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ABSTRACT
This manuscript presents some results on the conditions and characterizations of filters, congruence relations and homomorphisms in algebraic hoops. The paper investigates the structural properties of filters that are generated by a set in various ways and establishes several descriptions for such filters. It is demonstrated in this paper that the class of filters in hoops forms an algebraic lattice. This finding contributes to our understanding of the structural properties of filters and their relationship within hoops. Moreover, this manuscript explores congruence relations in hoops, revealing a fascinating connection between the lattice of filters and the lattice of congruences. In particular, it is shown that the lattice of filters is in one-to-one correspondence with the lattice of congruences in hoops, confirming that the variety of hoops is an ideal determined. Additionally, the manuscript delves into the properties of hoop homomorphisms in relation to congruences and filters, as well as to quotient structures. This paper presents several homomorphism and correspondence theorems for hoops.
1. Introduction
Hoops, originated by Bosbach (Bosbach, Citation1969, Citation1970), are algebras with a binary operation satisfying a set of axioms that have been an area of significant interest and research in algebraic logics. In recent years, there have been significant advancements in hoop theory, as evidenced by the introduction of deep structure theorems (see (Aaly Kologani & Borzooei, Citation2019; Aglianó et al., Citation2007; Borzooei & Aaly Kologani, Citation2014b; Namdar et al., Citation2017)). Many of these findings have had a significant impact on fuzzy logic. One noteworthy result is the structure theorem of finite basic hoops (as stated in (Aglianó et al., Citation2007) Corollary 2.10), which provides an elegant and concise proof of the completeness theorem for propositional basic logic (refer to (Aaly Kologani & Borzooei, Citation2019), Theorem 3.8). This completeness theorem was originally introduced by Hajek in (Hájek, Citation1998). Currently, the study of filter theory in hoops has gained considerable attention, leading to the discovery of important results. Notably, various types of filters, such as (positive) implicative filters and fantastic filters (as discussed in (Borzooei & Aaly Kologani, Citation2014a)), have been introduced, and their characteristics have been presented in (Alavi et al., Citation2017; Borzooei & Kologani, Citation2014; Kondo & Dudek, Citation2008; Namdar et al., Citation2017).
In this manuscript, we delve into the study of filters in hoops and derive several important characterizations and properties related to them. The concept of filters in hoops has been a subject of investigation due to its relevance in understanding the algebraic structure and properties of hoops. Filters are subsets of hoops that possess certain characteristics and play a fundamental role in the algebraic formulation and analysis of hoops. By understanding the conditions under which a set in a hoop can be classified as a filter, we can gain insights into the fundamental properties and structures within a hoop.
The primary focus of this manuscript is to obtain various characterizations of filters generated by a set in different ways. By deriving these conditions, we aim to provide a comprehensive understanding of the structures and properties of filters. This exploration leads us to establish that the class of filters in hoops forms an algebraic lattice. This result not only deepens our understanding of the relationships between filters but also lays the foundation for further investigation into the lattice structures within a hoop. We further investigate the structural properties of those implicative filters in hoops. Interestingly, we obtain the smallest implicative filter in a hoop
and we give an algebraic description for implicative filters generated by a set in general and by a filter in particular. It is also proved that the collection of all implicative filters in a hoop
is complete and is a filter in the lattice of all filters of
.
The other important aspect of our study involves the examination of congruence relations on hoops. Congruence relations, which are equivalence relations on a hoop preserving its algebraic operations, provide valuable insights into the structure and behavior of hoops. We show that the lattice of filters in hoops is in one-to-one correspondence with the lattice of congruences, establishing a strong connection between these two fundamental aspects of hoop theory. This correspondence further solidifies the idea that the variety of hoops is ideally determined.
In addition to exploring the relationships between filters and congruences, we thoroughly investigate the properties of hoop homomorphisms in connection with congruences and quotient structures. Homomorphisms are morphisms that preserve the algebraic structure between hoops, and by studying their interactions with congruences and filters, we can uncover additional insights into the relationships and correlations within hoops.
2. Preliminaries
In this section, we present the necessary background and introduce the fundamental concepts that form the basis of our study. A hoop is an algebraic structure that combines the properties of groups and semigroups, providing a framework for studying operations that are both associative and partially invertible.
Definition 2.1.
(Bosbach, Citation1969, Citation1970) A hoop is a system ; consisting of a non-empty set
, binary operations
and
and a constant 1 in
satisfying the following conditions:
(1) |
| ||||
(2) |
| ||||
(3) | |||||
(4) |
for all . A hoop
can be made into a partially ordered set together with a natural ordering ≤ on
given by
if and only if
.
