There exist subdirectly irreducible commutative basic algebras of an arbitrary infinite cardinality which are not MV-algebras

Abstract

It was recently proved by P. Wojciechowski that for any infinite cardinal there exists a linearly ordered MV-algebra of this cardinality. Since basic algebras are a (non-associative) generalization of MV-algebras, there rises a natural question if this is true also for basic algebras which are not MV-algebras. Using the construction by P. Wojciechowski and the modified construction by the first author, we can set up certain defectors which enable us to prove the result of the title.