A SPECIALIZATION semilattice is a join semilattice together with a coarser preorder ⊑,satisfying an appropriate compatibility condition. If 𝑋,is a topological space, then (P(𝑋, ), ∪, , ⊑, ) is a SPECIALIZATION semilattice, where 𝑥,⊑,𝑦,if 𝑥,⊆,𝐾, 𝑦, , for 𝑥, , 𝑦,⊆,𝑋, , and 𝐾,is closure. SPECIALIZATION semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In a former work we showed that every SPECIALIZATION semilattice can be embedded into the SPECIALIZATION semilattice associated to a topological space as above. Here we describe the universal embedding of a SPECIALIZATION semilattice into an additive closure semilattice.

Statement of Problem: Prosthetic treatments account for more than half of the all provided dental services, as well as the highest number of lawsuits in some countries. The quality of Prosthodontics postgraduate programs has profound effect on the improvement of provided patient health care and reduction of patient complaints. Goal: The goal of this study was to survey different universities around the globe for the duration and subjects of Prosthodontics specialty programs, the title of degree awarded and the programs that are offered in conjunction with other disciplines. Materials and Methods: This study surveyed Prosthodontics specialty programs offered by different universities available on their websites, and reviewed published research on this subject using search engines such as Google, Pubmed and Medline. The published Accreditation Standards for the program and minimum requirements for achieving competency after completion of the program was reviewed. Result: The results of survey showed that prosthodontics specialty is not recognized by all countries. In 42 universities surveyed, the duration of program ranged from two to four years. The title of awarded degree varied in different universities. In some universities, prosthodontics and other disciplines were integrated in one program. Few universities offered additional subjects like photography, geriatrics and pain in their educational curriculum. Conclusion: The core subjects of the programs seemed identical. However, programs differ in duration, title and extra subjects.