Gould, Victoria Completely right pure monoids. (English) Zbl 0608.20049 Proc. R. Ir. Acad., Sect. A 87, No. 1, 73-82 (1987). 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. Cited in 5 Documents MSC: 20M50 Connections of semigroups with homological algebra and category theory 20M15 Mappings of semigroups 20M14 Commutative semigroups Keywords:right S-systems; completely right pure monoids; right ideals; right congruences; completely right injective monoids; union of groups PDFBibTeX XMLCite \textit{V. Gould}, Proc. R. Ir. Acad., Sect. A 87, No. 1, 73--82 (1987; Zbl 0608.20049)