#### Abstract

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.

#### 1. Introduction

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. [1]) were defined by Imai and Iséki in [2]. The class of all BCK-algebras does not form a variety. To prove this fact, Komori introduced in [3] the new class of algebras called BCC-algebras. In view of strong connections with a -logic, BCC-algebras are also called -algebras (cf. [4] or [5]). 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. [6] or [7]). 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 [8].

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. [9] or [10]), but we save the first name because it coincides with the general concept of names presented in the book [11] 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 [6] or [7]) 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 [12]). 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 [12] 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 [13].

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 [14]. 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 ([8]).

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. [16]). In connection with this fact, Dudek and Zhang introduced in [16] 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. [16]). It is not difficult to see that is a BCC-ideal of each weak-BCC-algebra.

The equivalence classes of a congruence defined by (8), where , coincide with branches of ; that is, for any (cf. [14]). So,

In the following part of this paper, we will need those two propositions proved in [14].

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* [9], that is, weak-BCC-algebras containing only trivial branches. A special case of such algebras is group-like BCI-algebras described in [17].

From the results proved in [17] (see also [9]), it follows that such weak-BCC-algebras are strongly connected with groups.

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 [8], 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 [8].

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., [18] or [19]).

*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 44. *In the weak-BCC-algebras defined by (4) and (5), we have , .

*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. [18]). 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.*

*Proof. *By Proposition 65, a -nilradical of an -fold -ideal of is an ideal of . If , then . Hence, applying Theorem 16, we obtain
Thus, . So, .

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 means that and . Hence, . This implies . Consequently, .

is closed. Let . Then, . Thus,
So, .

is commutative. Let . Then, . From this, we obtain , which gives . Hence, .

For other types of ideals, the proof is very similar.