introduced. Its defining principle is that universally and existentially

quantified variables may not occur together in atoms. SF properly generalizes

both the Bernays-Sch\\"onfinkel-Ramsey (BSR) fragment and the relational monadic

fragment. In this paper the restrictions on variable occurrences in SF

sentences are relaxed such that universally and existentially quantified

variables may occur together in the same atom under certain conditions. Still,

satisfiability can be decided. This result is established in two ways: firstly,

by an effective equivalence-preserving translation into the BSR fragment, and,

secondly, by a model-theoretic argument.

Slight modifications to the described concepts facilitate the definition of

other decidable classes of first-order sentences. The paper presents a second

fragment which is novel, has a decidable satisfiability problem, and properly

contains the Ackermann fragment and---once more---the relational monadic

fragment. The definition is again characterized by restrictions on the

occurrences of variables in atoms. More precisely, after certain

transformations, Skolemization yields only unary functions and constants, and

every atom contains at most one universally quantified variable. An effective

satisfiability-preserving translation into the monadic fragment is devised and

employed to prove decidability of the associated satisfiability problem.

},\n}\n'