An example of a commutative basic algebra which is not an MV-algebra

Many algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval 0, 1] of the reals has to be an MV-algebra.