Notes on TQFT wire models and coherence equations for triangular cells.
For every finite Abelian group Γ and for all , if there exists a solution of the equation in non-negative integers , where are positive integers, then the number of such solutions is estimated from below in the best possible way.
We study numerical semigroups with the property that if is the multiplicity of and is the least element of congruent with modulo , then . The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups.