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We describe weak-BCC-algebras (also called BZ-algebras) in which the condition is satisfied only in the case when elements belong to the same branch. We also characterize ideals, nilradicals, and nilpotent elements of such algebras.
BCK-algebras which are a generalization of the notion of algebra of sets with the set subtraction as the only fundamental nonnullary operation and on the other hand the notion of implication algebra (cf. ) were defined by Imai and Iséki in . The class of all BCK-algebras does not form a variety. To prove this fact, Komori introduced in  the new class of algebras called BCC-algebras. In view of strong connections with a -logic, BCC-algebras are also called -algebras (cf.  or ). Nowadays, many mathematicians, especially from China, Japan, and Korea, have been studying various generalizations of BCC-algebras. All these algebras have one distinguished element and satisfy some common identities playing a crucial role in these algebras.
One of very important identities is the identity . It holds in BCK-algebras and in some generalizations of BCK-algebras, but not in BCC-algebras. BCC-algebras satisfying this identity are BCK-algebras (cf.  or ). Therefore, it makes sense to consider such BCC-algebras and some of their generalizations for which this identity is satisfied only by elements belonging to some subsets. Such study has been initiated by Dudek in .
In this paper, we will study weak-BCC-algebras in which the condition is satisfied only in the case when elements belong to the same branch. We describe some endomorphisms of such algebras, ideals, nilradicals, and nilpotent elements.
2. Basic Definitions and Facts
Definition 1. A weak-BCC-algebra is a system of type satisfying the following axioms: (i), (ii), (iii), (iv).
Weak-BCC-algebras are called BZ-algebras by many mathematicians, especially from China and Korea (cf.  or ), but we save the first name because it coincides with the general concept of names presented in the book  for algebras of logic.
A weak-BCC-algebra satisfying the identity (v) is called a BCC-algebra. A BCC-algebra with the condition (vi) is called a BCK-algebra.
One can prove (see  or ) that a BCC-algebra is a BCK-algebra if and only if it satisfies the identity (vii) .
An algebra of type satisfying the axioms (i), (ii), (iii), (iv), and (vi) is called a BCI-algebra. A BCI-algebra satisfies also (vii). A weak-BCC-algebra is a BCI-algebra if and only if it satisfies (vii).
Any weak-BCC-algebra can be considered as a partially ordered set. In any weak-BCC-algebra, we can define a natural partial order putting This means that a weak-BCC-algebra can be considered as a partially ordered set with some additional properties.
Proposition 2. An algebra of type with a relation defined by (1) is a weak-BCC-algebra if and only if for all the following conditions are satisfied: (i′),(ii′),(iii′),(iv′) and imply .
From (i′), it follows that in weak-BCC-algebras, implications are satisfied by all .
A weak-BCC-algebra which is neither BCC-algebra nor BCI-algebra is called proper. Proper weak-BCC-algebras have at least four elements (see ). But there are only two weak-BCC-algebras of order four which are not isomorphic: (6)
They are proper, because in both cases .
Since two nonisomorphic weak-BCC-algebras may have the same partial order, they cannot be investigated as algebras with the operation induced by partial order. For example, weak-BCC-algebras defined by (4) and (5) have the same partial order but they are not isomorphic.
The methods of construction of weak-BCC-algebras proposed in  show that for every , there exist at least two proper weak-BCC-algebras of order which are not isomorphic.
The set of all minimal (with respect to ⩽) elements of is denoted by . Elements belonging to are called initial.
In the investigation of algebras connected with various types of logics, an important role plays the so-called Dudek’s map defined by . The main properties of this map in the case of weak-BCC-algebras are collected in the following theorem proved in .
Theorem 3. Let be a weak-BCC-algebra. Then, (1), (2), (3), (4), (5), (6)for all .
Theorem 4. .
The proof of this theorem is given in . Comparing this result with Theorem 3(4), we see that is a subalgebra of ; that is, it is closed under the operation . In some situations (see Theorem 21), is a BCI-algebra.
