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Abstract
In this paper, as a generalization of the concepts of left filters, right filters and filters of ordered semigroups, the concepts, for any positive integers m and n, of left-m-filters, right-n-filters and -filters in ordered semigroups have been introduced and some properties of these generalized notions have been investigated. Finally left-m-filters (resp. right-n-filters,
-filters) of
-regular (resp.
-regular,
-regular) ordered semigroups have been characterized in terms of its prime generalized
-ideals (resp.
-ideals,
-ideals).
1. Introduction and preliminaries
In 1987, Kehayopulu [Citation1] introduced the concept of a filter on a poe-semigroup. Later on in 1990, Kehayopulu [Citation2] defined the relation on a po-semigroup as follows: Let S be a po-semigroup. Then
, where
denotes the filter of S generated by an element a of S. He showed that the relation
was the least semilattice congruence on S and was equal to the intersection of the semilattice congruences
on S, where P is a prime ideal of S. The study of the left (resp. right) filters on a po-semigroup was initiated by S. K. Lee and S. S. Lee [Citation3]. They proved some characterizations of these left (resp. right) filters on a po-semigroup in terms of its leftt (resp. right) prime ideals.
The notion of an -ideal of a semigroup was introduced by Lajos [Citation4] as a generalization of a one-sided ideal of a semigroup (see also [Citation5–8] for related notions and results on
-ideals in semigroups). Thereafter the notion of
-ideals was introduced in various algebraic structures and had been studied by many authors, for instance, Akram et al. [Citation9], Bussaban and Changphas [Citation10], Changphas [Citation11], Tilidetzke [Citation12], Yaqoob and Chinram [Citation13], and many others.
Let S be a non-empty set. The triplet is called an ordered semigroup (or a po-semigroup) if
is a semigroup and
is a partially ordered set such that
for all
.
For any non-empty subset A of an ordered semigroup S, we denote If
, we shall be writing, in whatever follows,
instead of
. For any non-empty subsets A and B of an ordered semigroup S, it is easy to verify that: (1)
; (2)
; (3) If
, then
; (4)
and (5)
, where the product of A and B is defined as usual by
.
Let S be an ordered semigroup. Then a non-empty subset A of S is called
a subsemigroup of S if
;
a left (resp. right) ideal of S if
and
;
an ideal of S if it is both a left and a right ideal of S.
Further a subsemigroup F of an ordered semigroup S is called
a left filter (resp. right filter) of S if for any
,
implies
(resp.
) and
implies
;
a filter if it is both a left filter and a right filter of S.
Throughout this paper, S will always stand for an ordered semigroup and for positive integers, unless and otherwise specified.
2. Main results
Definition 2.1
A subsemigroup F of an ordered semigroup S is called a left-m-filter (resp. right-n-filter) if
for any
,
implies
(resp.
);
implies
.
Further a subsemigroup F of an ordered semigroup S is called an -filter if it is both a left-m-filter and a right-n-filter of S.
Remark 1
In particular for , F is a left filter (resp. right filter). Clearly each left filter (resp. right filter, filter) of an ordered semigroup S is a left-m-filter (resp. a right-n-filter, an
-filter) for each positive integers m and n. Indeed; For any left filter F of S and
such that
, as F is a left filter,
. Therefore
. Thus, the concept of a left-m-filter (resp. right-n-filter,
-filter) is a generalization of the concept of a left filter (resp. right-filter, filter). Conversely a left-m-filter (resp. right-n-filter, an
-filter) need not always be a left filter (resp. right filter, filter) which has been shown by the following example:
Example 2.2
Let . Define an operation ○ and an order ≤ on S as follows:
The covering relation ≺ and the figure of S are as follows;
Then S is an ordered semigroup. Let . Since F is subsemigroup,
,
and
, we have that F is a left-3-filter of S but not a left filter of S.
Lemma 2.3
Let S be an ordered semigroup and T be a subsemigroup of S. Then for each left-m-filter (resp. right-n-filter) F of S, either or
is a left-m-filter (resp. right-n-filter) of T.
