Range sets and partition sets in connection with congruences and algebraic invariants
Let \(S\) be a semigroup of transformations on a set \(X\). The authors define relations closed under inclusion on \(R(X)\), the set of ranges of mappings from \(S\). A new notion of connectedness for semigroups of transformations, called range-connectedness, is introduced. It is shown that equivalences closed under inclusion on \(R(X)\) and left-zero congruences on \(S\) are dually isomorphic iff \(S\) is range-connected. Further the relationship between right-zero congruences on \(S\) and equivalences closed under inclusion on the partition sets of \(S\) is studied. Partition-connected and partition-\(n\)-connected semigroups are introduced. These notions can be used to investigate an arbitrary semigroup \(S\) by considering faithful representations of \(S\) by semigroups of transformations.
scientific article; zbMATH DE number 438886
scientific article