Corollary 5. for any weak-BCC-algebra .
Proof. Indeed, if , then for some . Thus, by Theorem 3, . Hence, ; that is, . So, .
Conversely, for , we have , where . Thus, , which completes the proof.
This means that an element is an initial element of a weak-BCC-algebra if and only if it is mentioned in the first row (i.e., in the row corresponding to 0) of the multiplication table of .
Let be a weak-BCC-algebra. For each , the set is called a branch of initiated by . A branch containing only one element is called trivial. The branch is the greatest BCC-algebra contained in a weak-BCC-algebra ().
According to [1, 15], we say that a subset of a BCK-algebra is an ideal of if , and imply . If is an ideal, then the relation defined by is a congruence on a BCK-algebra . Unfortunately, it is not true for weak-BCC-algebras (cf. ). In connection with this fact, Dudek and Zhang introduced in  the new concept of ideals. These new ideals are called BCC-ideals.
Definition 6. A nonempty subset of a weak-BCC-algebra is called a BCC-ideal if(1),(2) and imply .
By putting , we can see that a BCC-ideal is a BCK-ideal. In a BCK-algebra, any ideal is a BCC-ideal, but in BCC-algebras, there are BCC-ideals which are not ideals in the above sense (cf. ). It is not difficult to see that is a BCC-ideal of each weak-BCC-algebra.
In the following part of this paper, we will need those two propositions proved in .
Proposition 7. Elements are in the same branch if and only if .
Proposition 8. If , then also and are in .
One of the important classes of weak-BCC-algebras is the class of the so-called group-like weak-BCC-algebras called also antigrouped BZ-algebras , that is, weak-BCC-algebras containing only trivial branches. A special case of such algebras is group-like BCI-algebras described in .
Theorem 9. An algebra is a group-like weak-BCC-algebra if and only if , where , is a group. Moreover, in this case, .
Corollary 10. A group is abelian if and only if the corresponding weak-BCC-algebra is a BCI-algebra.
Corollary 11. is a maximal group-like subalgebra of each weak-BCC-algebra .
3. Solid Weak-BCC-Algebras
As it is well known in the investigations of BCI-algebras, the identity (vii) plays a very important role. It is used in the proofs of almost all theorems, but as Dudek noted in his paper , many of these theorems can be proved without this identity. Just assume that this identity is fulfilled only by elements belonging to the same branch. In this way, we obtain a new class of weak-BCC-algebras which are called solid.
Definition 12. A weak-BCC-algebra is called solid, if the equation (vii) is satisfied by all belonging to the same branch and arbitrary .
Any BCI-algebra and any BCK-algebra are solid weak-BCC-algebras. A solid weak-BCC-algebra containing only one branch is a BCK-algebra. To see examples of solid weak-BCC-algebras which are not BCI-algebras, one can find them in .
Theorem 13. Dudek’s map is an endomorphism of each solid weak-BCC-algebra.
Proof. Indeed, for all .
Corollary 14. is a maximal group-like BCI-subalgebra of each solid weak BCC-algebra.
Proof. Comparing Corollaries 5 and 11, we see that is a maximal group-like subalgebra of each weak BCC-algebra . Thus, by Theorem 9, there exists a group such that for . Since is solid, is its endomorphism. Hence, for ; that is, in the corresponding group. The last is possible only in an abelian group, but in this case, , which means that is a BCI-algebra.
Definition 15. For and nonnegative integers , we define
Theorem 16. In solid weak-BCC-algebras, the following identity is satisfied for each nonnegative integer .
Proof. Let . Then, by Theorem 3, implies . Suppose that for some nonnegative integer . Then, also , by (3). Consequently,
which means that because . So, is valid for all and each nonnegative integer .
Similarly and for and nonnegative integer . Thus, a weak-BCC-algebra satisfies the identity (12) if and only if holds for . But in view of Corollary 11 and Theorem 9 in the group , the last equation can be written in the following form: Since a weak-BCC-algebra is solid, by Corollary 14, is a BCI-algebra. So, the group is abelian. Thus, the above equation is valid for all . Hence, (12) is valid for all and all nonnegative integers .