Proof.
Suppose that . Now
and
. Therefore
. So
is a subsemegroup of T. Next suppose that, for any
,
. Therefore
. Since
and F is a left-m-filter of S,
. Also
. Thus
. Finally take any
and
such that
. As F is a left-m-filter of S and
,
. Therefore
. Hence
is a left-m-filter of T.
Corollary 2.4
Let S be an ordered semigroup and T be a subsemigroup of S. Then for every
-filter F of S, either
or
is an
-filter of T.
Lemma 2.5
Let S be an ordered semigroup and be a family of left-m-filters (resp. right-n-filters) of S. If
then
is a left-m-filter (resp. right-n-filter) of S.
Proof.
Assume that and let
Then
for each
. As, for each
,
is left-m-filter,
. Therefore
Thus
is a subsemigroup of S. Next take any
such that
. This implies that
for each
. As
's are left-m-filters,
for each
. So
. Finally take any element
and
such that
. Then
for each
. Since
's are left-m-filters,
for each
and, therefore,
. Thus
is a left-m-filter.
Corollary 2.6
Let S be an ordered semigroup and be a family of
-filters of S. If
then
is an
-filter of S.
Remark 2
Union of left-m-filters (resp. right-n-filters, -filters) of an ordered semigroup need not be a left-m-filter (resp. right-n-filter,
-filter) in general.
The following example shows that, in general, the union of left-m-filters (resp. right-n-filters, -filters) of an ordered semigroup is not always a left-m-filter (resp. right-n-filter,
-filter).
Example 2.7
Let . Define a binary operation ○ and an order relation ≤ on S as follows:
The covering relation ≺ and the figure of S are as follows;
Then S is an ordered semigroup. The subset of S is a left-2-filter of S because
and
implies
. Thus
is a left-2-filter of S and, as,
, F1 is not a left-filter of S. Similarly
is a left-2-filter of S. Now
. Since
,
is not a subsemigroup of S and, hence,
is not a left-2-filter of S.
Let be any family of ordered semigroups. Then the cartesian product
is an ordered semigroup [Citation14] under the operation
and order
Theorem 2.8
Let be a family of ordered semigroups and
for each
be a left-m-filter (right-n-filters) of
. Then
is a left-m-filter (right-n-filter) of the ordered semigroup
.
Proof.
First we show that F is a subsemigroup of S. To show this, assume that . Then
for each
. As each
is a left-m-filter,
for each
. Therefore
. Thus F is a subsemigroup of S.
Next assume that, for any ,
Therefore
for each
. As each
is a left-m-filter for each
,
for each
. Thus
.
Finally, take any and
such that
. Therefore
for each
. As for each
,
is a left-m-filter and
, we have
for each
. Thus
. Hence F is a left-m-filter of S.
Corollary 2.9
Let be a family of ordered semigroups and
for each
be an
-filter of
. Then
is an
-filter of the ordered semigroup
.
Definition 2.10
Let S be an ordered semigroup and be positive integers. Then a subsemigroup A of S is said to be an
-ideal of S if
; and
.
Similarly we may define a -ideal and a
-ideal of S.
If we drop the subsemigroup condition from the above definition, then A is called a generalized -ideal of S. Similarly a generalized
-ideal and a generalized
-ideal are defined.
Remark 3
It is easy to check that each -ideal (resp.
-ideal,
-ideal) of any ordered semigroup is always a generalized
-ideal (resp.
-ideal,
-ideal), but the converse is not true in general. This has been shown by the following example.
Example 2.11
Let be an ordered semigroup of Example 2.2. Then the subset
of
is a generalized
-ideal of
for all integers
and not an
-ideal of
. Further
is a generalized
-ideal and a generalized
-ideal of
for all integers
, but not an
-ideal and
-ideal of
.
A generalized -ideal (resp.
-ideal,
-ideal) A of an ordered semigroup S is called prime if for any
such that
, either
or
.
