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.