2025/07/11 by C. Matthew Evans, Evans, C. Matthew
Computer Science · Mathematics · #03G99 #06F35 #20F19 #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2507.08976
openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We recall the derived subalgebra of a BCK-algebra, and use this to define the derived ideal. Using the derived ideal, we show that the category of commutative BCK-algebras is a reflective subcategory of the category of BCK-algebras. After this, we introduce central series and define a notion of nilpotence for BCK-algebras and prove some properties of nilpotence. In particular, for any variety of BCK-algebras, the sub-class of nilpotent algebras is a sub-pseudovariety, though in general not a variety. We also show that the class of BCK-algebras of nilpotence class at most c is a sub-quasivariety of all BCK-algebras, and is a variety if and only if c=1. We close by showing that every finite BCK-algebra is nilpotent.