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Abstract
We consider the ring (integers modulo
) with the partial order ‘
’ given by ‘
if either
or
’. In this paper, we obtain necessary and sufficient conditions for the poset (
) to be a lattice.
1 Introduction
An element in a commutative ring
is said to be a generalized projection if
for some
with
(see [Citation1]); an element
is called a regular element if
for some element
in the ring. It is proved by L
szl
T
th [Citation2] that in
, an element is a generalized projection if and only if it is regular; in-fact the following result is proved.
Theorem 1.1
[Citation2], Theorem 1
Let be the prime factorization of
with
, for all
. For an integer
, the following assertions are equivalent:
(i) is regular (mod
); (ii) for every
, either
or
;
(iii) ; (iv)
and
;
(v) ; (vi) there exists an integer
such that
.
We denote by , the set of generalized projections (i.e. the set of regular elements) and
, the set of projections in
. Let
be a commutative ring, the relation ‘
’ defined by: for
, ‘
if and only if either
or
’ is a partial order on
(see [Citation1]). In particular,
is a poset with the smallest element
and the largest element
. It is known that
is a lattice. However, Khairnar and Waphare [Citation1] proved that for any finite commutative ring
,
is a lattice, hence in particular,
is a lattice for every
. Whenever
is a square-free integer, we get that
. In general,
is not a lattice, for example
(see ). In this paper, we give necessary and sufficient conditions for the poset (
) to be a lattice.
We denote by , the set of units in
and
, the set of nilpotents in
. In the following remark, we list observations required in a sequel.
Remark 1.2
Let be the prime factorization of
with
, for all
, and let
. Then,
(i) if and only if
unit (mod
) for all
.
(ii) if and only if
zero (mod
) for all
.
(iii) if and only if
zero or unit (mod
) for all
.
2 Upper covering projection and lower covering projection
In a poset ,
denotes
with
. We say that
is an upper cover of
or
is a lower cover of
(denoted by
), if
and there is no
such that
.
The following theorem gives an existence of the unique lower cover and the unique upper cover of any element in the poset
.
Theorem 2.1
[Citation1], Theorem 2.7
Let be a finite commutative ring. If
, then there exist a unique
and a unique
such that
. Further, these unique elements are projections and for
,
if and only if
; and
if and only if
.
With notations as in Theorem 2.1, the unique projection is called the lower covering projection of
and the unique projection
is called the upper covering projection of
. If
is a projection, then we assume that the lower covering projection of
and the upper covering projection of
is
itself.
Theorem 2.1 gives an existence of the upper covering projection and the lower covering projection of any element in . In this section, we determine the upper covering projection and the lower covering projection of elements in
.
The following lemma gives the conditions for strict comparability of a projection and a generalized projection.
Lemma 2.2
Let be the prime factorization of
with
, for all
. Let
and
. Then,
(1) if and only if for
,
implies that
.
(2) if and only if for
,
implies that
.
Proof
Let
and
be such that
. Then
and
. Therefore
. Conversely, suppose that for
,
implies that
. Then
. Thus
.
Let
and
be such that
. Then
. Therefore
. Conversely, suppose that for
,
implies that
. This gives
. Thus
. □
Remark 2.3
[Citation1], Remark 3
Let . Suppose
be the smallest integer such that
. Then
. Therefore
, and
. Clearly,
; and
if and only if
.
For any the following theorem gives a construction for
.
Theorem 2.4
Let be the prime factorization of
with
, for all
, and
. If
, then
. If
, then
where
.
Proof
Let . If
then by Remark 2.3,
. Suppose
. By Remark 1.2, there exists
such that
, and
for all
. Let
and
. Then
and by Lemma 2.2,
. Let
be such that
. If
, then
. Suppose
. Again by Lemma 2.2, for
,
implies that
. Thus
. □
For any the following theorem gives a construction for
.
Theorem 2.5
Let be the prime factorization of
with
, for all
, and
. Then
where
.
