Some Aspects of -Units in BCK-Algebras
We explore properties of the set of d-units of a -algebra. A property of interest in the study of -units in -algebras is the weak associative property. It is noted that many other -algebras, especially -algebras, are in fact weakly associative. The existence of -algebras which are not weakly associative is demonstrated. Moreover, the notions of a -integral domain and a left-injectivity are discussed.
Iséki and Tanaka introduced two classes of abstract algebras: BCK-algebras and BCI-algebras [1, 2]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. Neggers and Kim introduced the notion of -algebras which is another useful generalization of BCK-algebras and then investigated several relations between -algebras and BCK-algebras as well as several other relations between -algebras and oriented digraphs . After that some further aspects were studied [4–7]. As a generalization of BCK-algebras, -algebras are obtained by deleting two identities. Thus, one may introduce an additional operation and replace one of the deleted new algebras, for example, , to obtain new algebras for which the conditions (i) and (ii) implies , yielding a companion -algebra which shares many properties of BCK-algebras and such that not every -algebra is one. Allen et al.  developed a theory of companion -algebras in sufficient detail to demonstrate considerable parallelism with the theory of BCK-algebras as well as obtaining a collection of results of a novel type. Recently, Allen et al.  introduced the notion of deformation in -algebras. Using such deformations they constructed -algebras from BCK-algebras in such a manner as to maintain control over properties of the deformed BCK-algebras via the nature of the deformation employed and observed that certain BCK-algebras cannot be deformed at all, leading to the notion of a rigid -algebra and consequently of a rigid BCK-algebra as well.
In this paper we study properties of -units in -algebras , that is, elements of such that . Since , implies 0 is not a -unit of . Hence, -algebras such that every non-zero element is a -unit are special in the sense that they are “complete" with respect to this property. They are also not uncommon (see Proposition 3.2). It turns out that the property of weakly associativity in -algebras is an important property in this context. In addition, we consider the class of -integral domains and left-injective elements of -algebras (defined below) in analogy with the usual notions in the theory of rings and their modules, where again the -units investigated in this paper also play a significant role.
An (ordinary) -algebra  is an algebra where is a binary operation and such that the following axioms are satisfied: (I), (II), (III) and imply for all .
For brevity we also call a -algebra. In we can define a binary relation “” by if and only if .
A BCK-algebra  is a -algebra satisfying the following additional axioms: (IV) , (V) for all .
Example 2.1 (see ). (a) Every BCK-algebra is an ordinary -algebra. (b) Let be the set of all real numbers and define , , where and are the ordinary product and subtraction of real numbers. Then . If , then and , that is, ( or ) and ( or ), that is, ( and ) or ( and ) or ( and ) or ( and ); all imply . Hence, is an ordinary -algebra. But it is not a BCK-algebra, since axiom (V) fails: .
An algebra , where is a binary operation and , is said to be a strong -algebra  if it satisfies (I), (II) and () for all , where implies .Obviously, every strong -algebra is a -algebra, but the converse needs not be true. The -algebra in Example 2.1 (b) is not a strong -algebra, since implies either or .
Example 2.2 (see ). Let be the set of all real numbers and define , , where “” and “−" are the ordinary product and subtraction of real numbers. Then yields , and or , that is, , that is, is a -algebra.
However, is not a strong -algebra. If ( or ), then there exist and such that , that is, and . Hence, axiom fails and thus the -algebra is not a strong -algebra.
Theorem 2.3 (see ). The following properties hold in a BCK-algebra: for all , (B1) , (B2) .
A BCK-algebra is said to be bounded if there exists an element such that for all . We denote it by . Note that the usual notation is 1 rather than in literatures. We call such an element the greatest element of . In a bounded BCK-algebra, we denote by . A bounded BCK-algebra is called a -algebra  if it verifies condition : for all .
Theorem 2.4. If is a bounded commutative BCK-algebra, then for any .
It is well known that bounded commutative BCK-algebras, -posets and -algebras are logically equivalent each other (see [13, Page 420]).
Definition 2.5 (see ). Let be a -algebra and . is called a -subalgebra of if whenever and . is called a BCK-ideal of if it satisfies: , and imply .
is called a -ideal of if it satisfies () and and imply , that is, .
