Distributive Algebras, Isoclinism, and Invariant Probabilities
Stephen Buckley · Contemporary mathematics - American Mathematical Society · 2015
We develop a basic theory of distributive algebras, a certain class of universal algebras that generalize the class of (associative and nonassociative) rings. We then define and investigate isoclinism for distributive algebras—this is an equivalence relation among distributive algebras of a particular type—and we relate isoclinism to ring theory via isologism with respect to varieties of (possibly nonassociative) rings. Associated with any given ring variety is a map from rings to distributive algebras of a particular type, and we say that rings are isologic with respect to this variety if the associated distributive algebras are isoclinic. Certain probability functions on finite distributive algebras are invariant under isoclinism. These invariants allow us to derive some combinatorial consequences in ring theory by using an appropriate isologism.