Proof
Let . If
for all
then
. Hence
, a contradiction. Therefore there exists
such that
. Let
and
. We prove that
. Let
be such that
. Then
and hence
. This yields,
. Therefore
. Let
be such that
, and
be such that
. Then by Lemma 2.2, we get
. Consequently,
and hence
. This implies that
. Therefore
. Thus
. □
Let be a poset and
. The join of
and
if exists, denoted by
, is defined as
. The meet of
and
if exists, denoted by
, is defined as
.
We conclude this section with the following examples.
In the following example, is not square-free but the poset
is a lattice.
Example 2.6
Consider the ring . Then
and
. Note that
is not square-free but the poset
is a lattice (see ). Also, the nilpotent element
possess unique upper cover.
Example 2.7
Consider the ring . Then
and
. Note that
is not square-free but the poset
is a lattice (see ). Also, each of
and
possesses unique upper covers but
does not possess unique upper cover.
In the following example, the poset is not a lattice. Also, none of the nilpotent elements possess unique upper cover.
Example 2.8
Consider the ring . Then
and
. By , the poset
is not a lattice. Note that
and
do not exist. Also, each of
and
do not possess unique upper covers and each of
and
do not possess unique lower covers.
In the following example, is not square-free but the poset
is a lattice.
Example 2.9
Consider the ring . Then
and
. Note that
is not square-free but the poset
is a lattice (see ). Observe that, the nilpotent element
does not possess an unique upper cover.
Theorem 2.10
Let be a poset and
. If
is a dual atom then
.
Proof
If then
. Suppose
. So
is a dual atom if and only if
implies that
if and only if
implies that
if and only if
implies that
if and only if
is not a zero divisor if and only if
is an unit. Hence
. □
In the next section, we give a necessary and sufficient condition for the existence of supremum and infimum of any two elements of the poset .
3 Existence of ![](//:0)
and ![](//:0)
for ![](//:0)
![](//:0)
For , the ideal generated by
is denoted by
.
The following theorem characterizes the existence of for
.
Theorem 3.1
Let be the prime factorization of
with
, for all
. Let
be incomparable and
. Then,
exists if and only if the coset
has the smallest element.
Proof
For each , let
be the largest powers of prime
such that
and
respectively. Let
and
. Then
. Let
and for each
, let
be the largest powers of the prime
such that
. Then,
and
, if and only if
and
, if and only if
for all
, if and only if
, if and only if
.
Suppose exists. Since
and
are incomparable, we have
and
. This yields
. Let
. Then
and
. Therefore
. Thus
is the smallest element of the coset
. Conversely, suppose that the coset
has the smallest element, say
. This yields
and
. We claim that
. Let
be such that
and
. Then
. Therefore
. Thus
. □
From the proof of Theorem 3.1, it is clear that, if exists, then
is the least element of the coset
. Also, if the coset
has the smallest element
, then
.
The following corollary is an immediate consequence of Theorem 3.1.
Corollary 3.2
Let and
for some incomparable elements
. Then,
is a lattice if and only if every coset in
has smallest element.
In the following theorem, we characterize the existence of for
.
Theorem 3.3
Let be the prime factorization of
with
, for all
. Let
be incomparable and
. Then,
exists if and only if the ideal
has the largest element.
Proof
For each , let
be the largest powers of prime
such that
and
respectively. Let
and
. Then
. Let
and for each
, let
be the largest powers of prime
such that
. Then,
and
if and only if
and
if and only if
for all
if and only if
.
Suppose exists. Since
and
are incomparable, we have
and
. This yields
. Let
. Then
and
. Therefore
. Thus
is the largest element of the ideal
. Conversely, suppose that the ideal
has the largest element, say
. This yields
and
. We claim that
. Let
be such that
and
. Then
. Therefore
. Thus
. □
From the proof of Theorem 3.3, it is clear that if exists, then
is the largest element of the ideal
. Also, if the ideal
has the largest element
, then
.
The following corollary is an immediate consequence of Theorem 3.3.
Corollary 3.4
Let and
for some incomparable elements
. Then,
is a lattice if and only if every ideal in
has largest element.
