Proposition 5.1 The class of d-algebras and the class of pre-idempotent groupoids are Smarandache disjoint.
Proof Let be both a d-algebra and a pre-idempotent groupoid. Then and , by pre-idempotence and (D2), for any . By (D3), it follows that , which proves that . □
Proposition 5.2 The class of groups and the class of pre-idempotent groupoids are Smarandache disjoint.
Proof Let be both a group and a pre-idempotent groupoid. Then, for any , we have . It follows that , proving that . □
A groupoid is said to be an -groupoid if, for all ,
(L1) ,
(L2) .
Example 5.3 Let be the set of all real numbers. Define a map by , where is the ceiling function. Then . Define a binary operation ‘∗’ on X by for all . Then is an -groupoid. In fact, for all , . Moreover, .
Proposition 5.4 Let be a groupoid and let such that for some . If there exist such that , where for any , then is an -groupoid.
Proof Since , we have , where . It follows that . This shows that , , and hence and , proving the proposition. □
Proposition 5.5 Every -groupoid is pre-idempotent.
Proof Given , we have , proving the proposition. □
Proposition 5.6 The class of -groupoids and the class of groups are Smarandache disjoint.
Proof Let be both an -groupoid and a group with identity e. Then for all . Since any group has the cancellation laws, we obtain . If we apply this to (L2), then we have . This means that . It follows that , proving that . □
Proposition 5.7 The class of -groupoids and the class of BCK-algebras are Smarandache disjoint.
Proof Let be both an -groupoid and a BCK-algebra with a special element . Given , we have . Similarly, . Since X is a BCK-algebra, for all , proving that . □
Let . A groupoid is said to be a Fibonacci semi-lattice if for any , there exists in X depending on a, b, such that .
Note that every Fibonacci semi-lattice is a pre-idempotent groupoid satisfying one of the conditions , , , separately (and simultaneously).
Proposition 5.8 Let be a groupoid and let such that for some with . Then is a Fibonacci semi-lattice.
Proof Since , where with , we have , where . If we let , , then . It follows that . Hence , proving the proposition. □
A groupoid is said to be an -groupoid if for all ,
() ,
() .
Proposition 5.9 Let be a groupoid and let such that , . Then is an -groupoid.
Proof Since such that , , we have , where . If we let , , then . It follows that and , proving the proposition. □
Proposition 5.10 The class of -groupoids and the class of groups are Smarandache disjoint.
Proof Let be both an -groupoid and a group with a special element . Given , we have . Since every group has cancellation laws, we obtain . It follows that , and hence . This proves that , proving the proposition. □
A groupoid is said to be an -groupoid if for all ,
() ,
() .
Proposition 5.11 Let be a groupoid and let such that . Then is an -groupoid.
Proof Since such that , we have , where . If we let , , then . It follows that and , proving the proposition. □
Theorem 5.12 The class of -groupoids and the class of groups are Smarandache disjoint.
Proof Let be both an -groupoid and a group with a special element . Given , we have . Since every group has cancellation laws, we obtain . By applying (), we have , i.e., . By (), . This proves that , proving the proposition. □
Proposition 5.13 Every implicative BCK-algebra is an -groupoid.
Proof If is an implicative BCK-algebra, then and for any . It follows immediately that is an -groupoid. □
Remark The condition, implicativity, is important for a BCK-algebra to be an -groupoid.
Example 5.14 Let be a set with the following table:
Then is a BCK-algebra, but not implicative, since . Moreover, it is not a -groupoid since .
Theorem 5.15 Every BCK-algebra inherited from a poset is an -groupoid.
Proof If is a BCK-algebra inherited from a poset , then the operation ‘∗’ is defined by
Then the condition () holds. In fact, given , if , then and . It follows that . If , then and . It follows that . If x and y are incomparable, then and . It follows that .
We claim that () holds. Given , if , then , . It follows that . If , then , . It follows that . If x and y are incomparable, then and . It follows that . This proves that the BCK-algebra inherited from a poset is an -groupoid. □
Note that the BCK-algebra inherited from a poset need not be an implicative BCK-algebra unless the poset is an antichain [[14], Corollary 9].
Example 5.16 Let be a left-zero semigroup, i.e., for all . Then is an -groupoid, but not a BCK-algebra.