Abstract FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents. We show these not sufficient, giving example satisfying the properties, such does have This settles a problem presented at Conference Workshop on General Algebra Its Applications in 2013 more recently workshop NCS 2018....

A design pattern realization consists of a cluster of classes that work together to solve a particular problem using a well known, named solution. Developers may build systems out of several pattern realizations, and these pattern realizations may be interconnected, or, in other words, coupled. Coupled pattern realizations may represent a reasonable solution to software design problems, however...