![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
ABSTRACT
This paper introduces the concept of subalgebra and discusses some properties of subalgebra. We also examine different types of ideals and how they behave. We also describe some properties of a homomorphism. Finally, we present quotient algebra and discuss isomorphism theorems.
1. Introduction
Swamy (Citation1964b) introduced the notion of an autometrized algebra to obtain a unified theory of the then-known autometrized algebras: Boolean algebras (Blumenthal, Citation1952; Ellis, Citation1951), Brouwerian algebras (Nordhaus & Lapidus, Citation1954), Newman algebras (Roy, Citation1960), autometrized lattices (Nordhaus & Lapidus, Citation1954) and commutative lattice ordered groups or l-groups (Swamy, Citation1964a). Swamy &and Rao (Citation1977), Rachŭnek (Citation1987, Citation1989, Citation1990, Citation1998), Hansen (Citation1994), Kovář (Citation2000), and Chajda & Rachunek (Citation2001) developed the theory of autometrized algebra.
In addition, Subba Rao and Yedlapalli (Citation2018) and Rao et al. (Citation2019, Citation2021) studied the concept of representable autometrized algebra.
In Section 1.1., we recall some definitions and terms (Swamy & Rao, Citation1977; Swamy, Citation1964b). Section 2. introduces the concept of subalgebra and discusses the characteristics of subalgebra. Section 3. discusses different types of ideals and how they behave. We also discuss some properties of a homomorphism. In Section 4., we introduce quotient algebra and discuss isomorphism theorems.
1.1. Preliminaries
In this section, we look at some essential concepts, definitions, and terms that are important in other sections.
Definition 1.1 ([1])
A system A= is called an autometrized algebra if
is a commutative monoid.
is a partial ordered set, and
is translation invariant, that is,
.
is autometric on
, that is,
satisfies metric operation axioms:
(M1) and,
,
(M3) ;
.
Definition 1.2 ([7])
An autometrized algebraA = is called normal if and only if
(iv) For any and
in A,
such that
.
Definition 1.3 (Citation[7]) Let be a system. Then,
is said to be a lattice ordered autometrized algebra (or) autometrized l-algebra if
is a commutative semigroup with 0.
is a lattice, and
is translation invariant, that is,
;
(iii) is autometric on
, that is,
satisfies metric operation axioms:
,
and
.
Definition 1.4 ([McKenzie, Citation2018) Let be a closure operator on the set
.
is said to be algebraic if and only if
, for all
.
Theorem 1.5 ([18])
Let be a set. Let
be an algebraic closure operator on a set
. Then, the lattice of all
-closed elements (
) is an algebraic lattice, and the compact elements of
are exactly the finitely generated closed sets
, where
is a finite subset of
. Conversely, every algebraic lattice is isomorphic to the lattice of closed sets for some algebraic closure operator.
Definition 1.6 ([7])
A nonempty subset of a normal autometrized algebra A =
is called an ideal if and only if
imply
.
imply
.
Definition 1.7 (Citation[10])Let be an autometrized algebra. Let
be the set of all ideals of
. Let
. Then,
is called a regular ideal in A if
, where
for each
implies
for some
. Further regular ideal if and only if prime ideal.
Definition 1.8 (Citation[7]) Let A = and B =
be normal autometrized algebras. Let
be a map. Then,
is said to be a homomorphism from
to
if and only if
and
Definition 1.9 ([7])
Let A = and B =
be normal autometrized algebras. Let
be a homomorphism. Then,
where
is the zero element of
.
Definition 1.10 ([7])
Let be an autometrized algebra. Then,
is said to be semiregular if for any
,
.
2. Subalgebras
This section introduces the definition of subalgebra and discusses some properties of subalgebra.
Definition 2.1
Let A = be autometrized algebra. Let
. Then,
is said to be a subalgebra of
if;
is a commutative monoid.
is a subposet, and
is translation invariant, that is,
for any
.
is metric.
Remark 2.2 Any subalgebra of an autometrized algebra is an autometrized algebra.
Definition 2.3
Let be an autometrized algebra. Then, the set of all subalgebras of
is denoted by
.
Definition 2.4
Let be an autometrized algebra. Then,
is a subalgebra of
called trivial subalgebra of
and
is itself a subalgebra of
called improper subalgebra. Any other subalgebras of
are called non-trivial proper subalgebras.
