Semi-divisible residuated multilattices

Abstract In this paper, we investigate the notion of (semi)divisibility in the framework of residuated multilattices and determine all residuated multilattices of order seven. We start by proving that the natural extension of the notion of divisibility from residuated lattices to residuated multilattices is too stringent as it collapses the structure of residuated multilattice to that of residuated lattice. Weakening this notion yields the notion of semi-divisibility, which we show to present among pure residuated multilattices. We show that, of all the pure multilattices with seven elements, only one can be endowed with pocrim structures and indeed we prove that there are (up to isomorphism) six pocrim structures on this multilattice. Finally, we show that there are (up to isomorphism) two pure semi-divisible residuated multilattice with seven elements.