Solution to two problems on ideally conjunctive join-semilattices

In this paper, we solve two problems concerning the ideally conjunctive join-semilattices. First, we show that \(L/R^1({{\,\textrm{Id}\,}}L)|_L\) is ideally conjunctive for all join-semilattices L. Then we characterize those ideally conjunctive join-semilattices L such that \({{\,\textrm{coz}\,}}a\) is compact for all \(a\in L.\) Moreover, we give the definition of conjunctive posets and prove that the category of ideally conjunctive join-semilattices and join homomorphisms is reflective in the category of conjunctive posets and weakly ideal-continuous maps. As a corollary, we obtain the free ideally conjunctive join-semilattices over conjunctive posets.