Lemma 2.5
In an autometrized algebra , the intersection of any collection of subalgebras in
is again a subalgebra.
Proof.
Obvious.
Definition 2.6
Let be an autometrized algebra. Let
. Then, smallest subalgebra of
containing
is called the subalgebra generated by
and is denoted by
. That is;
.
If be a collection of subalgebra of
, then
is not a subalgebra of
.The subalgebra
generated by the set collection of subalgebra
is called the subalgebra generated by the collection of subalgebra
. It is easy to show that
is a lattice. Since if
, then
,
. Generally, if
be a family of subalgebra in
, then
Hence, is a complete lattice.
Example 2.7
Let with
and elements
are incomparable. Define
and
by the following tables.
Table
Table
So is an autometrized algebra. Let the set
and
. Clearly
are subalgebras of
. Since
and
; and hence,
is not a subalgebra of
. But
is a subalgebra of
.
Definition 2.8
Let be an autometrized algebra. Let
be a family of subalgebras of
. Then,
is said to be a directed family of subalgebras, if for any
there exists
such that
are subalgebras of
.
Theorem 2.9
Let be an autometrized algebra. The union of any directed family of subalgebras of
is again a subalgebra of
.
Proof.
Let be a directed family of subalgebras of
. Let
.
To show that
is a commutative monoid. Since
; imply that
.
Let . Suppose
and
for some
. Since
is directed; there exists
such that
are subalgebras of
. Then,
. Since each
are subalgebra;
. Since
; therefore,
.
(ii) Since
; implies
is a subposet and
is translation invariant.
(iii) Let
. Suppose
and
for some
. Since
is directed; there exists
such that
are subalgebras of
. Then,
. Since each
are subalgebra;
. Since
; therefore,
. Hence,
is a subalgebra.
Remark 2.10 Let be an autometrized algebra. Let
be a chain of subalgebras. Then,
is a subalgebra in
.
Remark 2.11 Let be an autometrized algebra. Then,
is closed under directed unions.
Theorem 2.12
Let be an autometrized algebra. Suppose
for
. For any finite subset
of
, define
= subalgebra of
generated by
. Then,
is a directed family of subalgebras of
and
.
Proof.
To show that is a directed family of subalgebras of
.
Let where
are finite subsets of
. Let
. This implies that
is a finite subset of
. Therefore,
.
Since ;
are subalgebras of
. Hence,
is a directed family of subalgebras of
. By Theorem 2.9;
is a subalgebra of
.
To show that .
Since each ; therefore,
Let . Let
. Then,
is a finite subset of
. This implies that;
and
. Therefore,
. Thus,
. So,
. Then,
By Equationequations (1)(1)
(1) and (Equation2
(2)
(2) );
.
Corollary 2.13
Every autometrized algebra can be written as a union of its directed family of subalgebras.
Proof.
From the above theorem; take to conclude that every autometrized algebra is the directed union of its finitely generated subalgebras.
Theorem 2.14
Let be an autometrized algebra. For any
, the following properties hold:
If
, then
.
Proof.
Suppose
. Since
and
; implies that
.
Clearly
.
By Theorem 2.12,
.
Theorem 2.15
Let be an autometrized algebra. Let
= Power set of
and
. Define
by
= subalgebra generated by
. Then,
is an algebraic closure operator on
.
Proof.
By (i), (ii) and (iii) of Theorem 2.14, is a closure operator on
. And again by (iv) of Theorem 2.14;
is a closure operator on
.
Remark 2.16 The set of all -closed elements is given by
Theorem 2.17
Let be an autometrized algebra. Let
= Power set of
and
. Define
by
= subalgebra generated by
is a closure operator on
. Then,
is an algebraic lattice.
Proof.
Clearly, is an algebraic closure operator on
. And
where
is a finite subset of
is a compact element. By Theorem 2.14,
implies that
is compactly generated. Therefore,
is an algebraic lattice.
Definition 2.18
Let be an autometrized algebra. Then,
is an algebraically closed set system if and only if
is closed under arbitrary intersections and unions of chains.
Theorem 2.19
Let be an autometrized algebra. Then,
is an algebraically closed set system.
Proof.
Since is closed under arbitrary intersection by lemma (2.5) and unions of chains by remark (2.11). Hence,
is an algebraically closed set system.
3. Ideals and Homomorphism Theorems
In this section, we will discuss the different types of ideals and their behavior. We also discuss some properties of a homomorphism.
