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Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 3, September 2011
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A Note on Maximal Ideals in Ternary Semigroups
Thawhat Changphas and Boonyen Thongkam
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayuttaya Rd., Bangkok 10400, Thailand
Abstract
In a commutative ring with identity element any maximal ideal is prime. Similar results hold
in a commutative ordered semigroup with identity element ([2], [5]). In this note, we follow
the idea in [2] by showing that the result holds on a commutative ternary semigroup with
identity element. Moreover, we prove by an example that the converse of the statement does
not hold, in general.
Mathematics Subject Classification: 06F05
Keywords: semigroups; ternary semigroups; maximal and prime ideals.
1. Introduction and Preliminaries
There are a number of relations between maximal and prime ideals in a commutative ring with identity
element. Such as, in a commutative ring with identity element any maximal ideal is prime. In 1969, Schwarz
proved the similar result on a semigroup, in [5, Theorem 1, p. 73]. In 2003, Kehayopulu, Ponizovskii and
Tsingelis generalized the Schwarz' result, they have done on any ordered semigroups, in [2, p. 34]. In this paper,
we follow the idea of [2] by showing that in any commutative ternary semigroup with identity element any
maximal ideal is prime.
Ternary algebraic systems, a nonempty set with a ternary operation, have been introduced by Lehmer in
1932 [3] and studied by Siomon in 1965 [6]. Ternary semigroups were first introduced by Banach, he showed by
an example that a ternary semigroup does not necessary reduce to an ordinary semigroup.
Definition 1.1 Let be a nonempty set. Then is called a ternary semigroup if there exists a ternary operation
, written as
, such that
for all
.
Let
be a semigroup. For
, define a ternary operation on
by
.
Then
is a ternary semigroup. The notion of ternary semigroups have been widely studied.
For nonempty subsets
and
of a ternary semigroup
, let
.
For
, let
. For any other cases can be defined analogously.
Definition 1.2 A ternary semigroup
is said to be commutative if for any bijection
for all
Definition 1.3 Let
on
,
.
be a ternary semigroup. A nonempty subset
.
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of
is called an ideal of
if
Thawhat Changphas & Boonyen Thongkam
In [1], p. 14, the intersection of all ideal of a ternary semigroup
is an ideal of . It is called the ideal of generated by , denoted by
the form:
containing a nonempty subset
. In a ternary semigroup ,
.
Definition 1.4 Let
implies
be a ternary semigroup. An ideal
or
.
Definition 1.5 An ideal
implies
of a ternary semigroup
or
.
Let
of ternary semigroups
of
is of
(1)
of
is said to be prime if for any
is called a maximal ideal of
,
if for any ideal
of
,
be a nonempty family of ternary semigroups. Consider the Cartesian product
for all
. Defone a ternary operation
,
written as
,
by
.
Then
is a ternary semigroup.
2. Main Results
The following lemmas are required.
Lemma 2.1 Let
, then the set
Proof. Since
Let
for every
be a nonempty family of ternary semigroups. If
is an ideal of
.
for all
and
, there exists
for each
. Since
is an ideal of
. Since
for each
,
.
, it follows that
.
Then
. In the same manner, we have that
and
.
Therefore, the set
is an ideal of
.
Let
be the closed interval of real numbers. Using the usual multiplication,
under the ternary operation defined by
for all
. Next, we show that any closed interval
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where
is a ternary semigroup
is an ideal of
.
A Note on Maximal Ideals in Ternary Semigroups
Lemma 2.2 If
, then the set
Proof. Let
. Since
. Since
. Similarly,
An element
for all
is an ideal of
, we have
and
and
.
. We shall show that
. Let
, we obtain
. Thus
. Then
. Therefore,
is an ideal of .
of a ternary semigroup
is called an identity element of
and
if
.
The following theorem shows that in a commutative ternary semigroup with identity element any
maximal ideal is prime.
Theorem 2.3 Let
, then
is prime.
be a commutative ternary semigroup with identity element. If
Proof. Let be an identity element of
ideal of , let
be such that
. Assume that
and
is a maximal ideal of
is a maximal ideal of . To show that
is a prime
. Since is commutative, by (1), we have
.
Since
, it follows that
Since
, we have
.
Since
is maximal, we obtain
Let
and
. By (2),
such that
.
. Then
.
If
, then
,
so
. If
, because
is commutative, then
.
So
. Therefore,
is a prime ideal of
.
Remark. The converse of Theorem 2.3 does not holds, in general. The following example shows. Let
be a ternary semigroup mentioned above. We consider the ternary semigroup
. Clearly,
is
commutative and has the identity
. Let
By Lemma 2.2,
is an ideal of
. Let
, we have
. Since
is an ideal of
be such that
or
. This implies that
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, by Lemma 2.1, we obtain
. Then
or
. Thus
is an ideal of
. Since
is a prime
Thawhat Changphas & Boonyen Thongkam
ideal of
. By Lemma 2.2,
is not maximal.
that
is an ideal of
. Since
, we have
ACKNOWLEDGEMENTS. The first author is supported by the Centre of Excellence in Mathematics, the
Commission on Higher Education, Thailand.
References
[1] A. Iampan, On Ordered Ideal Extensions of Ordered Ternary Semigroups, Lobachevskii Journal of
Mathematics, Vol. 31 (2010), No. 1, 13-17.
[2] N. Kehayopulu, J. Ponizovskii, M. Tsingelis, A note on maximal ideals in ordered semigroups, Algebra and
Discrete Mathematics, 2003, No. 2, 32-35.
[3] D. H. Lehmer, A ternary analogue of abelian groups, Am. J. Math., Vol. 54 (1932), 329.
[4] M. Petrich, Introduction to Semigroups, Merrill, Columbus, 1973.
[5] St. Schwarz, Prime ideals and maximal ideals in semigroups, Czechoslovak Mathematical Journal, Vol. 19
(1969), No. 1, 72-79.
[6] F. M. Siomon, Ideal theory in ternary semigroups, Math. Jap., Vol. 10 (1965), 63.
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