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There has been some recent interest in applying topological structure on BCH algebras. In this paper Lee and Ryu initiates the study of topological BCK algebras. Motivated by this, in this paper we define the notion of topological d-algebras, give some examples and prove some important theorems

Nagamani, N

Kandaraj, N

Journal of Physical Sciences, Vol. 24, 2019, 9-19 
ISSN: 2350-0352 (print), www.vidyasagar.ac.in/publication/journal 
Published on 6 June 2019 
 
9 
 
Topological Structures on d-Algebras 
N Nagamani1 and N. Kandaraj2 
1PG and Research Department of Mathematics 
Saiva Bhanu Kshatriya College Aruppukottai -626 101, Tamil Nadu, India   
Email: nagamanin28@gmail.com 
2PG and Research Department of Mathematics 
Saiva Bhanu Kshatriya College, Aruppukottai- 626 101, Tamil Nadu, India 
email: n.kandarajsbkc1998@gmail.com 
Received 12 December 2018; accepted 30 January 2019 
ABSTRACT 
There has been some recent interest in applying topological structure on BCH algebras. 
In this paper Lee and Ryu initiates the study of topol gical BCK algebras. Motivated by 
this, in this paper we define the notion of topological d-algebras, give some examples and 
prove some important theorems. 
 
Keywords: d-algebra, topological d-algebra, topological d-sub algebra, topological 
d-ideal. 
 
Mathematical Subject Classification (2010): 06F35, 03G25 
 
1. Introduction 
Imai and Iseki introduced two classes of abstract algebras BCK algebras and 
BCI-algebras [5, 6, 7, 8]. It is known that the class of BCK-algebras is a proper subclass 
of the class of BCI-algebras. In [3, 4] Hu and Li introduced a wider class of abstract  
algebras BCH algebras. They have shown that the class of BCI algebras is a proper 
subclass of the class of BCH-algebras. Neggers and Kim introduced d algebras [12]. Dar 
introduced the notions of left and right mapping on BCK-algebras in [1]. Further in [10] 
the authors discussed notions of endomorphism’s on BCH-algebras. Lots of paper have 
been published in related algebraic structure [7-19]. Left and Right mapping over 
BCI-algebras have been discussed in [2]. In [9] the author studied topological structure 
on BCH algebras and proved some theorems that determine the relationship between 
them. There has been some recent interest in applying topological notions to non 
mainstream algebras. Motivated by this, in this paper, we study the connection between 
topology and d-algebras. 
 
2. Preliminaries 
Definition 2.1. [1, 3, 4] A -algebra (,∗ ,0) is a non empty set  with a constant 
 and a binary operation satisfying the following conditions. 
1. 
 ∗ 
 = 0 
2. (
 ∗ ) ∗  = (
 ∗ ) ∗  
N Nagamani and N. Kandaraj 10 
 
3. 
 ∗  = 0   ∗ 
 = 0⇒ 
 =  ∀ 
, ,  ∈ . 
Definition 2.2. [8] Let  be a set with a binary operation  and a constant 0. Then 
(,∗ ,0) is called a -algebras if it satisfies the following axioms. 
1. (
 ∗ ) ∗ (
 ∗ ) ∗ ( ∗ ) = 0 
2. 
 ∗ (
 ∗ ) ∗  = 0 
3. 
 ∗ 
 = 0 
4. 
 ∗  =  and  ∗ 
 = 0⇒ 
 =  ∀ 
,  ∈ . 
Definition 2.3. [2, 7] Let X be a set with a binary operation  and a constant 0. Then 
(,∗ ,0) is called a -algebras if it satisfies the following axioms. 
1. (
 ∗ ) ∗ (
 ∗ ) ∗ ( ∗ ) = 0 
2. 
 ∗ (
 ∗ ) ∗  = 0 
3. 
 ∗ 
 = 0 
4. 
 ∗  = 0 and  ∗ 
 = 0 ⇒ 
 =  ∀ 
, ,  ∈  
5. 0 ∗ 
 = 0. 
Definition 2.4. [12] A -algebra is a non empty set  with a constant 0 and binary 
operation  and satisfying the following axioms. 
1. 
 ∗ 
 = 0 
2. 0 ∗ 
 = 0 
3. 
 ∗  = 0 and  ∗ 
 = 0 ⇒ 
 =  ∀ 
,  ∈ . 
Example 2.5. Let ( = 0,1,2,3,∗ ,0) be a set with the following cayley table.  
 
