“…Some of these generalizations are equivalent (see the discussion in Fleischer [8]), but they fall into two groups depending whether we ask the projections of a subdirect product just to be onto on elements of the domain as in [6,10,12], or we ask them to be full also, that is, onto on atomic relations as in [3,13,15]. The first type of subdirectly irreducible structures form a subclass of the irreducibles of the second type; usually there is a tradeoff between simpler decompositions in the first case and tighter decompositions in the second.…”