Corollary 17. The map is an endomorphism of each solid weak-BCC-algebra.
Definition 18. A weak-BCC-algebra for which is an endomorphism is called -strong. In the case , we say that it is strong.
A solid weak-BCC-algebra is strong for every . The converse statement is not true.
Example 19. The weak-BCC-algebra defined by (4) is not solid because , but it is strong for every . Indeed, in this weak-BCC-algebra, we have for , for , and for all . So, it is -strong and -strong. Since in this algebra for even , and for odd , it is strong for every .
Example 20. Direct computations show that the group-like weak-BCC-algebra induced by the symmetric group (Theorem 9) is -strong for and but not for .
Theorem 21. A weak-BCC-algebra is strong if and only if is a BCI-algebra, that is, if and only if is an abelian group.
Proof. Indeed, if is strong, then holds for all . Thus, in the group , we have , which means that the group is abelian. Hence,
for all . So, is a BCI-algebra.
On the other hand, according to Theorem 3, for any , , we have and . So, if is a BCI-algebra, then for any , we have . Consequently, because . This completes the proof.
Corollary 22. A strong weak-BCC-algebra is -strong for every .
Proof. In a strong weak-BCC-algebra , the group is abelian and for every . Thus, for all and .
Example 20 shows that the converse statement is not true; that is, there are weak-BCC-algebras which are strong for some but not for .
Corollary 23. A weak-BCC-algebra in which is a BCI-algebra is strong for every .
Corollary 24. In any strong weak-BCC-algebra, we have for every and every natural .
4. Ideals of Weak-BCC-Algebras
To avoid repetitions, all results formulated in this section will be proved for BCC-ideals. Proofs for ideals are almost identical to proofs for BCC-ideals.
Theorem 25. Let be a weak-BCC-algebra. Then, is an ideal (BCC-ideal) of if and only if the set theoretic union of branches , , is an ideal (BCC-ideal) of .
Proof. Let denote the set theoretic union of some branches initiated by elements belonging to ; that is,
By Corollary 11, is a weak-BCC-algebra contained in .
If is a BCC-ideal of , then obviously . Consequently, because . Now let and for some . Then, , , , and for some and . Thus, , which means that since two branches are equal or disjoint. Hence, , so . Therefore, . This shows that is a BCC-ideal of .
Conversely, let be a BCC-ideal of . If for some and , then , . Hence, . Since and , the above implies . Thus, is a BCC-ideal of .
is a subalgebra of each weak-BCC-algebra , but it is not an ideal, in general.
Example 26. It is easy to check that in the weak-BCC-algebra defined by (22) is not an ideal because , but .
The above example suggests the following.
Theorem 27. If is a proper ideal or a proper BCC-ideal of a weak-BCC-algebra , then has at least two nontrivial branches.
Proof. Since , at least one branch of is not trivial. Suppose that only has more than one element. Then, for any and , , we have . But, by Corollary 11, is a maximal group-like subalgebra contained in . Thus, and , because in the case in the corresponding group , we obtain which is impossible for . Therefore, and has only one element. So, . Hence, , which according to the assumption on implies . The obtained contradiction shows that cannot be an ideal of . Consequently, it cannot be a BCC-ideal, too.
Definition 28. A nonempty subset of a weak-BCC-algebra is called an -fold -ideal of if it contains and
An -fold -ideal is called an -fold -ideal. Since -fold -ideals coincide with BCK-ideals, we will consider -fold -ideals only for and . Moreover, it will be assumed that because for we have , which implies . So, for every -fold -ideal of . Note, that the concept of -fold -ideals coincides with the concept of -ideals studied in BCI-algebras (see e.g.,  or ).
Example 29. It is easy to see that in the weak-BCC-algebra defined by (4), the set is an -fold -ideal for every . It is not an -fold -ideal, where is odd and is even because in this case and , but .