Definition 3.1
A nonempty subset of an autometrized algebra
is called an ideal if and only if
imply
.
imply
.
Definition 3.2
Let be an autometrized algebra. Let
be an ideal of
.
is called maximal if and only if whenever
is an ideal such that
, then either
or
.
Theorem 3.3
Let be an autometrized algebra. Let
be an ideal of
. If
is a maximal ideal, then
is a prime ideal.
Proof.
To show that is a prime ideal. Suppose
is a maximal ideal. We know that regular ideal if and only if prime ideal.
To show that is regular ideal.
Suppose . Therefore,
. Since
is maximal ideal;
. Thus,
is regular. Hence,
is prime.
Example 3.4
Let with
. Define
and
by the following tables.
Table
Table
It is clear to show that is an autometrized algebra. Here
and
are ideals of
. Clearly
is a maximal ideal of
. But
is prime but not maximal.
Definition 3.5
Let be an autometrized algebra. Then, radical of
is the set
.
Theorem 3.6
Let be an autometrized algebra. Let
be a prime ideal of
. Let
are maximal ideals. Assume that
. Then,
for some
.
Proof.
We know that . So,
or
. Therefore,
for some
by induction on
. Since
is maximal; and hence,
for some
.
Definition 3.7
Let be an autometrized algebra. An ideal
of
is called a strong ideal if
and
for
.
Remark 3.8 Here and
are not true for any ideal of
. Consider the following examples.
Example 3.9
Let with
. Define
and
by the following tables.
Table
Table
Clearly, is an autometrized algebra and
is an ideal of
. So,
;
and
. We see that
but
. Thus,
is not a strong ideal.
Example 3.10
Let with
and elements
are incomparable. Define
and
by the following tables.
Table
Table
Clearly, is an autometrized algebra. Here
is an ideal of
, which is a strong ideal.
Definition 3.11
Let be an autometrized algebra. Let
be the set of all ideals of
. Let
. Then,
(i) the sum of and
, denoted
, is the set
(ii) the distance of and
, denoted
, is the set
Theorem 3.12
Let be an autometrized algebra and
.
If
is an ideal, then
.
If
is semiregular and
is an ideal, then
.
Proof.
Suppose that
is an ideal. We know that
. So,
. Therefore,
. Conversely, we know that
. Therefore,
. Hence,
Suppose that
is an ideal. Let
. Then,
. Since
is semiregular
. This implies that
. Similarly,
. Therefore,
. So,
. Therefore,
. Conversely, we know that
. Therefore,
. Hence,
.
Remark 3.13 Let A be a semiregular autometrized algebra. By Theorem 3.12; if and
are ideals, then
.
Definition 3.14
An autometrized algebra is called monoid if and only if
.[Associative]
.[Identity]
Then, we say that is a monoid autometrized algebra.
Example 3.15
In Example 2.7, it is easily to show that is associative and
is identity with respect to
. Therefore,
is a monoid autometrized algebra.
Theorem 3.16
Let be a monoid autometrized algebra. Then, every ideal of
is strong.
Proof.(i) Let is an ideal of
. To show that
Suppose . To show that
. Let
where
. Since
and
is monoid; we get
Therefore,
. Thus,
.
Conversely, let . So,
. We know that
. Clearly,
. Since
is associative;
. Again
is monod; implies
. As a result,
. Therefore,
. Hence,
.
Now to show . Suppose
. To show that
. Let
. This implies that
for some
. Therefore,
. Clearly,
. Since
is monoid;
. Hence,
.
(ii) To show that .
Let . Clearly,
by
. So,
for some
. Therefore,
Hence, .
Conversely, suppose . Therefore,
for some
. Therefore,
Therefore, . Hence,
by
.
Definition 3.17
Let be an autometrized algebra. A strong ideal
of
is called a distant ideal if there exists a strong ideal
of
such that
and
. The set of all distant ideals of
is denoted by
.
If is a monoid autometrized algebra, then
.
Theorem 3.18
Let be an autometrized l-algebra. Let
and
. If
is prime, then
is prime.
Proof.
Let . Suppose
. To show that either
or
. Since
. Since
is prime; either
or
. Since
; either
or
.
