Abstract
The construction of a wreath product of monoids with small categories is a generalization of wreath products of monoids, and of endomorphism monoids of free, projective, and even arbitrary acts over monoids. In this paper, we give a complete description of wreath products of monoids with small symmetric categories whose principal right (left) ideals from a tree with respect to inclusion. Such tree conditions are fulfilled by monoids whose (finitely generated) right (left) ideals are projective or injective. In particular, the characterization of (semi)hereditary endomorphism monoids of projective acts becomes a special case of the results obtained here.
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