Putting in (23), we see that each -fold -ideal of a weak-BCC-algebra is an ideal. The converse statement is not true since, as it follows from Theorem 30 proved below, each -fold ideal contains the branch which for BCC-ideals is not true.
Theorem 30. Any -fold -ideal contains .
Proof. Let be an -fold -ideal of a weak-BCC-algebra . Since for every from it follows that , we have which, according to (23), gives . Thus, .
Corollary 31. An -fold -ideal together with an element contains whole branch containing this element.
Proof. Let and be an arbitrary element from the branch containing . Then, according to Proposition 7, we have . Since is also an ideal, the last implies . Thus, .
Corollary 32. For any -fold -ideal from and , it follows that .
Theorem 33. A nonempty subset of a solid weak-BCC-algebra is its -fold -ideal if and only if (a) is an -fold -ideal of , (b).
Proof. Let be an -fold -ideal of . Then, clearly is an -fold -ideal of . By Corollary 31, is the set theoretic union of all branches such that . So, any -fold -ideal satisfies the above two conditions.
Suppose now that a nonempty subset of satisfies these two conditions. Let . If , , , and , then , which, by , implies . This, by , gives . So, . Hence, .
Note that in some situations, the converse of Theorem 30 is true.
Theorem 34. An ideal of a weak-BCC-algebra is its -fold -ideal if and only if .
Proof. By Theorem 30, any -fold -ideal contains . On the other hand, if is an ideal of and , then from and , by (i′), it follows that so and , as comparable elements, are in the same branch. Hence, , by Proposition 7. Since and is a BCC-ideal (or a BCK-ideal), implies . Consequently, . So, is an -fold -ideal.
Corollary 35. Any ideal containing an -fold -ideal is also an -fold -ideal.
Proof. Suppose that an ideal contains some -fold -ideal . Then, , which completes the proof.
Corollary 36. An ideal of a weak-BCC-algebra is its -fold -ideal if and only if the implication is valid for all .
Proof. Let be an -fold -ideal of . Since , from and by Corollary 32, we obtain . So, any -fold -ideal satisfies this implication.
The converse statement is obvious.
Theorem 37. An -fold -ideal is a -fold -ideal for any .
Proof. Similarly, as in the previous proof, we have for every . Thus, and are in the same branch. Hence, if is an -fold -ideal and , then, by Corollary 31, also . This, together with , implies . Therefore, is a -fold ideal.
Theorem 38. is the smallest -fold -ideal of each weak-BCC-algebra.
Proof. Obviously, . If , then , and Thus, . Since means , from the above, we obtain . So, . Hence, is an -fold -ideal. By Theorem 30, it is the smallest -fold -ideal of each weak-BCC-algebra.
Theorem 39. Let be a weak-BCC-algebra. If has elements and divides , then is an -fold -ideal of .
Proof. By Corollary 11, is a group-like subalgebra of . Hence, if has elements, then in the group connected with (Theorem 9), we have for every and any integer .
At first, we consider the case . If for some , , , then, by (i′), we have . Hence, and , as comparable elements, are in the same branch. Consequently, (Proposition 7). Since, is an ideal in each weak-BCC-algebra, from the last, we obtain , and consequently, . But, , so ; that is, . This in the group connected with gives . So, .
Now let . Then . This, similarly as in the previous case, for gives . Consequently, . So, . This in the group implies . Hence, .
The proof is complete.
The assumption on the number of elements of the set is essential; if is not a divisor of , then may not be an -fold -ideal.
Example 40. The solid weak-BCC-algebra defined by (30) is proper, because . The set has three elements. The set is an -fold -ideal for every natural but it is not a -fold ideal because and .
In the case when has only one element, the equivalence relation induced by has one-element equivalence classes. Since these equivalence classes are branches, a weak-BCC-algebra with this property is group-like. Direct computations show that in this case, is an -fold -ideal for every natural .
This observation together with the just proved results suggests simple characterization of group-like weak-BCC-algebras.
