A bottleneck algebra is a linearly ordered set (B,≤) with two operations a⊕b=max{a,b} and a⊗b=min{a,b}. A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way. In 1996, Cechl´arov´a and Pl´avka posed an open problem: Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2B-independent. In this paper, we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2B-independent and answer this open problem.

A bottleneck algebra is a linearly ordered set (B,≤) with two operations a⊕b=max{a,b} and a⊗b=min{a,b}. A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way. In 1996, Cechl´arov´a and Pl´avka posed an open problem: Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2B-independent. In this paper, we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2B-independent and answer this open problem.