## The Monotone Catenary Degree of Block Monoids

##### Abstract

The block monoid B(G) of a finite abelian group G is the set of zero-sum sequences g1 · · · gn
such that
!n
i=1
gi = 0 with operation given by concatenation. A factorization z = α1 · · · · · αn
of length |z| = n of α ∈ B(G) is a formal product of n zero-sum sequences each containg
no proper zero-sum subsequences. The monotone catenary degree, cmon(G), is the smallest
m ∈ N0 ∪ {∞} such that for each α ∈ B(G) and every pair of factorizations z, z′ of α with
|z| ≤ |z′|, there are factorizations z = z0, z1, . . . , zk = z′ of α such that for each i ∈ [0, k − 1]
|zi| ≤ |zi+1| and zi+1 is constructed by replacing at most m atoms of zi with at most m new
atoms.
Recently, Geroldinger and Yuan provided bounds for cmon(G) leaving open precise bounds
when G is one of the following groups:
Zn, Z32, Z42
, Z23
, Z33
, Z43
, Z53
, Z2 ⊕ Z4, Z2 ⊕ Z6.
We calculate cmon(Z2⊕Z2) and cmon(Z4) and introduce results giving bounds for cmon(B(G)).
We investigate cmon(Z2⊕Z4) using a recursive argument for computing the adjacent catenary
degree.