It is observed in Bosbach (Citation1970) that infimum of any two elements (with respect to the natural order) in a hoop always exists, and it is given as:
So that together with the natural ordering can be viewed as a meet semi-lattice. Moreover, the binary operations
and
satisfies the adjointness property; i.e. for any
it holds that
Theorem 2.2.
Every hoop satisfies the following properties:
(1) |
| ||||
(2) |
| ||||
(3) |
| ||||
(4) |
| ||||
(5) |
| ||||
(6) |
|
Lemma 2.3.
(Aglianó et al., Citation2007) Given a hoop , the pseudo-join of h and g in
, denoted
, is defined by:
Then for any, ;
(1) |
| ||||
(2) |
| ||||
(3) |
|
Lemma 2.4
(Aglianó et al., Citation2007). The following are equivalent in a hoop :
(1) |
| ||||
(2) | for all | ||||
(3) | for all | ||||
(4) | for all |
Definition 2.5.
(Aglianó et al., Citation2007) By a join hoop, we mean a hoop satisfying one and hence all other conditions of Lemma 2.4.
3. Some new results on filters
Filters in hoops extend the notion of filters in Boolean algebras and serve as a powerful tool for characterizing subsets that exhibit certain desirable properties.
Definition 3.1.
(Aglianó et al., Citation2007). A filter in a hoop is a subset U of
such that:
(F1) | |||||
(F2) |
|
The collection of all filters in will be denoted by
.
Theorem 3.2.
A set U contained in is a filter if and only if the following conditions are satisfied:
(F1) | |||||
(F2) | |||||
(F3) |
|
for all .
Proof.
Assume that U is a filter. We show that U satisfies and
. Let
. Then
. This implies that
and so
Therefore
holds. Again
and
imply
. Thus
and hence
holds.
Conversely, suppose that U satisfies the conditions ,
and
. We claim that U is a filter. It is enough to show that U satisfies
. Let
Then, by
,
Since
it follows from
that
and hence U is a filter.
Lemma 3.3.
Let and
. Then, we have:
(1) |
| ||||
(2) |
| ||||
(3) |
| ||||
(4) |
|
Lemma 3.4.
Let V be a filter in . If U is a filter of V (considering V as a hoop with the restricted operations of
), then U is a filter in
.
Theorem 3.5.
A nonempty set U contained in is a filter if and only if
for all
; where
Proof.
Let
and
Then
and so by assumption it is clear that
.
Since U is nonempty, we can choose
. As
, it follows from our assumption that
Let
Then, by assumption,
Since,
It is routine to verify that the class of all filters of
is closed under arbitrary intersection. It follows then that for any set
always we can find the smallest filter of
containing X. Usually, it is called the filter of
generated by X, and it is denoted by Fg(X). If X is empty, then it is trivial to check that
. The following theorem gives an equational description for filters generated by a non-empty set.
Theorem 3.6.
Let X be a non-empty set contained in . Then,
Proof.
Put We show that W is the least filter in
containing X. Clearly
and
Let
Then
and
and so
Therefore W is a filter. Suppose that Z is any filter containing X and let . Then, we can choose
with
Then it is clear that
This proves that
In order to get another description for Fg(X), let us define sets inductively as follows:
Then we have the following theorem.
Theorem 3.7.
For any :
where denotes the set of positive integers.
Proof.
Put Then, it can be easily shown that
and
. Then
and
Let
. Then
such that
and
Let
Then
and so
Therefore T is a filter. Let K be any filter such that
Now we show
for all
by induction. Since
and
we have
Thus the statement is true for n = 1. Let
and assume that
Now
implies that
and
and thus
Therefore
and so by mathematical induction,
for all
and hence
This proves that
For any define sets
inductively as follows:
Then we have the following theorem.
Theorem 3.8.
For any set X contained in :
where denotes the set of positive integers.
Proof.
Put Then, we can easily show that clearly
and
Thus
and
Let
Then there exist
such that
and
Let
Then
and since
Therefore T is a filter. Let K be any filter such that
Now we show
for all
We use induction on n. Since
and
we have
Thus the statement is true for n = 1. Let
and assume that
Now
implies that
such that
Thus
implying
and hence
Therefore
Thus, by mathematical induction,
for all
and hence
. This proves that
.
Lemma 3.9
For any the infimum
and supremum
of U and V are given, respectively, by:
Proof.
Put Now we claim that T is the least filter containing
Clearly,
. Let
be such that
There exist
such that
and
Thus we have
Since and
we have
. Therefore, T is a filter. Let K be any filter such that
. Then
implies that
for some
and
Thus
and
and so
. This proves that
Corollary 3.10.
forms a lattice.
