In the book "Combinatorial theory" by Martin Aigner (from 1979), the standard algebra of a poset is introduced as the subalgebra of the incidence algebra of a poset consisting of the functions that are constant on isomorphic intervals.

(It was answered in the comments by Darij Grinberg that standard algebras are called reduced incidence algebras today (which answered one original question), so I removed a part of the questions.)

Questions: Are the quiver and relations of a reduced incidence algebra of a poset known? Are there some examples in some textbooks for the algebraic structure of those algebras?