- 原书名：Monoids, Acts and Categories
Ⅰ Elementary properties of monoids，acts and categories
1 Sets and relations
2 Groupoids，semigroups and monoids
3 Some classes ofsemigroups
4 Acts over monoids（monoid automata）
5 Decompositions and components
1 Products and coproducts
2 Pullbacks and pushouts
3 Free objects and generators
4 Cofree objects and cogenerators
6 Wreath products of monoids and acts
7 The wreath product of a monoid with a small category
Ⅲ Classes of acts
For finitely generated and arbitrary right ideals we present some results without proofs.Similarly to the case of rings we use the following
Definition 11.17.A monoid S is called rt：ght（8emi—）hereditary if all（finitely generated）right ideals of S are projective.
Theorem 11.18（Kilp（KIL71l，Dorofegva（Dor72））.A monoid S is right semihereditary if and only if S is right PP and all its incomparable principal right ideals a're disjoint.
Theorem 11.19（Kilp（KIL71），Dorofeeva（Dor72））.A mon.oid S is right hereditary if and only if S is right semihereditary and satisjies the ascending chain condition for right ideals.
Theorem 11.20（Dorofeeva（Dor72））.For a monoid S the following condi tions are equivalent：
（i）S is right（semi—）hereditary.
（ⅱ）All factor acts of（fg—）weakly injective right S—acts are（fg—）weakly injec tive.
（iii）All factor acts of injective right S—acts are（fg—）weakly in，jective.
Applications to endomorphism monoids of acts
Without proofs we give some results connecting hereditarity of endomorphism monoids of free，projective and also injective S—acts to hereditarity of S itself.