Theorem 3.11
The map is an algebraic closure operator.
Proof.
Clearly, the given map forms a closure operator on . Now we show that for any
where is to mean E is a finite subset of
Let
Then
for some
. Put
Then
and
Thus,
, and the other inclusion is obvious. Therefore, the given map is an algebraic closure operator.
Corollary 3.12.
The pair forms an algebraic lattice.
Proof.
It is proved in the above theorem that the map is an algebraic closure operator on
. Then, the class
of closed elements of
with respect to this closure operator is an algebraic lattice (see Theorem 5.5 of (Burris & Sankappanavor, Citation1981)). It is also clear that
if and only if U is a filter in
, and hence we have
. Therefore,
is an algebraic lattice.
4. Implicative filters
In this section, we recall the definition of implicative filters in hoops, and we present some results mainly focusing on the characterization of implicative filters generated by a set.
Definition 4.1.
A set M in is defined to be an implicative filter if:
(IF1) | |||||
(IF2) |
|
Theorem 4.2.
Every implicative filter is a filter.
Proof.
Let M be an implicative filter in and
We first show that
. Put
and
Then, it can be shown that
and
Thus by
,
i.e.
Now we claim that
Put
and
Then
Also
Thus, by
so
Next let
such that
and
Then
Put
and
Then
and
Thus, by
There fore M is a filter.
The following theorem gives a set of equivalent conditions for a set to be an implicative filter and the proof is ilar to that of Haveshki et al. (Citation2006).
Theorem 4.3
For any set M contained in the following are equivalent.
(1) | M is an implicative filter; | ||||
(2) | M is a filter and | ||||
(3) | M is a filter and | ||||
(4) |
|
Theorem 4.4
Any filter U of is an implicative filter if and only if
for all
; where
.
Given a set X in , in the following theorem we describe the least implicative filter containing X.
Theorem 4.5
For a set X contained in , let us define an indexed family
of sets as follows:
and for
;
Then is the least implicative filter containing
Proof.
We first show that for all
. Clearly,
We use induction on
Let
and assume that
We claim that
As
we can write 1 as
and
This implies that
Therefore
Next, we show that for all
. Taking n = 1,
It is observed that
. Thus, it remains to show that
Let
Then
and
This implies that
Therefore
and hence
Now let
and assume that the result holds for
i.e.
Now we claim that
Let
Then
for some
such that
and
for some
Since, by induction assumption,
we get
and
Then
Therefore
Put . We claim to show that S is an implicative filter. Clearly
Let
such that
and
Then
and
for some
Since
is a chain, assume without loss of generality
and hence
and
implying that
Therefore S is an implicative filter. Finally, it remains to show that S is the least such a filter containing X. Let G be any implication filter such that
It suffices to show that
We use induction on
If
then
Let
and assume that
Now we claim that
Let
Then
where
This implies that
Since G is an implicative filter
Therefore
and hence
Therefore
Hence proved.
Definition 4.6.
For any filter U in , define a set
to be:
Observe that is contained in
containing U and is closed with respect to
. Moreover, U is an implicative filter if and only if
Furthermore,
is an idempotent hoop if and only if
Notation: For let us denote by
the element
. It is clear that
for all
, and
for all
if and only if
is an idempotent hoop.
Definition 4.7.
Define a set in
by:
We call the implicative center
.
Theorem 4.8
is the least implicative filter in
. Moreover,
is idempotent hoop if and only if
.
Proof.
One can easily observe that is the filter generated by the set
and hence it is an implicative filter. Moreover, if J is any other implicative filter in , then it follows from Theorem 4.4 that
and hence
is the least among all implicative filters.
Corollary 4.9.
For any filter U in , the smallest implicative filter
of
containing U can be described as follows:
where the supremum being computed over all filters of .
Lemma 4.10
Let be the collection of all implicative filters in
Then the following are true.
(1) |
| ||||
(2) |
|
Proof.
(1) It is clear that itself is an implicative filter. So
has the largest element. Now it is enough to show that
is closed under arbitrary intersection. Let
be a family of implicative filters in
Now we claim that
Clearly
Let
such that
and
Then
and
for all
Then
Therefore
and hence
is an implicative filter of
Thus
is a complete lattice.
(2) Let . Then, by (1), we get
. Moreover let
and N be any filter in
such that
Now our aim is to show that M is an implicative filter. As N is given to be a filter, it is enough to show that
for all
. As M is an implicative filter taking
and
we get
and
Since
it follows that
i.e.
