A set of integers is said to be symmetric if there is an integer g such that the mapping on β«ήβ¬ given by n Β¬ g y n carries the set bijectively onto its complement in β«.ήβ¬ A numerical monoid A is a submonoid of the additive Γ 4 monoid of natural numbers β«ήβ¬ s 0, 1, 2, . . . whose complement β«ήβ¬ y A is finite. The type of a numerical monoid A is a positive integer that provides a measure of the deviation of A from symmetry. So, for example, symmetric numerical monoids are characterized as those whose type equals one. Numerical monoids arise as the value semigroups 1 of one-dimensional analytically-irreducible Noetherian local rings R for which the residue class field of R and the residue class field of its integral closure R coincide, and the symmetry properties of the value semigroups play an important role in the study of these rings. For instance, R will be a w x Gorenstein ring if and only if its value semigroup is symmetric K and, more generally, the CohenαMacaulay type of R is linked to the type of its 1 The value-semigroup is the image of R under the valuation from the integral closure of R into β«.ήβ¬ In this paper we are careful to refer to semigroups which contain 0 as monoidsαan exception is made here due to the prevalent usage of the term ''value semigroup.'' 636