×

Completely right pure monoids. (English) Zbl 0608.20049

A monoid S is completely right pure if all its right S-systems are absolutely pure; we show that this is equivalent to all right S-systems satisfying a weaker notion of purity that we call almost pure. This approach produces a new characterization of completely right pure monoids in terms of right ideals and right congruences that is analogous to a characterization of completely right injective monoids given by several authors. Using this result we prove that a monoid S is a completely right pure union of groups if and only if it has local left zeros and is such that for each finitely generated right ideal I and each finite subset A of \(S\setminus I\), there is an idempotent generator of I that commutes with all elements of A.

MSC:

20M50 Connections of semigroups with homological algebra and category theory
20M15 Mappings of semigroups
20M14 Commutative semigroups
PDFBibTeX XMLCite