Therefore
(*). Now let
such that
and
This implies that
and
By (*) we have
Thus by transitivity
Therefore
Note. During our investigation, we noticed that most of the results in this section can also be readily extended to those positive implicative filters.
5. Congruence relations
An equivalence relation Φ on is said to be a congruence relation on
provided that it satisfies the substitution property for
and
; i.e. for all
:
The collection of all congruence relations on will be denoted by
. For
and
let
In other words, is an equivalence class of Φ to which h belongs. Put
Let us define the binary operations and
on
as follows. For
:
For all . Then, it is routine to check that the binary operations are well defined and
is a hoop called the quotient of
modulo Φ.
Lemma 5.1.
For any ,
is a filter of
.
Proof.
Since
Let
Then
Therefore is a filter in
.
It is evident that the class of all congruence relations on
contains
and is closed under arbitrary intersection so that it is a complete lattice under the usual inclusion order. Moreover, for any
:
where is the congruence relation on
generated by
. It is described in (Burris & Sankappanavor, Citation1981) for general universal algebras in the following way:
where is defined as follows:
where is the relational product; i.e.
if and only if there is
such that
and
. We say that two congruences Φ and Ψ permute if
, and an algebra A is called permutable, provided that all congruences of A permute each other; i.e.
for all
. Moreover, a class
of algebras of a fixed type is permutable if each of its member is permutable. It was proved by Mal’cev in (Mal’cev, Citation1954) that a variety
of algebras is permutable if and only if it has a 3-ary term
satisfying the identities
The term p is called the Ma’cev term. In this regard, the class of hoops is congruence permutable, where its Mal’cev term is:
Therefore every hoop is congruence permutable.
Corollary 5.2.
For any :
.
Lemma 5.3.
For any :
Proof.
Suppose that Then we have the following:
Therefore . Conversely, suppose that
Similarly Since
, by transitivity we get
Therefore
Remark. It can be deduced from the above lemma that for all
, and hence every hoop is 1-regular.
Lemma 5.4.
For any :
(1) | |||||
(2) |
Proof.
(1) It is obvious.
(2) Since it follows from Lemma 5.3 that
. Thus
To prove the other inclusion let
Then
Therefore Hence (2) holds.
Definition 5.5.
For a given filter U in consider the binary relation on
denoted by
and given by:
Theorem 5.6
for all
. And conversely if
is a congruence relation on
, then U is a filter.
Proof.
Let . Clearly
is reflexive and symmetric. Now we proceed to show that
is transitive. Let
and
Then
and
Then, by Lemma 3.4 (2), we have
and
and hence
Next we prove the substitution property for Let
and
Then
and
Then, by Lemma 3.4(1), we have
and
and hence by Lemma 3.4(2), we have
In a ilar fashion it can be shown that
. Thus,
. Finally, we prove the substitution property for
Let
and
Then
and
Then, by Lemma 3.4(1), we have
and
and hence by Lemma 3.4(2), we have
In a ilar fashion, it can be shown that
. Thus
. Therefore
Conversely suppose that is a congruence relation on
Then, by Lemma 4.1
is a filter of
Now we claim that
Now let
Then
That is,
and hence
. Again, to show the other inclusion, let
. Then
and
Therefore
and hence
. Thus
. This proves the result.
Lemma 5.7.
For any and
we have:
(1) | |||||
(2) |
Proof.
(1) It follows from the converse part of the above theorem.
(2) Let Then
and so
and
This implies that
and
. Since Φ is symmetric and
by transitivity, we get
and so
On the other hand let
. Then
and
This implies that
and so
Therefore
Theorem 5.8.
There is a lattice isomorphism between and
.
Proof.
Define a map by
for all
If
is a map defined by
, then
and
. Therefore, π is invertible, and hence, it is bijective. Now, it remains to show that π is a lattice homomorphism.
Let . Then
and
Thus π is a lattice homomorphism and hence it is a lattice isomorphism. Therefore .
Corollary 5.9.
The map is a lattice isomorphism of
onto
.
Proof.
Since the map is the inverse of π and π is a lattice isomorphism of
onto
, the given map is a lattice isomorphism.
Corollary 5.10
The following holds for all filters U and V in :
(1) | |||||
(2) | |||||
(3) |
|
Proof.
It is a direct consequence of Corollary 5.10.
Next, we see distributivity
Theorem 5.11.
Every hoop is congruence distributive.
Proof.
Applying Theorem 4.8 it suffices to show that the lattice of filters of
is distributive. Let
and W be any filters in
. Then
and
for some
It is sufficient to verify that
Now
Therefore is distributive.