Note that, by axiom (I) and definition of -subalgebra, can be deduced easily.
Example 2.6 (see ). Let be a -algebra which is not a BCK-algebra with the following table: Then is a -ideal of .
In a -algebra, a BCK-ideal need not be a -subalgebra, and also a -subalgebra need not be a BCK-ideal. Clearly, is a -subalgebra of any -algebra and every -ideal of is a -subalgebra .
Let be a -algebra and . Define . is said to be edge if for any , . It is known that if is an edge -algebra, then for any .
3. -Units and Weakly Associativity
Let be a -algebra. An element of is said to be a -unit if , where .
Example 3.1. Let be a set with the following table: Then is a -algebra. It is easy to show that 1, 2 are -units of .
Proposition 3.2. Let be a field. Define a binary operation “” on by for any . Then is a -algebra such that every non-zero element of is a -unit.
Proof. It is easy to show that is a -algebra. Given a non-zero element and any element in , the equation has a solution .
Note that the -algebra in Proposition 3.2 is not a BCK-algebra, since .
A -algebra is said to be weakly associative if for any , there exists a such that .
It is known that if is a BCK-algebra with condition , then for all , where is the greatest element of the set . Hence every BCK-algebra with condition is weakly associative.
Proposition 3.3. The -algebra defined in Proposition 3.2 is weakly associative.
Proof. Given , we let . Then we have proving the proposition.
Example 3.4. Let be a set with the following table: Then is a -algebra which is not a BCK-algebra. We know that , but there is no element such that . Hence is not weakly associative.
Proposition 3.5. Let be a weakly associative -algebra and . If is a -unit, then is also a -unit.
Proof. Let be an arbitrary element of . Then for some . Since is weakly associative, there exists such that , proving , which shows that is a -unit.
Proposition 3.6. Let be a poset with minimal element 0. Define a binary operation “” on by Then is a weakly associative BCK-algebra.
Proof. Given , we have either or . Now, assume . Since is a BCK-algebra, and hence . If we take , then we obtain . Assume . If we take , then , proving is weakly associative.
Let be a -algebra and let . We define the set of all monomials , and by where is an indeterminate. We may regard for all .
Define a binary operation “” on by where and are fixed elements of . Then we obtain the following.
Proposition 3.7. If is a -algebra and if in , then is a -algebra.
Proof. For any , we have and .
Assume that . Then and . Since is a -algebra, we obtain and . It follows that either or . Since , we have and , that is, . This proves the proposition.
Using Proposition 3.7, we construct a BCK-algebra which is not weakly associative.
Example 3.8. Define a binary operation “” on by for all . Then it is easy to see that is a BCK-algebra. If we take , , then by Proposition 3.7, is a -algebra, where for all .
We claim that is not weakly associative. By routine calculation, we obtain and for any . If is weakly associative, then for some . It follows that and , and hence 7 divides 18, a contradiction.
We claim that is a BCK-algebra. For any , we have and for some . Hence is a BCK-algebra which is not weakly associative.
4. -Units in BCK-Algebras
Proposition 4.1. If is a BCK-algebra and is a -unit of , then is a bounded BCK-algebra.
Proof. Let be a -unit of . Then , which means that for any , there exists such that . Hence we have for any . This proves that is a bounded BCK-algebra.
The converse of Proposition 4.1 need not be true in general.
Example 4.2. Consider the BCK-algebra [11, Page 252]) with the following table: Then is a bounded BCK-algebra not verifying condition , but 4 is not a -unit, since .
Proposition 4.3. Let be a -algebra and let . Then is the greatest element of if and only if is a -unit of .
Example 4.4. Consider the BCK-algebra with the following table: Then is a -algebra [11, Page 253]), namely a bounded commutative BCK-algebra, and 4 is both the greatest element of and a -unit of .
5. -Integral Domains
Let be a -algebra. An element is said to be a -zero divisor of if there exists an element in such that .
Example 5.1. Let be a set with the following table: Then is a -algebra and are -zero divisors of .
