TL;DR
This paper investigates BCK-sequences and n-commutative BCK-algebras, providing counterexamples that resolve two open problems about their properties and characterizations.
Contribution
It demonstrates that BCK-sequences are not always prolongable and that n-commutative BCK-algebras cannot be characterized by a single identity.
Findings
Counterexamples show BCK-sequences are not always prolongable.
Counterexamples show n-commutative BCK-algebras are not characterized by one identity.
Answers to two open problems are negative.
Abstract
BCK-sequences and n-commutative BCK-algebras were introduced by T. Traczyk, together with two related problems. The first one, whether BCK-sequences are always prolongable. The second one, if the class of all n-commutative BCK-algebras is characterised by one identity. W. A. Dudek proved that the answer to the former question is positive in some special cases, e.g. when BCK-algebra is linearly ordered. T. Traczyk showed that the answer to the latter is affirmative for n = 1, 2. Nonetheless, by providing counterexamples, we proved that the answers to both those open problems are negative.
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