Some remarks on Maltsev and Goursat categories
Let \({\mathcal A}\) be a regular category, i.e., a finitely complete category with a regular epimorphic-monomorphic factorization stable under pulling back. A relation on an object \(A\) of \({\mathcal A}\) is a subobject \(R\) of \(A \times A\). Two such relations \(R\) and \(S\) can be composed to get a relation \(RS\) on \(A\). If \(R\) and \(S\) are equivalence relations on \(A\), we get two increasing sequences of relations on \(A\): \[ \Delta_ A \leq R \leq RS \leq RSR \leq RSRS \leq \dots \] \[ \Delta_ A \leq S \leq SR \leq SRS \leq SRSR \leq \dots \] whose terms are generally distinct. However it may happen that these sequences are stationary, for any relations \(R\) and \(S\) on any object \(A\) of \({\mathcal A}\). In this case, the category is said to be \(n\)-permutable for some adequate \(n \in \mathbb{N}\). For \(n = 2\), i.e., \(RS = SR\), the category is called a Maltsev category. For \(n = 3\), i.e., \(RSR = SRS\), the category is called a Goursat category. Numerous and various characterizations and examples of these categories are given.
scientific article; zbMATH DE number 554359
scientific article