In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?

在本文中，我们研究交换 BCK 代数的素数谱。我们使用有根树给出了交换 BCK 代数的新结构，并确定了此类代数的理想格和素理想格。我们证明了任何可交换 BCK 代数的谱都是局部紧的广义谱空间，当且仅当代数作为理想有限生成时，它才是紧的。此外，我们证明如果一个交换 BCK 代数是对合的，那么它的谱是一个 Priestley 空间。最后，我们考虑了谱的函子性质，并定义了一个从交换 BCK 代数范畴到零分布格范畴的函子。我们给出了这个问题的部分答案：这个函子的图像中有哪些分布格？