Numbered Distributive Semilattices

In this article, we consider several definitions of a Lachlan semilattice; i. e., a semilattice isomorphic to a principal ideal of the semilattice of computably enumerable $m$-degrees. We also answer a series of questions on constructive posets and prove that each distributive semilattice with top and bottom is a Lachlan semilattice if it admits a $\Sigma^0_3$-representation as an algebra but need not be a Lachlan semilattice if it admits a $\Sigma^0_3$-representation as a poset. The examples are constructed of distributive lattices that are constructivizable as posets but not constructivizable as join (meet) semilattices. We also prove that every locally lattice poset (in particular, every lattice and every distributive semilattice) possessing a $\Delta^0_2$-representation is positive.