The isomorphism problem for graph magma algebras

Abstract

(One-value) graph magma algebras are algebras having a basis $\mathcal{B}=V\cup \left\{1\right\}$ such that, for all $u,v\in V,uv\in \{u,0\}.$ Such bases induce graphs and, conversely, certain types of graphs induce graph magma algebras. The equivalence relation on graphs that induce isomorphic magma algebras is fully characterized for the class of associative graphs having only finitely many non-null connected components. In the process, the ring-theoretic structure of the magma algebras induced by those graphs is given as it is shown that they are precisely those graph magma algebras that are semiperfect as rings. A complete description of the semiperfect rings that arise in this fashion, in ring theoretic and linear algebra terms, is also given. In particular, the precise number of isomorphism classes of one-value magma algebras of dimension n is shown to be $\sum _{j\le n}p(j)$ where, for any $i\in {\mathit{Z}}^{+},$ p(i) is the number of partitions of i. While it is unknown whether uncountable dimensional algebras always have amenable bases, it is shown here that graph magma algebras do.

Keywords:

• Basic rings
• contracted semigroup algebras: graph magmas
• isomorphism problem
• semiperfect rings
• semiprimary rings

2020 Mathematics Subject Classification:

• Primary: 16L30
• Secondary: 16S99

Acknowledgment

We wish to thank the anonymous referee for a careful analysis of our paper. Their insightful and helpful comments were instrumental in getting a much-improved final version.