Theorem 41. A weak-BCC-algebra is group-like if and only if for some and all
Proof. Let be a weak-group-like BCC-algebra. Then, , which means that has a discrete order; that is, implies . Since for we have , a group-like weak-BCC-algebra satisfies the identity . In particular, for , we have . So, implies .
Conversely, if the above implication is valid for all , then gives . This, according to the assumption, implies . Hence, , which means that is group-like.
Remember that an ideal of a weak-BCC-algebra is called closed if for every , that is, if .
Theorem 42. For an -fold -ideal of a solid weak-BCC-algebra , the following statements are equivalent: (1) is a closed -fold -ideal of , (2) is a closed -fold -ideal of , (3) is a subalgebra of , (4) is a subalgebra of .
Proof. The implication follows from Theorem 33.
Observe first that is a closed BCK-ideal of and for any . Since is a group-like subalgebra of (Corollary 11), in the group , we have (Theorem 9), which means that . Thus, Hence, . But and is a BCK-ideal of ; therefore . Consequently, for every . So, is a subalgebra of .
, so . Let , . If , then , and by the assumption . From this, we obtain , which together with Theorem 33 proves . Hence, is a subalgebra of .
The implication is obvious.
5. Nilpotent Weak-BCC-Algebras
A special role in weak-BCC-algebras play elements having a finite “order,” that is, elements for which there exists some natural such that . We characterize sets of such elements and prove that the properties of such elements can be described by the properties of initial elements of branches containing these elements.
Definition 43. An element of a weak-BCC-algebra is called nilpotent, if there exists some positive integer such that . The smallest with this property is called the nilpotency index of and is denoted by . A weak-BCC-algebra in which all elements are nilpotent is called nilpotent.
By , we denote the set of all nilpotent elements such that . denotes the set of all nilpotent elements of . It is clear that .
Example 45. In the weak-BCC-algebra defined by (35) there are no elements with , but there are three elements with and three with .
Proposition 46. Elements belonging to the same branch have the same nilpotency index.
Proof. Let . Then , which, by Theorem 3, implies . This together with gives . Hence, . In the same manner from , it follows that , which by induction proves for every and any natural . Thus, implies . On the other hand, from , we obtain . This implies since and elements of are incomparable. Therefore, if and only if . So, for every .
Corollary 47. A weak-BCC-algebra is nilpotent if and only if its subalgebra is nilpotent.
Corollary 48. .
The above results show that the study of nilpotency of a given weak-BCC-algebras can be reduced to the study of nilpotency of its initial elements.
Proposition 49. Let be a weak-BCC-algebra. If is a BCI-algebra, then is a subalgebra and a BCC-ideal of for every .
Proof. Obviously, for every . Let . Then and , for some . Since is a BCI-algebra, by Theorem 16, we have . Hence, . Consequently, . So, is a subalgebra of .
Now let , , . If , then also . Thus, and which implies . This together with Corollary 48 implies . Therefore, is a BCC-ideal of . Clearly, it is a BCK-ideal, too.
Corollary 50. is a subalgebra of each solid weak-BCC-algebra.
Proposition 51. is a subalgebra of each weak-BCC-algebra in which is a BCI-algebra.
Proof. Since and for every , the set is nonempty. Let , . If and , , then . From this, by Proposition 46, we obtain , which in the group can be written in the form . But is a BCI-algebra; hence, is an abelian group. Thus, by Theorem 9. Hence, . This implies . Therefore, is a subalgebra of .
Corollary 52. is a subalgebra of each solid weak-BCC-algebra.
Corollary 53. Any solid weak-BCC-algebra with finite is nilpotent.
Proof. Indeed, is a maximal group-like BCI-algebra contained in any solid weak-BCC-algebra. Hence, the group is abelian. If it is finite, then each of its element has finite order . Thus, for every . Consequently, for every . Therefore, .
Corollary 54. A solid weak-BCC-algebra is nilpotent if and only if each element of the group has finite order.
Corollary 55. In a solid weak-BCC-algebra , the nilpotency index of each is a divisor of .
