\(p\)-pseudosimple algebras
An algebra \({\mathfrak A}\) is called simple if it has only two congruences, 0 and 1; it is called pseudosimple if for any of its congruences \(\mu\neq 1\), we have \({\mathfrak A}\cong {\mathfrak A}/\mu\); it is called quasisimple if for any of its congruences \(\mu\neq 1\), there exists \(\mu'\geq \mu\) with \({\mathfrak A}\cong {\mathfrak A}/\mu'\). The last two notions were studied by H. Andréka, J. Németi and A. G. Pinus, respectively. In this paper, the authors introduce the notion of \(p\)-pseudo-simplicity by the requirement that \({\mathfrak A}\cong {\mathfrak A}/\mu\) for all principal congruences \(\mu\) of \({\mathfrak A}\). This is a generalization of pseudo-simplicity which is independent of quasi-simplicity. The authors give some properties of this notion and in particular they prove that \({\mathfrak A}\) is \(p\)-pseudosimple iff any of its homomorphic images is an epilimit of some system of isomorphic copies of \({\mathfrak A}\).
scientific article; zbMATH DE number 536682