Positioned Numerical Semigroups with Small Gender

An M -semigroup (respectively an N -semigroup) is a positioned numerical semigroup S , such that g ( S ) = F ( S ) + 3 2 (respectively, g ( S ) = F ( S ) + 4 2 ). In this paper, we describe and characterize this class of semigroups; in particular, we show that the type of an M -semigroup (respectively an N -semigroup) is equal to 2 or 3 (respectively 2, 3 or 4). Moreover, we give algorithms for computing the whole set of M -semigroups and N -semigroups.