A proof of Lyndon's finite basis theorem

A new proof is given of the theorem, originally proved by R.C. Lyndon, that any two element algebra of finite similarity type has a finite basis for its equations. We also provide a new proof of a result of W. Taylor that any equational class generated by a two element algebraic system contains only a finite number of subdirectly irreducible members, each of which is finite. The original proofs of these two theorems relied on E.L. Post's classification of the two element algebraic systems. Our paper uses instead some recent results from universal algebra.