Definition 3.19
Let and
be autometrized algebras. Let
be a map. Then,
is said to be a homomorphism from
to
if and only if
(i)
(ii)
(iii)
A homomorphism is called
an epimorphism if and only if
is onto.
a monomorphism (embedding) if and only if
is one-to-one.
an isomorphism if and only if
is a bijection.
Definition 3.20
Let and
be autometrized algebras. Let
be a map. If
, then
is said to be an order-embedding of
into
. That is; f is both order-preserving and order-reversing.
Remark 3.21 If is an order-embedding of
into
, then
is necessary injective. Since
implies
and
and in turn
according to antisymmetry of
.
Definition 3.22
Let and
be autometrized algebras. Let
be a homomorphism. Then,
where
is the zero element of
.
Clearly, is one-to-one if and only if
.
Theorem 3.23
Let be autometrized algebras. Let
be a homomorphism. Let
be an ideal of B. Then,
is an ideal of
.
Proof.
To show that is an ideal of A.
Let . Therefore,
. Since
is an ideal;
. Then
. And thus,
.
Let and
. Therefore,
. Suppose
. Since
is a homomorphism;
and implies
. Clearly,
. Since
is ideal; implies that
. Therefore,
. Hence,
is an ideal of
.
Theorem 3.24
Let be autometrized algebras. Let
be an epimorphism and order-reversing. Let
be an ideal of
. Then,
is an ideal of B.
Proof.
To show that is an ideal of A.
Let . Therefore, there exists
such that
. Clearly,
. Since
is ideal;
. Now consider
. Then,
. And thus,
.
Let and
. Since
is an onto; there exists
such that
. Clearly,
. Suppose
. Therefore,
. Since
is homomorphism;
. Since
is an order-reversing;
. Since
is ideal; implies that
. Therefore,
. Hence,
is an ideal of
.
Theorem 3.25
Let and
be autometrized algebras. Let
and
. If
and
are homomorphisms and
. Then,
.
Proof.
Let . Clearly,
.
To show that is a subalgebra of
.
Let . Therefore,
and
. Consider
Therefore, .
Since ; clearly
is a subposet and
is translation invariant.
Let . Therefore,
and
. Consider
Therefore, .
Therefore, is a subalgebra of
. Since
;
. So,
. Therefore,
. As result,
. Hence,
.
Theorem 3.26
Let A, B and C be autometrized algebras. Let f : A → B and g : B → C be homomorphisms. Then, g ◦ f : A → C is also a homomorphism.
Proof.
Let , .
(i) Consider
g f (a + b) = g(f (a + b)).
= g(f (a) + f (b)).[Since f is a homomorphism]
= g(f (a)) + g(f (b)).[Since g is a homomorphism]
= g f (a) + g
f (b).
(ii) Consider
g f (a
b) = g(f (a
b)).
= g(f (a) f (b)). [Since
is a homomorphism]; = g(f (a))
g(f (b)). [Since
is a homomorphism];
.
(iii) Suppose Then,
Hence, .Therefore, g
f is a homomorphism.
Theorem 3.27
Let and
be autometrized algebras and
be a homomorphism. Then,
is a subalgebra of
is a subalgebra of
.
Proof.
Suppose is a subalgebra of
. To show that
is a subalgebra of B. Let
. Therefore,
and
for
.
(i) Since f is a homomorphism; a + b = f (s) + f (t) = f (s + t). Since S is a subalgebra; Thus,
(ii) Since
; implies
is a subposet and
is translation invariant.
(iii) Since is a homomorphism;
. Since
is a subalgebra;
. Thus,
.
Hence, is a subalgebra of
.
Theorem 3.28
Let and
be autometrized algebras and
be a homomorphism. Then,
is a subalgebra of
is a subalgebra of
.
Proof.
Suppose is a subalgebra of
. To show that
is a subalgebra of
. Let
. Therefore,
.
Since
is a subalgebra of
;
. Therefore,
.
Since
; implies
subposet and
is translation invariant.
Since
is a subalgebra of
;
. Therefore,
.
Hence, is a subalgebra of
.
Theorem 3.29 Let and
be autometrized algebras. Let
be a homomorphism. Then,
for any
.
Proof.
Since ; implies that
. Again since
is a subalgebra of A and by Theorem 3.27
is a subalgebra; implies
.
Conversely, let . This implies that
. Clearly,
and
. Therefore,
implies that
. Hence,
. Thus,
.
