Algebras from diagrams (Q1093710)

The notion of algebra diagram \({\mathcal R}={\mathcal P}_ 1\dot\cup...\dot\cup {\mathcal P}_ n\) is introduced which is closely connected with semigroup algebras \({\mathcal S}=\cup_{ij}e_ i{\mathcal S}e_ j\) (the last is a finite semigroup with zero and orthogonal idempotents \(e_ 1,...,e_ n)\) as well as with quivers. For every field K the semigroup algebra \(R=K{\mathcal R}\) is a basic K-algebra with basic set of idempotents \(e_ 1,...,e_ n\) and \({\mathcal R}\) is the diagram for \({}_ RR.\) A method of construction of algebras corresponding to the diagram \({\mathcal R}\) is shown under different restrictions on \({\mathcal R}\). Varying conditions on \({\mathcal R}\) one can obtain algebras \(K{\mathcal R}\) with given properties. In particular, in such a way serial algebras and zero relation algebras are obtained.