Monoids in the fundamental groups of the complement of logarithmic free divisors in C3

We study monoids generated by certain Zariski–van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C3 listed by Sekiguchi. These monoids admit positive homogeneous presentations (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaußian and, hence, are neither Garside nor Artin (Theorem 2). However, we introduce the concept of fundamental elements for positive homogeneously presented monoids, and we show that all 17 monoids possess fundamental elements (Theorem 3). As an application of the study of monoids, we solve some decision problems for the fundamental groups except in three cases.