 
 
 
 
 
 
Then (,∗ ,0) is a -algebra. 
 
Definition 2.6. [11] A topology, on a -algebra  is -algebra topology and  
furnished with, is a topological -algebras if (
, )→ 
 ∗  is continuous from 
 ×  with its Cartesian product topology to  with the topology. In this case abbreviate 
we call  a -algebra.  
Definition 2.7. [9] Let (,∗) be a -algebra and   a topology on . Then  =
(,∗,  )  is called a topological  -algebra, if the operation  is continuous, or 
equivalently, for any 
,  ∈  and for any 
,  ∈  and for any open set ! of 
 ∗  
there exist two open sets " and # respectively such that " ∗ # is a subset of !.   
*  0 1 2 3 
0 0 0 0 0 
1 1 0 1 0 
2 2 2 0 0 
3 3 3 1 0 
Topological Structures on d-Algebras 
11 
 
Definition 2.8. [1] Let  be a -algebra. If  is satisfies the condition, 
 (
 ∗ ) ∗  = (
 ∗ ) ∗ ( ∗ ), ∀ 
, ,  ∈ . Then  is called positive implicative 
-algebra. 
 
Definition 2.9. [10] Let (,∗)  be a  -algebra, and  ∈  . Define a left map 
$%:  →  by, $%(
) =  ∗ 
 ∀ 
 ∈ . 
Let (,∗)  be a  -algebra, and  ∈  . Define a right map (%:  →   by, 
(%(
) = 
 ∗  ∀ 
 ∈ . 
Let (,∗) be a -algebra. The set of all left maps on  is defined as, $().     
 
3. Topological structures on d-algebras 
Definition 3.1. Let  be a -algebra and ", # are any non empty subsets of X. We 
define a subset ) ∗ * of   as follows. " ∗ # = 
 ∗ | 
 ∈ ",  ∈ # 
Example 3.2.  Let  = (0, , ,, -,∗ ,0)  be a -algebra the binary operation  will 
be defined as follows. 
   
.  
 
 
 
 
 
Let " = 0,   and # = , , be two subsets of .              
" ∗ # = 0 ∗ , 0 ∗ ,,  ∗ ,  ∗ ,                                                         
           = 0,0,0, ,  
           = 0, ,                                                                                                                     
subsets of . 
Definition 3.3. Let (,∗) be a -algebra. Let  be the collection of subsets of .   is 
said to be a topology on . If  
1. , ∅ ∈  .  
2. arbitrary union of members of   is in  .  
3. finite intersection of members of   is in  . 
Definition 3.4. Let (,∗) be a  -algebra and  a topology on . Then  = (,∗
,0,  ) is called a topological 0-algebra, (it is denoted by 10-algebra) if the operation 
" ∗ " is continuous or equivalently for any 
,  ∈  and for any open set ! of 
 ∗  
there exist two open sets " and # respectively such that " ∗ # is a subset of !. 
Example 3.5. Let ( = 0,1,2,∗ ,0) be a -algebra the binary operation  will be 
* 0 a b c 
0 0 0 0 0 
a a 0 b b 
b b b 0 0 
c c c a 0 
N Nagamani and N. Kandaraj 
12 
 
defined as follows: 
 
 
 
 
 
 
Define a topology = ∅, 0, 1, 0,1, . Let 
 = 2,0 and  = 1,2 then 
 
 ∗  = 2 ∗ 1,2 ∗ 0,0 ∗ 1,0 ∗ 2 = 1,0,0,0 = 0,1 = ! . Put " = 0  and 
# = 1, 
 " ∗ # = 0 ∗ 1 = 0⊆ !.  Therefore 0⊆ 0,1.  Hence (,∗ ,0,  )  is a 
-algebra. 
Example 3.6. Let ( = 0,1,2,∗ ,0) be a -algebra the binary operation  will be 
defined as Define a topology  = ∅, 0, 1, 0,1, .  
 
