On generating sets of finite algebras (Q895876)

Let \(A\) be a finite algebra, and let \(\Phi (A)\) be the intersection of its maximal subalgebras. \(A\) is called a \(\mathcal{B}\)-algebra if all subsets \(X\) that are minimal with respect to \(\subseteq\) among those for which \(X \cup \Phi (A)\) generates \(A\) have the same cardinality. An element \(u\) of \(A\) is a \(j\)-element if the subalgebra generated by \(u\) is a join irreducible element of the lattice of subalgebras of \(A\). An algebra is called a \(\mathcal{B}_j\)-algebra if all minimal subsets \(Y\) of \(j\)-elements such that \(Y \cup \Phi (A)\) generates \(A\) have the same cardinality. If \(A\) is an expansion of a finite group with some of its endomorphisms and \(A\) has property \(\mathcal{B}\) or \(\mathcal{B}_j\), then all of its homomorphic images have the same property (Theorem~3.1). The property \(\mathcal{B}_j\) is also preserved by forming certain direct products (Theorem~3.3). Concerning the question raised by Proposition~1.2(2) whether homomorphic images of \(j\)-elements are again \(j\)-elements, the reviewer would like to point out the following example: let \(A = (\{1, a, a^2,\ldots, a^{7}\}, *, 1)\) be the eight element monoid presented by \(\langle a \mid a^8 = a^2 \rangle\), \(B = (\mathbb{Z}_6, +, 0)\), and \(\varphi (a^t) = [t]_6\) for \(t \in \mathbb{N}_0\). Then, \(a\) is a \(j\)-element of \(A\), but \(\varphi (a) = [1]_6\) is not a \(j\)-element of \(B\).