Every element with finite extent
in a meet-continuous semilattice with complete chains is the meet of a finite number
of meet irreducibles. This includes both semilattices with the ascending chain
condition and compact topological semilattices with finite breadth. By applying this
decomposition to topological lattices on an n-cell, the following results are obtained:
If L and M are topological lattices on n and m-cells respectively and there is an
order isomorphism between the boundaries of L and M, then L and M
are homeomorphic. If, in addition, L and M are distributive, L and M are
isomorphic.
