Multipotent GBL-algebras
GBL-algebras are residuated lattices satisfying a divisibility law. An element of a GBL-algebra \(\mathbf{A}\) is \(n\)-potent if \(a^{n+1}=a^{n}\) holds. Thus, \(\mathbf{A}\) is \(n\)-potent if there is an \(n\in \omega\) such that every \(a\in A\) is \(n\)-potent. An algebra is called multipotent if every element is \(n\)-potent for some \(n\), but the algebra itself is not \(n\)-potent for any \(n\). Every finite GBL-algebra is \(n\)-potent for some \(n\), and every \(n\)-potent algebra is multipotent, but neither of the converses holds. A GBL-algebra is normal if every filter of \(\mathbf{A}\) is normal. The aim of this paper is to present some properties of the multipotent GBL-algebras. First, we have that every multipotent GBL-algebra \(\mathbf{A}\) is normal, and if \(\mathbf{A}\) is subdirectly irreducible then it contains a greatest idempotent element \(a<1\). Next, it is shown that subdirectly irreducible multipotent GBL-algebras with a unique idempotent element \(a<1\) are pseudo-MV-algebras. More precisely, they become pseudo-MV-algebras when expanded with a constant interpreted as the bottom element. If \(\mathbf{A}\) is a subdirectly irreducible multipotent GBL-algebras, then \(\mathbf{A}\) is decomposable as ordinal sum. Also, the authors slightly extend the results of \textit{P. Jipsen} and \textit{F. Montagna} [ibid. 55, No.~2--3, 227--238 (2006; Zbl 1109.06011); ibid. 60, No.~4, 381--404 (2009; Zbl 1192.06011)] on conditions implying commutativity (these results explicitly state that any \(n\)-potent GBL-algebra is commutative). In this sense we have that the variety of GBL-algebras generated by its multipotent members is commutative. This implies that every multipotent GBL-algebra satisfying preliniarity is commutative. But if the variety of integral GBL-algebras is noncommutative then it is not generated by its multipotent members. Finally, the authors explain what happens with the results presented here in other (sub)varieties: FL, pseudo BL-algebras, \(l\)-groups or (atomic) pseudo-MV-algebras, and they conclude that there are continuum many varieties of noncommutative pseudo-MV-algebras generated by algebras with a single coatom.
scientific article; zbMATH DE number 5833771