The isomorphism problem for graph magma algebras

Abstract

(One-value) graph magma algebras are algebras having a basis B = V boolean OR {1} such that, for all u,v is an element of V, uv is an element of {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-the-oretic 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 Sigma(j <= n) p(j) where, for any i is an element of 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.