Definition 3.30
Let be autometrized algebras. Let
and
be homomorphisms. Then,
is said to be an extension homomorphism of g(denoted by
) if
is subalgebra of
.
Definition 3.31
Let be autometrized algebras. Let
be homomorphisms. Then,
is said to be a directed family of homomorphism if for any
there exists
which is a common extension of
. That is,
and
.
Theorem 3.32
Let be an autometrized algebra. Any directed family of homomorphisms that maps into
(
- valued homomorphisms) has a common extension.
Proof.
Let be homomorphism. Suppose
is a directed family.
To show that is a directed family of autometrized algebras.
Let . We have
,
are homomorphisms. Therefore, there exists
such that
and
. So,
are subalgebras of
and
. Whence,
is a directed family. By Theorem 2.9;
is an autometrized algebra.
Define a map by
. Clearly,
is well defined.
To show that is a homomorphism.
Let . Say
and
. Then, there exists
such that
. Therefore,
.
(i) Consider
(ii) Consider
(iii) Suppose . Since
is a homomorphism;
. Since
; implies that
.
Hence, is a homomorphism. Since
;
is a common extension homomorphism of
.
Definition 3.33
Let and
be autometrized l-algebras. Let
be a map. Then,
is said to be a homomorphism from
to
if and only if
, and
Theorem 3.34
Let be autometrized l-algebras. Let
be a homomorphism. Let
be a prime ideal of
. Then,
is a prime ideal of
.
Proof.
By Theorem 3.23; is an ideal. To show that
is prime. Let
. Suppose
.
To show that either or
.
Since and
is a homomorphism;
and implies that
. Since
is an ideal;
. Also
is a prime ideal; either
or
. Therefore, either
or
. Hence,
is a prime ideal of
.
Theorem 3.35
Let be autometrized l-algebras. Let
be an epimorphism and order-reversing. Let
be a prime ideal of
. Then,
is a prime ideal of
.
Proof.
By Theorem 3.24; is an ideal. To show that
is prime.
Let . Since
is an onto; there exists
such that
. Suppose
. Therefore,
.
To show that either or
.
Since and
is a homomorphism;
. Since
is an ideal; implies that
. Therefore,
. Since
is prime; implies that either
or
. Thus, either
or
. Hence,
is a prime ideal of
.
4. Quotient Algebra
In this section, we introduce quotient algebra and discuss isomorphism theorems.
Theorem 4.1
Let be an autometrized algebra. Let
be an ideal of
. Let
. For any
, define the operations:
Then, is an autometrized algebra is called the quotient algebra of
by ideal
.
Proof.
To claim that is an autometrized algebra.
(a) To show that is a commutative semigroup with additive identity
.
Let . Therefore,
.
:
. Since
; implies that
.
Commutative:
.
:
.
Therefore, is identity for
.
(b) To show that is a poset and translation invariant.
: Let
. Clearly
. Since
; implies
.
: Let
. Therefore,
. Suppose
and
. Therefore,
and
. Since
is an autometrized algebra;
. Hence,
.
: Let
. There-fore,
.
Suppose and
. Therefore,
and
. Since
is an autometrized algebra;
. It follows that
. Therefore,
is a poset.
Suppose . Then,
Therefore, is translation invariant.
Hence, is a partially ordered set such that the semigroup translations are invariant under inclusion.
(c) To show that is a metric.
(M1) Let . Therefore,
. Consider
. We know that;
. Therefore,
.
Suppose . So,
. Therefore,
.
Conversely, suppose
We proved that and
.
(M3) Let . Therefore,
. Consider
Therefore, is metric on
. And thus
is an autometrized algebra.
Example 4.2
We saw in Example 2.7 that is an autometrized algebra. Now consider an ideal
in
. Clearly,
and
. We can easily verify that
satisfies
of Theorem 4.1. Hence,
is an autometrized algebra.
Theorem 4.3
Let be an autometrized algebra. Let
be a strong ideal of
. Define a map
by
. Then,
is an epimorphism and
.
Proof.
Clearly, is an onto map.
To show that is a homomorphism.
Let . Consider
(i)
(ii)
(iii) Suppose . Therefore,
. Hence,
.
Thus, is a homomorphism.
Now, we shall prove that .
Therefore, is an epimorphism and
.
Theorem 4.4 (First Isomorphism Theorem)
Let be a monoid autometrized algebra. Let
be an autometrized algebra. Let
be a homomorphism. Then,
.
