Minimal But Inefficient Presentations for Self Semidirect Products of the Free Abelian Monoid on Two Generators

Abstract

Let A and K both be copies of the free Abelian monoid on two generators. For any connecting monoid homomorphism θ: A → End(K), let M = K ⋊_θ A be the corresponding monoid semidirect product. In Çevik (2005), it has been proved necessary and sufficient conditions for the efficiency of a standard presentation for M in terms of the matrix representation for θ. As a main result of this article, we give sufficient conditions for this presentation of M to be minimal but not efficient. To do that we will use the same method as given in Çevik (2003b).

Key Words:

• Efficiency
• Free Abelian monoids
• Minimality
• Semidirect products

2000 Mathematics Subject Classification:

• 20L05
• 20M05
• 20M15
• 20M50
• 20M99

ACKNOWLEDGMENT

I am very grateful to the referee for the kind help in improving the presentation of this work.

Notes

Communicated by A. Yu. Olshanskii.