The corner element of generalized numerical semigroups

TL;DR

This paper introduces the corner element concept for generalized numerical semigroups, extending the conductor idea to higher dimensions, and provides properties, algorithms, and bounds related to this new invariant.

Contribution

It defines the corner element for generalized numerical semigroups, explores its properties, and develops algorithms and bounds for fixed corner elements.

Findings

Properties of the corner element established

Algorithm for enumerating semigroups with fixed corner

Bounds on the number of semigroups with given corner

Abstract

In this paper we introduce the concept of corner element of a generalized numerical semigroup, which extends in a sense the idea of conductor of a numerical semigroup to generalized numerical semigroups in higher dimensions. We present properties of this new notion and its relations with existing invariants in the literature, and provide an algorithm to compute all the generalized numerical semigroups with fixed corner. Besides that, we provide lower and upper bounds on the number of generalized numerical semigroups having a fixed corner element.