Order structure of U-semiabundant semigroups and rings. Part II: two-sided Lawson’s order

In Part II, we deal with the “two-sided” Lawson order, which is the intersection of his orders ⩽l and ⩽r, on U-semiabundant semigroups (presented as certain biunary semigroups). It is shown that, to a great extent, the order structure of these semigroups is determined by that of their set of projections. Our main topics of interest are existence of meets in such semigroups and rings and the possible lattice structure of their lower sections. In particular, every lower section of a U-semiabundant ring is shown, under certain simple assumptions, to be an orthomodular lattice, and explicit descriptions of the corresponding lattice operations and orthocomplementation are given.