In particular, if is onto; then,
.
Proof.
Let . By a known theorem,
is a strong ideal of
. Therefore,
is an autometrized algebra.
Define a map by
.
Now we need to show if is well-defined. Let
. Suppose
.
Therefore, is well defined.
Let . Clearly, there exists
such that
. Now
implies that
. Therefore,
. Hence,
is an onto map.
To show that is a homomorphism.
Let . Consider
(i)
(ii)
(iii) Suppose . Since
is homomorphism;
. Therefore,
.
Thus, is a homomorphism.
Now, we shall prove that .
Therefore, is one-to-one. Hence,
. If
is onto; then
.
Theorem 4.5
Let be a semiregular autometrized algebra. Let
be an ideal of
. Let
be a subalgebra of
. Define
for
and
such that for any
;
and
. Then,
is a subalgebra of
.
Proof.
To show that is a subalgebra of A.
(i) To show that is a commutative monoid. Since
and
; imply that
.
Let where
and
. Therefore,
and
. Then,
by definition.
(ii) Clearly
is a subposet, and
is translation invariant.
(iii) Let
where
and
. Therefore,
by definition. Since
and
. Hence,
is a subalgebra of
.
Theorem 4.6
Let be autometrized algebra. Let
be an ideal of A. Let
be a subalgebra of
. Then,
is ideal of
.
Proof.
To show that is an ideal of
.
Let
. Therefore,
and
. Clearly,
and
. Hence,
.
Let
and
. Suppose
. So,
and
. Since
is an ideal; implies that
. Therefore,
. Hence,
is an ideal.
Remark 4.7 Let be a semiregular autometrized algebra. Let
be an ideal of A. Let
be a subalgebra of
. Clearly,
is an ideal of
.
Theorem 4.8 (Second Isomorphism Theorem)
Let be a monoid autometrized algebra. Let
be an ideal of
. Let
be a subalgebra of
. Then,
.
Proof.
Clearly, and
are strong ideals.
Define a map by
. It is obvious that;
is well defined and onto map.
To show that is a homomorphism. Let
. Consider
(i)
(ii)
(iii) Suppose .
Thus, is a homomorphism.
Now, we shall prove that .
Therefore, by first isomorphism theorem: . Hence,
.
Theorem 4.9
Let be autometrized algebra. Let
be ideas of
. Suppose
. Then,
is ideal.
Proof.
To show that is ideal.
Let
. Therefore,
. Therefore,
. Hence,
.
Let
and
. Therefore,
.
Suppose that . Therefore,
. This implies that
. Since
is an ideal and
; imply that
. Hence,
. Thus,
is ideal.
Theorem 4.10 (Third Isomorphism Theorem)
Let be a monoid autometrized algebra. Let
be ideas of
. Suppose
. Then,
.
Proof.
Clearly, and
are strong ideals. Define a map
by
.
is well defined because
;
Therefore, is well defined.
To show that is onto. It is clear that
for any
. This implies that
such that
. Hence,
is onto map.
To show that is a homomorphism.
Let . Therefore,
. Consider
(i)
(ii)
(iii) Suppose . Therefore,
. As a result
. Thus,
is a homomorphism.
Now, we shall prove that .
Therefore, by first isomorphism theorem: . Hence,
.
Theorem 4.11 (Correspondence Theorem)
Let be autometrized algebra. Let
be an idea of
.
Let = set of all ideals of
containing
.
Let = set of all ideals of
.
Define by
. Then,
is one-to-one and onto. That is;
and
are in one-to-one correspondence.
Proof.
To show that is well defined.
Let . That is;
are ideals of A and
.
Suppose . Therefore,
. Thus,
. Hence,
is well defined.
To show that is one-to-one.
Let . Suppose
. Therefore,
. Let
. Therefore,
. This implies that
for some
. Since
and
; imply that
. Since
;
. Therefore,
. Similarly,
. Therefore,
. Hence,
is one-to-one.
To show that is onto.
Let . Therefore,
is an ideal of
. Consider the natural homomorphism:
defined by
. Therefore, by Theorem 3.23,
is an ideal of
containing
. Thus,
and
.
Let . This implies that
. So,
. Therefore,
. Hence,
. Conversely, let
. So,
. This implies that
. Therefore,
. Thus,
. Therefore,
. This shows that
is onto.
