$k$-furcus semigroups

Abstract

A bifurcus semigroup is a semigroup in which every nonunit nonatom can be written as the product of exactly two atoms. We generalize this notion to $k$-furcus semigroups: every element that can be factored as the product of at least $k$ nonunits can be factored as the product of exactly $k$ atoms. We compute some factorization-theoretic invariants of $k$-furcus semigroups that generalize the bifurcus results. We then define two variations on the $k$-furcus property, one stronger (presumabaly strictly) and the other strictly weaker than the $k$-furcus property.