Note that if , then 0 is a -zero divisor of .
Let be a BCK-algebra. If with , then in the induced order, that is, is not a maximal element of . This shows that every non-maximal element of a BCK-algebra is a -zero divisor of .
A -algebra is said to be a -integral domain if every non-zero element is not a -zero divisor, that is, implies .
Proposition 5.2. The -algebra in Proposition 3.2 is a -integral domain.
Proof. If and , then , that is, . Hence is not a -zero divisor and the conclusion follows.
Example 5.3. Let be a set with the following table: Then is a -integral domain which is not a BCK-algebra, since every non-zero element is not a -zero divisor.
Given two posets and , we construct the ordinal sum of and if for all and .
Proposition 5.4. Let be an ordinal sum , where is an anti-chain. If one defines a binary operation “" on by then the BCK-algebra is a -integral domain.
Proof. If in , then . Since is a BCK-algebra, we have , and if , then . Hence implies , proving the proposition.
In the situation of Proposition 5.4, it follows that is a -unit if . Indeed, for all , and implies . If , then is a -integral domain having no -units.
Let be a -algebra. A non-zero element is said to be left-injective if for all implies . A -algebra is said to be left-injective if every non-zero element of is left-injective.
Proposition 6.1. Let be a -algebra and . If is left-injective, then it is not a -zero divisor of .
Proof. If we assume that is a -zero divisor of , then for some in . Since is a -algebra, . It follows from is left-injective that , a contradiction.
Proposition 6.2. Let be a finite -algebra and . If is left-injective, then it is a -unit.
Proof. Given a left-injective element , we define a map by . Then it is injective mapping, since is left-injective. Since is finite, is onto, which proves that . This proves that is a -unit.
In the infinite case, Proposition 6.2 need not be true. We give an example of -algebra such that every non-zero element is a left-injective, but not a -unit element.
Example 6.3. Let be the set of real numbers and let for any . Then it is easy to show that is a -algebra which is not a BCK-algebra, since in general. Let in . Assume . Then . Since is a bijective mapping, we obtain , whence implies and , that is, if , it is a left-injective element. Since in that case, it follows that does not have a solution in such a case. Hence , that is, is not a -unit.
By Proposition 6.2, the -algebra described in Example 6.3 is a -integral domain such that every element is not a -unit. The following example shows that there is a -algebra such that every non-zero element of is a -unit, but not left-injective.
Example 6.4. Let . Define a binary operation “” on by for any . Then it is easy to show that is a -algebra. We claim that every non-zero element of is a -unit. Given , if we take in as follows: then , which proves that is a -unit. We claim that is not left-injective, since , but .
A non-empty subset of a -algebra is said to be a left-ideal of if it satisfies the condition . Every left-ideal of a -algebra contains 0, since for some . Hence, a left-ideal of satisfies . Every -ideal of a -algebra is a left-ideal of , but the converse may not be true in general.
Example 6.5. Let be a -algebra which is not a BCK-algebra with the following table: Then is a left-ideal, but not a -ideal of , since and , but .
A -algebra is said to be simple if its only left-ideals are and . A -algebra is said to be -proper if for all , is a left-ideal of .
Example 6.6. In Example 6.3, for any in , we have , that is, is a left-ideal of . This shows that is -proper. It is not simple, since for any with , that is, is a left-ideal of .
Proposition 6.7. If is a weakly associative -algebra, then it is -proper.
Proof. For any , if , then for some . Since is weakly associative, for some , that is, , proving that is a left-ideal of .
Theorem 6.8. Let be a -algebra. Then is simple and -proper if and only if every non-zero element of is a -unit.
Proof. Since is -proper, is a left-ideal of for any non-zero element of . Moreover, implies and hence . By the simplicity of it follows that , and thus is a -unit.
Assume that every non-zero element of is a -unit. We claim that is -proper. For any , if , then is a left-ideal of . Assume . Since is a -unit, we have and hence , proving that is a left-ideal of . We claim that is simple. Assume that is a left-ideal of such that . If we let in , then since is a -unit. This proves that is simple.
The authors would like to express their great thanks to the referee’s careful reading and valuable suggestions.
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