  The stable model (SM) semantics lacks the properties of existence, relevance
and cumulativity. If we prospectively consider the class of conservative
extensions of SM semantics (i.e., semantics that for each normal logic program
P retrieve a superset of the set of stable models of P), one may wander how do
the semantics of this class behave in what concerns the aforementioned
properties. That is the type of issue dealt with in this paper. We define a
large class of conservative extensions of the SM semantics, dubbed affix stable
model semantics, ASM, and study the above referred properties into two
non-disjoint subfamilies of the class ASM, here dubbed ASMh and ASMm. From this
study a number of results stem which facilitate the assessment of semantics in
the class ASMh U ASMm with respect to the properties of existence, relevance
and cumulativity, whilst unveiling relations among these properties. As a
result of the approach taken in our work, light is shed on the characterization
of the SM semantics, as we show that the properties of (lack of) existence and
(lack of) cautious monotony are equivalent, which opposes statements on this
issue that may be found in the literature; we also characterize the relevance
failure of SM semantics in a more clear way than usually stated in the
literature.
