Title: Infinite Dimensional Algebras with no Simple Bases
Speaker: Pınar Aydoğdu (Hacettepe University, Turkey)
Abstract: Following {ALM}, a basis $B$ over an infinite dimensional $F$-algebra $A$ is amenable if for all $r\in A$, the set of the coordinate vectors of the family $\{rb|b\in B\}$ with respect to $B$ is summable. A basis $B$ is said to be congenial to a basis $C$ if the coordinate vectors of the elements of $B$ represented with respect to $C$ is summable. If $B$ is congenial to $C$ but $C$ is not congenial to $B$, then we say that $B$ is properly congenial to $C$. An amenable basis $B$ is called simple if it is not properly congenial to any other amenable basis. In {ALM}, the fundamental question whether all algebras have simple bases has been raised. In this work, using a construction inspired by that in {KS} and {OW}, we introduce a family of algebras granting us examples of algebras without simple bases and of one-sided simple bases. This is a joint work with Sergio R. L\'opez-Permouth and Rebin A. Muhammad.
Bibliography:
{ALM} L.M. Al-Essa, S.R. Lopez-Permouth and N.M. Muthana (2017): \emph{Modules over infinite dimensional algebras}, Linear and Multilinear Algebra, DOI:10.1080/03081087.2017.1301365.
{KS} A.V. Kelarev and O.V. Sokratova (2000): \emph{Syntactic Semigroups and Graph Algebras}, Bull. Austral. Math. Soc., 62, 471-477.
{OW} S. Oates-Wiliams (1984): \emph{On the variety generated by Murskii's algebra}, Algebra Universalis, 18, 175-177.
Seminar file: 2018-Feb 23 OSU-OU-RingTheorySeminar-pinar.pdf