 
 
 
 
 
 
Let 
 = 2 and  = 1 then 
 ∗  = 2 ∗ 1 = 1 = !.  
Put " = 0 and # = 1, " ∗ # = 0 ∗ 1 = 0 ⊄ !.  
Therefore (,∗ ,0,  ) is  not a -algebra. 
Remark 3.7. The member of   are called an open set in . The complement of 3 ∈  , 
that is \3 is called a closed set in , denoted by 35. The open set of an element 
 ∈ , 
is a member of   containing .  
Definition 3.8. Let  and 6 be  a -algebra. A map 7:  → 6 is continuous, if for 
every 
 ∈  and any open set ! of 7(
), there exist an open set # of 
 such that 
7(#) subset of !. 
Example 3.9. Let ( = 0,1,2,3,∗ ,0,  )  and (6 = 0,1,2,∗ ,0,  )  are the 
-algebra the binary operation  will be defined an  and 6 are as follows respectively. 
The topology  8 = ∅, 1,0, 2,3,   is on   and the topology 
 9 = ∅, 2, 1,0,  is on 6. 
* 0 1 2 
0 0 0 0 
1 2 0 2 
2 1 1 0 
* 0 1 2 
0 0 0 0 
1 2 0 2 
2 1 1 0 
Topological Structures on d-Algebras 
13 
 
 
 
 
 
 
 
The map 7:  → 6 is defined by,  7(
) = 1, for all 
 ∈  is continuous. 
Definition 3.10. Let  be a -algebra, and 
 ∈ . Define a left map  $%:  →   
by  $%(
) =  ∗ 
, ∀   ∈ .  
 
Example 3.11. Let ( = 0,1,2,∗ ,0,  ) be a -algebra the binary operation  will 
be defined as follows, 
 
 
 
 
 
 
 = ∅, 0,1, 2,3,  is a topology on . For 
 ∈ , the map  $9:  → 6 is a left 
map defined by, $9() = 2 ∗ . 
Theorem 3.12. Let   be a -algebra. Every left map on   is continuous. 
 Proof. Let 
 ∈ . Define a left map $::  →  by, $%() = 
 ∗  ∀  ∈ . Since 
 is - algebra, for every
,  ∈  and any open set ! of 
 ∗ , there exist a two 
open sets " and # of 
 and  respectively, such that " ∗ # is a subset of !.  
Clearly, 
 ∗ #⊆ " ∗ #. $:(#) subset of !. Since 
 is arbitrary, $: is continuous. 
Definition 3.13. Let  be a -algebra. If  satisfies the condition, 
 (
 ∗ ) ∗  = (
 ∗ ) ∗ ( ∗ ) , for all 
, ,  ∈  . Then   is called positive 
implicative 10-algebra.  
Definition 3.14. Let  be a positive implicative -algebra and 3 be any subset of . 
Define $; = $;| ∈ 3. 
Definition 3.15. Let  be a -algebra, define a map <:  → $() by <(
) = $: . 
Remark 3.16.  Let  be a positive implicative -algebra, and 3,  be any subset of 
 . If 3⊆   then <(3) = <(). Let   be a positive implicative  -algebra and 
=8, =9 any subset of . If = = =8 ∪ =9 then <(=) = <(=8) ∪ <(=9) = <(=8 ∪
=9).  
* 0 1 2 3 
0 0 0 0 0 
1 1 0 3 2 
2 2 3 0 1 
3 3 2 1 0 
* 0 1 2 
0 0 0 0 
1 1 0 2 
2 2 2 0 
* 0 1 2 
0 0 0 0 
1 2 0 2 
2 1 1 0 
N Nagamani and N. Kandaraj 
14 
 
Theorem 3.17. The map < is a  -isomorphism. 
Proof. Clearly, < is 1-1 and onto. Consider, <(
 ∗ ) = $(:∗?) imply 
 $:∗?(@) = (
 ∗ ) ∗ @ = (
 ∗ @) ∗ ( ∗ @) = <(
) ∗ <(), ∀ 
,  ∈ .  
Hence < is a -isomorphism. 
 