Hence, and
are in one-to-one correspondence.
Example 4.12
We saw in Example 3.4 that is an autometrized algebra. Also,
and
are ideals of
. Clearly,
is contained in the ideals
and
. Therefore,
and
We also see that
,
and
. Thus,
and
are in one-to-one correspondence.
Remark 4.13 Let be an autometrized l-algebra. Let
be an ideal of
. Let
. For any
, define the operations:
Then, is an autometrized l-algebra is called the quotient algebra of
by ideal
.
5. Conclusion
This paper introduced the concept of subalgebra and discussed some properties of subalgebra. We also examined different types of ideals and how they behaved. We also described some properties of a homomorphism. Finally, we presented quotient algebra and discussed isomorphism theorems.
Acknowledgements
The authors are very grateful to the anonymous reviewers for their helpful comments and suggestions that helped for improving the paper’s quality.
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Funding
Notes on contributors
Gebrie Yeshiwas Tilahun
Gebrie Yeshiwas Tilahun is a PhD scholar at the Department of Mathematics, College of Natural Sciences, Arba Minch University, Ethiopia. His research interest includes the development of an autometrized algebra.
References
- Blumenthal, L. M. (1952). Boolean geometry. I. Bulletin of the American Mathematical Society, 58(4), 501.
- Chajda, I., & Rachunek, J. (2001). Annihilators in normal autometrized algebras. Czechoslovak Mathematical Journal, 51(1), 111–14. https://doi.org/10.1023/A:1013757805727
- Ellis, D. (1951). Autometrized boolean algebras I: Fundamental distance-theoretic properties of B. Canadian Journal of Mathematics, 3, 87–93. https://doi.org/10.4153/CJM-1951-011-x
- Hansen, M. E. (1994). Minimal prime ideals in autometrized algebras. Czechoslovak Mathematical Journal, 44(1), 81–90. https://doi.org/10.21136/CMJ.1994.128442
- Kovář, T. (2000). Normal autometrized lattice ordered algebras. Mathematica slovaca, 50(4), 369–376.
- McKenzie, R. N., McNulty, G. F., & Taylor, W. F. (2018). Algebras, lattices, varieties 1em Plus 0.5em Minus 0.4em. American Mathematical Soc, 383.
- Nordhaus, E., & Lapidus, L. (1954). Brouwerian geometry. Canadian Journal of Mathematics, 6, 217–229. https://doi.org/10.4153/CJM-1954-023-7
- Rachŭnek, J. (1987). Prime ideals in autometrized algebras. Czechoslovak Mathematical Journal, 37(1), 65–69. https://doi.org/10.21136/CMJ.1987.102135
- Rachŭnek, J. (1989). Polars in autometrized algebras. Czechoslovak Mathematical Journal, 39(4), 681–685. https://doi.org/10.21136/CMJ.1989.102344
- Rachŭnek, J. (1990). Regular ideals in autometrized algebras. Mathematica slovaca, 40(2), 117–122.
- Rachŭnek, J. (1998). Spectra of autometrized lattice algebras. Mathematica Bohemica, 123(1), 87–94. https://doi.org/10.21136/MB.1998.126293
- Rao, B., Kanakam, A., & Yedlapalli, P. (2019). A note on representable autometrized algebras. Thai Journal of Mathematics, 17(1), 277–281.
- Rao, B., Kanakam, A., & Yedlapalli, P. (2021). Representable autometrized semialgebra. Thai Journal of Mathematics, 19(4), 1267–1272.
- Roy, K. R. (1960). Newmannian geometry I. Bullutin Calcutta Math Society, 52, 187–194.
- Subba Rao, B., & Yedlapalli, P. (2018). Metric spaces with distances in representable autometrized algebras. Southeast Asian Bulletin of Mathematics, 42(3), 453–462.
- Swamy, K. L. N. (1964a). Autometrized lattice ordered groups I. Mathematische Annalen, 154(5), 406–412. https://doi.org/10.1007/BF01375523
- Swamy, K. L. N. (1964b). A general theory of autometrized algebras. Mathematische Annalen, 157(1), 65–74. https://doi.org/10.1007/BF01362667
- Swamy, K. L. N., & Rao, N. P. (1977). Ideals in autometrized algebras. Journal of the Australian Mathematical Society, 24(3), 362–374. https://doi.org/10.1017/S1446788700020383