Let S be an ordered semigroup and be non-negative integers. Then S is said to be
-regular if for each
, there exists
such that
i.e., if
or equivalently for each non-empty subset A of S,
with the convention that
.
Lemma 2.12
Let S be an -regular (resp.
-regular) ordered semigroup and F be a non-empty subset of S. Then the following are equivalent:
F is a left-m-filter ( resp. right-n-filter) of S;
or
is a prime generalized
-ideal (resp.
-ideal) of S, where
is the complement of F in S.
Proof.
. Assume that
. If
, then, as S is
-regular,
, a contradiction. Therefore
Let
. If
, then, as F is a left-m-filter, we have
. Thus
. Therefore
is a generalized
-ideal of S. Now to show that
is a prime generalized
-ideal of S, take any
such that
. If
and
, then
, a contradiction. Therefore either
or
. Hence
is a prime generalized
-ideal of S.
. Let
is a prime generalized
-ideal of S. To show that F is a left-m-filter of S, take any
. If
, by hypothesis, either
or
, a contradiction. Thus
and, so, F is a subsemigroup of S. Next suppose that for any
. If
, then, by
-regularity of S,
for some
. So
, a contradiction. Thus
. Finally take any element
and
such that
. If
, then
which is a contradiction. Thus
. Hence
is a left-m-filter of
.
Corollary 2.13
Let S be an -regular ordered semigroup and F be any non-empty subset of S. Then the following are equivalent:
F is
-filter of S;
or
is prime generalized
-ideal of S, where
is the complement of F in S.
Lemma 2.14
A -regular (resp.
-regular) ordered semigroup S does not contain proper left-m-filters (resp. right-n-filters) if and only if S does not contain proper prime generalized
-ideals (resp.
-ideals).
Proof.
Assume that S does not contain a proper left-m-filters. Let A be any proper prime generalized -ideal of S. Then, by Lemma 2.12,
is a proper left-m-filter of S which is a contradiction. Therefore S does not contain any proper prime generalized
-ideal.
Conversely assume that S does not contain proper prime generalized -ideals. Let F be any proper left-m-filter of S. Then, by Lemma 2.12,
is a proper prime generalized
-ideal of S which is a contradiction. Hence S does not contain proper left-m-filters.
Corollary 2.15
An -regular ordered semigroup S does not contain proper
-filters if and only if S does not contain proper prime generalized
-ideals.
Let and
be two ordered semigroups respectively and
be a mapping from S into T. Then φ is called an isotone if for all
implies
. Further φ is called a reverse isotone if
implies
. A mapping
is called a homomorphism if φ is an isotone and
Each reverse isotone mapping
is clearly one-one. Further φ is called an isomorphism if
φ is homomorphism;
φ is a reverse isotone; and
φ is onto.
Lemma 2.16
Let and
be ordered semigroups,
any homomorphism and F a left-m-filter (resp. right-n-filter) of T. Then
is a left-m-filter (resp. right-n-filter) of S.
Proof.
First we show that is a subsemigroup of S. If
, then
. As φ is a homomorphism and F is a left-m-filter of T,
. So
and, thus,
is a subsemigroup of S. Next take any
such that
. Then
If
and
be such that
, then
and
. Therefore
and, so,
Hence
is a left-m-filter of S.
Corollary 2.17
Let and
be ordered semigroups,
a homomorphism, F an
-filter of T. Then
is an
-filter of S.
Application: Since an ordered-Γ-semigroup is a generalization of an ordered semigroup, all the above results hold true for an ordered-Γ-semigroup.
Problems: . Let
and
be two ordered semigroups,
an onto homomorphism, F a left-m-filter (resp. a right-n-filter, an
-filter) of S. Does
is a left-m-filter (resp. a right-n-filter, an
-filter) of T?
. Under what conditions, the concept of a left-m-filter (a right-n-filter, an
-filter) of an ordered semigroup coincides with the concept of a left-filter (a right-filter, a filter)?
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Ahsan Mahboob http://orcid.org/0000-0003-3305-2029
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