Definition 3.18. Let (,∗ ,0,  ) be a positive implicative -algebra, and the collection 
of subsets of $(),  A = $B⊆ $(
)| = ∈  } is a called a C(D)-topology on the set 
$(). 
Example 3.19. Let ( = {0,1,2,3},∗ ,0,  ) be a positive implicative -algebra the 
binary operation  will be defined as follows.  
 
 
 
 
 
 
 
 
 = {∅, {2}, {3}, {0,1}, {2,3}, {0,1,3}, {0,1,2}, } is a topology on , Now, 
 $() = {$E, $8, $9, $F} and the $G-topology.  
 A = {∅, <({2}), <({3}), <({0,1}), <({2,3}), <({0,1,3}), <({0,1,2}), $()} 
Theorem 3.20. Let (,∗ ,0,  )  be a positive implicative  -algebra, then A =
{<(=)|= ∈  } is a topology on $(). 
Proof. It is trivial that, $()  and < ∈  A  .Let {<(=H)} ∈  A , implies that {=H} ∈
 .Then "H∈I=H ∈  , "H∈I<(=H) ∈  A , similarly, we can prove a finite intersection of 
element of  A is in  A. 
 
Definition 3.21. Let G be a non empty subset of a -algebra , then G is called  
10-subalgebra of  if 
 ∗  ∈ G ∀ 
,  ∈ G. 
 
Example 3.22. Let  = {0,1,2,3} be a -algebra in which the operation  is defined 
as follows: 
 
 
 
 
 
* 0 1 2 3 
0 0 0 0 0 
1 1 0 0 1 
2 2 2 0 2 
3 3 3 3 0 
Topological Structures on d-Algebras 
15 
 
 
  
 
 
 
 
 
 
The topology = {∅, {0}, {1}, {2}, {1,2}, {1,0}, {2,0}, {1,2,0}, }.  
Let 
 = {1,2},  = {2,3},  
,  ∈ ,  

 ∗  = {(1 ∗ 2), (1 ∗ 3), (2 ∗ 2), (2 ∗ 3)} 
        = {1,0,0,0}  
            = {1,0}  
            = !,  
Let " = {1,0} and # = {2},  
clearly, " ∗ # = {(1 ∗ 2), (0 ∗ 2)} 
                = {1,0}⊆ !  
Therefore (,∗ ,0,  ) is a -algebra. 
In   the sets G8 = {0,1,2},  G9 = {0,1}, GF = {0,1,3} GJ = {0,2}  are 
-subalgebra of , while G = {0,2,3} is not a -subalgebra of . 
Definition 3.23. Let   be a -algebra and   be a subset of  , then   is called a 
10-ideal of . If it satisfies the following conditions. 
1.0 ∈  
2.
 ∗  ∈  and  ∈ ⇒ 
 ∈  
3.
 ∈  and  ∈ ⇒ 
 ∗  ∈  (K. L)  ∗ 
 ⊆  
Example 3.24. Let  = {0,1,2,3} be a -algebra in which the operation  is defined 
as follows. 
 
 
 
 
 
 
 
 
In  the sets 8 = {0,1} and 9 = {0,2} are -ideals of . While J = {1,2} are 
* 0 1 2 3 
0 0 0 0 0 
1 1 0 1 0 
2 2 2 0 0 
3 3 3 1 0 
* 0 1 2 3 
0 0 0 0 0 
1 1 0 1 0 
2 2 2 0 0 
3 3 3 1 0 
N Nagamani and N. Kandaraj 
16 
 
not -ideals of  = {0,1,2,3}. 
Remark 3.25. Let (,∗ ,0,  )  be a  -algebra. Every  -ideal of   is a  - 
subalgebra of  but the converse need not be true. For the example, Let  = {0, , ,, -} 
is a -algebra the binary operation  will be defined as follows,   
 
  
 
 
 
 
 
 = {0, } is a -subalgebra but not a -ideal of  for  ∗ - = ,∉ . 
Definition 3.26. Let (,∗ ,0,  ) be a -algebra define a binary relation on  by 
taking 
 ≤  if and only if 
 ∗  = 0. In this case (, ≤) is a partially ordered set. 
Definition 3.27. Let (,∗ ,0,  )  be a  -algebra and 
 ∈   define 
 ∗  = {
 ∗
| ∈ }.  is said to be edge 10-algebra if for any 
 ∈ , 
 ∗  = {
, 0}. 
 
