Join-irreducible clones of multiple valued logic algebra

We study a problem of representation for the lattice of clones of multiple valued logic (MVL) functions. It is a problem of description of a generating system of clones, from which the whole lattice, or a given sublattice, can be reconstructed by synthesis. Here, the “synthetic means” considered is the join operation (V) for lattice elements. The generating set of clones is defined to be the set of all join irreducible elements. All atoms of the lattice are among them, which “automatically” makes the problem difficult, since finding all atoms is already a hard problem. We give a criterion for a clone to be join irreducible. The problem is easier for the particular case of transformation monoids, i.e. for the sublattice of clones that contain only essentially unary functions. We find constructive criteria for monoids, that are join irreducible, and particularly, for atoms. We show a graph theoretical property of the lattice, which in general case is different from that of the binary case. Finally, we see that any one variable function of k valued logic, with k<5, generates a join irreducible clone