Factorization under local finiteness conditions

It has been recently observed that fundamental aspects of the classical theory factorization can be greatly generalized by combining languages monoids and preorders. This led to various theorems on existence certain factorizations, herein called ⪯-factorizations, for ⪯-non-units a (multiplicatively written) monoid H endowed with preorder ⪯, where an element u∈H is ⪯-unit if u⪯1H⪯u ⪯-non-unit otherwise. The “building blocks” these factorizations are ⪯-irreducibles (i.e., a∈H cannot written as product two each which strictly ⪯-smaller than a); it interesting look sufficient conditions ⪯-factorizations bounded in length or finite number (if measured counted suitable way). precisely kind questions addressed present work, whose main novelty study interaction between minimal refinement used counter “blow-up phenomena” inherent non-commutative non-cancellative monoids) some finiteness describing “local behavior” pair (H,⪯). Besides examples remarks, paper includes many arithmetic results, part new already basic case ⪯ divisibility (and hence setup theory).