. (2017). Some results on Arf numerical semigroups with multiplicity 8. International Conference on Mathematics and Mathematics Education (ICMME-2017), 11-13 Mayıs 2017, Şanlıurfa#### Abstract

A numerical semigroup is a subset of the set of nonnegative integers (denoted
here by
¥
) closed under addition, containing the zero element and with finite
complement in
¥
. Note also that up to isomorphism the set of numerical semigroups
classify the set of all submonoids of
( , ) ¥ + . Let S be submonoid of
¥
, the condition
of having finite complement in
¥
is equivalent to saying that the greatest common
divisor (gcd for short) of its elements is one.
Those positive integers which do not belong to
S
are called gaps of
S
. The
number of gaps of
S
is called the genus of
S
and it is denoted by
G S( )
. The
largest gaps of
S
is
F S( )
if
S
is different from
¥ .
m S s S s ( ) min : 0 = Î > { } are called multiplicity of
S
, respectively. Also,
n S Card F S S ( ) 0,1,2,..., ( ) = Ç ({ } )
is called the number determine of
S .
If
a Î ¥
and
a S Ï , then
a
is called gap of
S
. We denote the set of gaps
of
S
, by
H S( )
, i.e,
H S S ( ) \ = ¥
.The
G S Card H S ( ) ( ( )) =
is called the genus of
S .
Also, It known that
G S F S n S ( ) ( ) 1 ( ) = + - .
A numerical semigroup S is called Arf if
x y z S + - Î
for all
x y z S , , Î
, where
x y z ³ ³
. This definition was first given by C. Arf in 1949.
In the study, we intend to examine the Arf numerical semigroup with multiplicity
eight and fixed conductor. We will also able to compute the notable elements and
special sets of these numerical semigroups.