Multiple constraints on three and four words

In this paper we consider several questions related to the defect theorem for sets of three and four words. We start by investigating how large systems of pairwise independent or pairwise non-equivalent equations over three unknowns possessing purely non-periodic solutions can be. In other words, we analyze how weak the cumulative defect effect of such systems is. Then, we investigate the maximal size of chains of equations over three or four words such that every time we add a new equation the set of solutions strictly decreases.