Example 3.28. Let  = {0,1,2} be a set with the following cayley table.  
 
 
 
 
 
 
(X, *) is a edge -algebra. 
Lemma 3.29. Let (,∗ ,0,  ) be an edge -algebra, then 
 ∗ 0 = 
 for any 
 ∈ . 
Proof. Since (,∗ ,0,  )  is an edge  -algebra, either 
 ∗ 0 = 
 N 
 ∗ 0 = 0 for 
any 
 ∈ . Let  ≠ 0 and 
 ∗ 0 = 0. By condition two of the -algebra 0 ∗ 
 = 0. 
Thus we have 
 ∗ 0 = 0 and 0 ∗ 
 = 0. Hence by the condition three of the -algebra 

 = 0 a contradiction to the fact that 
 ≠ 0. Hence we have 
 ∗ 0 = 
 ∀ 
,  ∈ . 
Proposition 3.30. If (,∗ ,0,  ) is an edge -algebra, then the condition 
 
 ∗ (
 ∗ ) ∗  = 0 ∀ 
,  ∈  holds. 
Proof. If 
 = 0 then 
 ∗ (
 ∗ ) ∗  = 0 (since (0 ∗ 
 = 0)). Let 
 ≠ 0, 
 ∗  ∈ .  
Since 
 ∗  = 
 N 0, Now 
 ∗  ≠ 0, 
 ∗  = 
. 
 ∗ (
 ∗ ) ∗  = (
 ∗ 
) ∗  = 0.  
 
* 0 a b c 
0 0 0 0 0 
a a 0 0 b 
b b c 0 0 
c c c c 0 
* 0 1 2 
0 0 0 0 
1 1 0 1 
2 2 2 0 
Topological Structures on d-Algebras 
17 
 
Definition 3.31. A -algebra  is said to be commutative if 
∧ = ∧
 for all 

,  ∈ . 
Example 3.32. Let  = {0,1,2} be a -algebra which the operation  is defined as 
follows, 
 
 
 
 
 
 
 
For 1∧2 = 2 ∗ (2 ∗ 1) = 2 ∗ 1 = 1    →  (1) 
       2∧1 = 1 ∗ (1 ∗ 2) = 1 ∗ 0 = 1    →  (2)  
From (1) and (2), ( ,*)  is a commutative -algebra.  
Example 3.33.  Let  = {0,1,2} be a -algebra which the operation  is defined as 
follows, 
 
 
 
 
 
 
 
 
( ,*) is a commutative -algebra. 
 
Example 3.34. Let  = {0,1,2,3} be a -algebra which the operation  is defined as 
follows,  
 
 
 
 
 
 
 
 
 
For 1∧2 = 2 ∗ (2 ∗ 1) = 2 ∗ 2 = 0    →  (1)                                                                              
* 0 1 2 
0 0 0 0 
1 1 0 0 
2 2 1 0 
* 0 1 2 
0 0 0 0 
1 1 0 1 
2 2 2 0 
*  0 1 2 3 
0 0 0 0 0 
1 1 0 0 1 
2 2 2 0 0 
3 3 3 3 0 
N Nagamani and N. Kandaraj 
18 
 
      ∧1 = 1 ∗ (1 ∗ 2) = 1 ∗ 0 = 1     →  (2) 
From (1) ≠ (2). 
( ,*) is not a commutative -algebra. 
Definition 3.35. A -algebra (,∗ ,0,  ) is said to be 10-transitive if 
 ∗  = 0 
and  ∗  = 0 ⇒ 
 ∗  = 0.  
Example 3.36. Consider the following -algebra  with the table. 
 
 
 
 
 
 
 
We can easily see that 1 ∗ 2 = 0, 2 ∗ 3 = 0 but 1 ∗ 3 = 0 and hence (,∗ ,0,  ) is 
a -transitive. 
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1 1 0 0 0 
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