Minimal Presentations of Full Subsemigroups of ${\bf N}^2$

J.C. Rosales, P.A. García-Sánchez
2001 Rocky Mountain Journal of Mathematics  
We show that the cardinality of a minimal presentation for a two-dimensional full affine subsemigroup of N 2 minimally generated by p elements is , where G(S) denotes the subgroup of Z 2 spanned by S. In this paper we are going to assume that S is a full subsemigroup of N 2 such that rank (G(S)) = 2. (The case when rank (G(S)) ≤ 1 has no interest, because under this assumption with respect to the ordering a ≤ b if and only if b − a ∈ N 2 , then S is minimally generated by M . Furthermore, we
more » ... assume that the elements in M are ordered so that a 1 < a We define the map and denote its kernel congruence by σ. Clearly, S ∼ = N p /σ. We say that ρ is a minimal system of generators for σ if ρ generates σ and ρ has minimal cardinality among the generating systems of σ. It can be shown that #ρ ≥ p − 2 (see [5]). Given s ∈ S \{0}, we define the graph G s as the graph whose vertices are
doi:10.1216/rmjm/1021249446 fatcat:pp4uovwhkbegvg6gnzrybbi2va