6. -Nilradicals of Solid Weak-BCC-Algebras
The theory of radicals in BCI-algebras was considered by many mathematicians from China (cf. ). Obtained results show that this theory is almost parallel to the theory of radicals in rings. But results proved for radicals in BCI-algebras cannot be transferred to weak-BCC-algebras.
In this section, we characterize one analog of nilradicals in weak-BCC-algebras. Further, this characterization will be used to describe some ideals of solid weak-BCC-algebras.
We begin with the following definition.
Definition 56. Let be a subset of solid weak-BCC-algebra . For any positive integer by a -nilradical of , denoted by , we mean the set of all elements of such that ; that is,
Example 57. In the weak-BCC-algebra defined in Example 44 for and any natural , we have , . But for , we get , . The set is empty.
Example 58. The solid weak-BCC-algebra defined by (40) is proper, because . In this algebra, each -nilradical of is equal to ; each -nilradical of is empty.
The first question is when for a given nonempty set its -nilradical is also nonempty? The answer is given in the following proposition.
Proposition 59. A -nilradical of a nonempty subset of a weak-BCC-algebra is nonempty if and only if contains at least one element .
Proof. From the proof of Theorem 16, it follows that for every and any positive . So, if and only if . The last means that because is a subalgebra of .
Corollary 60. for every .
Proof. Indeed, for every . Thus, .
Corollary 61. If has elements, then for any subset of containing , and if .
Proof. Similarly, as in previous proofs, we have for every and any . Since and is a group-like subalgebra of , in the group (Theorem 9). If has elements, then obviously . Hence, . This completes the proof.
Corollary 62. Let . Then if and only if .
Proof. Since , we have .
Corollary 63. .
Proposition 64. Let be a solid weak-BCC-algebra. Then for every positive integer and any subalgebra of a -nilradical is a subalgebra of such that .
Proof. Let . Then and , by Theorem 16. Hence, . Clearly .
Proposition 65. In a solid weak-BCC-algebra, a -nilradical of an ideal is also an ideal.
Proof. Let be a BCC-ideal of . If and , then and , by Theorem 16. Hence, . Thus, .
Note that the last two propositions are not true for weak-BCC-algebras which are not solid.
Example 66. The weak-BCC-algebra induced by the symmetric group is not solid because is not an abelian group (Corollary 14). Routine calculations show that is a subalgebra and a BCC-ideal of this weak-BCC-algebra, but is neither ideal nor subalgebra.
Theorem 67. In a solid weak-BCC-algebra, a -nilradical of an -fold -ideal is also an -fold -ideal.
Note that in general, a -nilradical of an ideal does not save all properties of an ideal . For example, if an ideal is a horizontal ideal, that is, , then a -nilradical may not be a horizontal ideal. Such situation takes place in a weak-BCC-algebra defined by (34). In this algebra, we have for all elements. Hence, means that and which is also true for .
Nevertheless, properties of many main types of ideals are saved by their -nilradicals. Below, we present the list of the main types of ideals considered in BCI-algebras and weak-BCC-algebras.
Definition 68. An ideal of a weak-BCC-algebra is called (i)antigrouped, if (ii)associative, if (iii)quasiassociative if (iv)closed, if (v)commutative, if (vi)subcommutative, if (vii)implicative if (viii)subimplicative if (ix)weakly implicative if (x)obstinate, if (xi)regular, if (xii)strong, if for all .
Definition 69. We say that an ideal of a weak-BCC-algebra has the property if it is one of the above types, that is, if it satisfies one of implications mentioned in the above definition.
Theorem 70. If an ideal of a solid weak-BCC-algebra has the property , then its -nilradical also has this property.
Proof. is antigrouped. Let . Then . Since, by Theorem 3, is an endomorphism of each weak-BCC-algebra, we have
Thus, , which according to the definition implies . Hence, .
is associative. If , then and which, in view of Theorem 16, means that and . Since an ideal is associative, this implies ; that is, .
is quasiassociative. Similarly as in the previous case