On the variety of \(m\)-generalized Łukasiewicz algebras of order \(n\)
The paper is well written and very nice. The variety \({\mathcal L}_n^m\) of \(m\)-generalized Łukasiewicz algebras of order \(n\) (introduced by \textit{T. Almada} and the author [Stud. Log. 69, No.~3, 329--338 (2001; Zbl 0995.03047)] contains the variety \({\mathcal L}_n\) of Łukasiewicz (Łukasiewicz-Moisil) algebras of order \(n\) (\(m,n \in {\mathbb N}, \; n \geq 2\)). For \({\mathbf A}=(A; \wedge, \vee, f, D_1, \dots, D_{n-1},0,1) \in {\mathcal L}^m_n\), the set \(S_A=\{x \in A \mid f^2(x)=x\}\) is the universe of \({\mathbf S_A}\), the greatest subalgebra of \({\mathbf A}\) which belongs to \({\mathcal L}_n\). The author describes the congruences on \({\mathbf A} \in {\mathcal L}^m_n\) via the Stone filters of \({\mathbf S_A} \in {\mathcal L}_n\), finds a finite set of polynomial equations that give equationally definable principal congruences for \({\mathcal L}^m_n\) and characterizes the principal congruences on \({\mathbf A}\) via the principal Stone filters of \({\mathbf S_A}\). Next, the author proves that \({\mathcal L}^m_n={\mathcal L}_n\) if and only if \(m=1\) or \(n=2\), and that \({\mathcal L}^m_n\) is a discriminator variety; consequently, the variety \({\mathcal L}^m_n\) is congruence-permutable, congruence-regular, congruence-uniform, congruence-coherent and filtral. Finally, the variety \({\mathcal L}^m_n\) is finitely generated.
scientific article; zbMATH DE number 5701679