Corollary 3.5
Let ,
for some incomparable elements
and
for some incomparable elements
. Then, every ideal in
has largest element if and only if every coset in
has smallest element.
Proof
Follows from Corollary 3.2 and Corollary 3.2. □
The following two lemmas relate the largest element of an ideal with the smallest element of a coset and vice versa.
Lemma 3.6
Let with
and
,
. Then, the largest element of the ideal
becomes the smallest element of the coset
.
Proof
Since , we have
. Therefore
and
are distinct elements in
. Let
be the largest element of
. Then
for all
. This yields
and
. That is
or
; and
or
. If
and
, then
, a contradiction to the fact that
. Thus, either
or
. Suppose
. That is
. This implies that
, hence
. Similarly,
implies that
. Thus, in any case,
. Let
be any element. Then
. Thus
is the smallest element of
. □
Lemma 3.7
Let with
and
,
. Then the smallest element of the coset
becomes the largest element of the ideal
.
Proof
Since , we have
. Therefore
and
are distinct elements in the coset
. Let
be the smallest element of
. Then
for all
. This yields
and
. That is
or
; and
or
. If
and
, then
, a contradiction to the fact that
. Thus, either
or
. Suppose
. That is
. This implies that
, hence
. Similarly,
implies that
. Thus, in any case,
. Let
be any element. Then
. Thus
is the largest element of
. □
Remark 3.8
Let and
be the ideal generated by
. Then
is the largest element of
. For
,
, hence
.
If then
and
both exist in the poset
(see [Citation1]). The following two theorems give the existence of
and
in the poset
where
.
Theorem 3.9
If then
exists in the poset
. Further,
.
Proof
If and
are comparable then clearly
exists and
. Suppose
and
are incomparable. Let
,
be the ideal generated by
and
be the ideal generated by
. As,
, by Remark 1.2 (iii),
. Therefore
. By Remark 3.8, the ideal
possesses the largest element, say
and
. Since
and
, we have
. As,
and
are incomparable, we have
. By Lemma 3.6,
becomes the smallest element of the coset
. By Theorem 3.1,
exists and
. Thus,
. □
Theorem 3.10
Let be such that
. Then
exists in the poset
and
.
Proof
If and
are comparable then clearly
exists and
. Suppose
and
are incomparable. Let
and
be the ideal generated by
. As,
, by Remark 1.2 (iii),
. Therefore
, and hence
. By Remark 3.8, the ideal
possesses the largest element, say
and
. By Theorem 3.3,
exists and
. Thus
. □
In the following theorem, we give a necessary and sufficient condition for the poset to be a lattice.
Theorem 3.11
Let . Then,
is a lattice if and only if for every
with
,
possess the largest element.
Proof
Suppose is a lattice. Let
with
. If
does not possess the largest element, then
. Let
; and
, where
is a prime such that
and
. Then
and
. If
, then
. That is
. This yields
. This implies that
. Hence
. Consequently
. By Remark 3.8,
possess the largest element, a contradiction. Therefore
. Similarly,
. Thus
and
are incomparable. Observe that
. As,
does not possess the largest element. By Lemma 3.7, the coset
does not possess the smallest element. By Corollary 3.2,
is not a lattice, a contradiction. Therefore
possess the largest element. Conversely, suppose for every
with
,
possess the largest element. If
is not a lattice, then by Corollary 3.2, there exist incomparable elements
such that
and
does not possess the smallest element. Therefore
and
. Hence
. Let
and
. Then
and
. By Lemma 3.6,
does not possess the largest element, a contradiction. Thus
is a lattice. □
Remark 3.12
Let . If
be such that
then
for any
. Since
, we have
. Hence
. From this, it follows that, if
is an ideal generated by a nilpotent element of
such that
, then
does not possess the largest element. Thus
is not a lattice.
Notes
Peer review under responsibility of Kalasalingam University.
References
- Khairnar Anil Waphare B.N. Order properties of generalized projections Linear Multilinear Algebra 65 7 2017 1446 1461
- Tóth László Regular integers modulo n Ann. Univ. Sci. Budapest. Comp. 29 2008 263 275