Finitely Generated Varieties of Commutative BCK-algebras: Covers

The article aims at describing all covers of any finitely generated variety of cBCK-algebras. It is known that subdirectly irreducible cBCK-algebras are rooted trees (concerning their order). Also, all subdirectly irreducible members of finitely generated variety are subalgebras of subdirectly irreducible generators of that variety. The first part of the article focuses on subalgebras of finite subdirectly irreducible cBCK-algebras. In the second part of the article, a construction is presented that provides all the covers of any finitely generated variety.

💡 Research Summary

The paper provides a complete description of all covers of finitely generated varieties of commutative BCK‑algebras (cBCK‑algebras). It begins by recalling that a subdirectly irreducible cBCK‑algebra can be represented as a rooted tree when viewed as a partially ordered set, and that in the finite case the tree determines a unique BCK‑operation. Using Jónsson’s Lemma, the authors observe that for a finitely generated variety V = V(A₁,…,Aₙ) the set of its subdirectly irreducible members Si(V) consists exactly of the subalgebras of the generators A₁,…,Aₙ. Consequently, the problem of describing covers reduces to analysing subalgebras of finite subdirectly irreducible cBCK‑algebras.

The first technical part classifies subalgebras of a finite subdirectly irreducible cBCK‑algebra A. Two families appear: (i) down‑sets, which are automatically subalgebras, and (ii) sets of elements whose heights are divisible by a fixed integer k>1, denoted A_k, together with the maximal elements m(A). Proposition 9 proves that A_k ∪ m(A) is a subalgebra if and only if every branching element and every maximal element of A belong to A_k. Lemmas 4–6 give the necessary height‑based calculations, showing that the BCK‑operation on a tree behaves like truncated subtraction on ℕ, and that a finite cBCK‑algebra is generated by its maximal elements together with any atom.

Having identified all possible subdirectly irreducible subalgebras, the second part constructs every cover of V. The construction is visual and elementary: given a subdirectly irreducible tree T, one creates a new tree T′ by attaching a fresh leaf to an existing maximal element (or, equivalently, by extending a chain by one level). The resulting algebra B = T′ is again subdirectly irreducible, strictly contains T, and the variety generated by B is a minimal proper extension of V, i.e., a cover. The authors prove that any cover of V must arise in this way, because any proper extension must introduce a new minimal element whose removal returns the original generators, and the only way to do this while preserving the tree structure is precisely the leaf‑addition operation.

The paper also discusses the relationship with Łukasiewicz BCK‑algebras (ŁBCK‑algebras) and MV‑algebras, noting that every finite cBCK‑chain is an ŁBCK‑chain and can be identified with the interval