Recall from (Chajda et al., Citation2003) that a permutable and regular variety of algebras is called arithmetic. Therefore, it is evident from the above theorem that the variety of hoops is arithmetic.
Notation. Given a filter U of and
; we write
to denote the congruence class of
containing a, and by
we mean the quotient
.
6. Hoop Homomorphisms
In this section, we study homomorphism and isomorphism Theorems in hoops.
Definition 6.1.
A map is said to be a hoop homomorphism if it satisfies the following conditions: for all
(1) | |||||
(2) |
Note that if is a hoop homomorphism, then one can easily check that
A hoop homomorphism
is called a hoop monomorphism (respectively, a hoop epimorphism) if it is injective (respectively, surjective), and it is called a hoop isomorphism if it is both hoop monomorphism and epimorphism. The following result follows from the definition of hoop homomorphism.
Theorem 6.2.
Let, f be a hoop homomorphism from to
. Then
(1) | If V is a filter in | ||||
(2) | If f is onto and U is a filter in |
Definition 6.3.
Given a hoop homomorphism , its kernel is defined to be the set:
Lemma 6.4.
For any hoop homomorphism , its kernel ker(f) is a filter in
.
Proof.
One can easily observe that , and hence it follows from (1) of Theorem 6.2 that ker(f) is a filter in
.
Theorem 6.5.
A hoop homomorphism is a monomorphism if and only if
.
Proof.
Suppose that f is a monomorphism. Then, f is an injection and therefore
Theorem 6.6
[The first isomorphism Theorem] Let be a hoop homomorphism. Then
Proof.
Put Define
by:
Now we show that
is a hoop isomorphism. Let
such that
Then
and
and so
This shows that
is well defined. Now for any
, we have
and
and so
is a hoop homomorphism. Since any element of
is of the form
for some
and
,
is a surjection. Also, for any
Therefore is an injection. Thus,
is a hoop isomorphism and so
Note that, in particular, if f is surjective, then
Theorem 6.7
[The second Isomorphism Theorem] Let U and V be filters in. Then
Proof.
Define a map by
, for all
Now for any
,
and similarly we can show
and hence f is a hoop homomorphism. Let
Then
and so there exists
and
such that
Now
implies that
Thus
Put
and we claim that
Now
Therefore Since
we have
and
Therefore and hence f is onto. Therefore, by first isomorphism theorem, we have
But
This proves the result.
Theorem 6.8
[The third Isomorphism Theorem] Let U and V be filters in such that
Then
Proof.
Define a map by:
Let such that
Then
and
This implies that
and hence
. Therefore,
is well defined. It is also clear that f is a hoop epimorphism. Thus, by first isomorphism theorem,
But
Hence proved.
Theorem 6.9
(Correspondence Theorem) Let U be a filter in Then there is a one-to-one correspondence between the set
of all filters of
containing U and the set
of all filters of
.
Proof.
Note that every filter of is of the form
where W is a filter of
which contains U. Define a map
by:
Let such that
Then
. Now
Therefore Similarly, we have
Therefore
and hence g is injective. Let
Put
. Now we claim that W is a filter which contains
Now let
Then
Since P is a filter of
, we have
This implies that
therefore W is a filter of
Let
. Then
and
and hence
This implies that
Therefore
and
Therefor g is bijective, i.e. there is a one-to-one correspondence between
and
7. Conclusions
This manuscript has presented several new and significant results on filters in hoops. We have derived conditions for sets in a hoop to be filters and have provided multiple characterizations for filters generated in different ways. Furthermore, we have demonstrated that the class of filters in hoops forms an algebraic lattice. Additionally, our investigation into congruence relations on hoops has revealed an interesting correspondence between the lattice of filters and the lattice of congruences. This finding confirms that the variety of hoops is indeed ideally determined. Moreover, our exploration of hoop homomorphisms in connection with congruences and quotient structures has enabled us to establish various homomorphism and correspondence theorems. Through these theorems, we have developed a deeper understanding of the relationships between different structures within hoops. Overall, this manuscript contributes to the existing body of knowledge on hoops by providing new insights into the properties of filters, congruences, and hoop homomorphisms. We believe that these results have the potential to impact future research in this field, and we hope that they will inspire further investigations and applications in the study of hoops.
Authors contribution
All listed authors have made a significant scientific contribution to the findings of this manuscript.
Human and animal rights
This article does not contain any studies with human participants or animals performed by any of the authors.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
The work was supported by the University of Gondar.
Data availability
No data were used to support the results of this study.
Additional information
Funding
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