Geodesic
Generators and relations for the semigroups of increasing functions on $\mathbb N$ and $\mathbb Z$
Algebra and discrete mathematics, no. 4 (2005), pp. 1-15.

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The semigroups of all increasing functions over $\mathbb{N}$ and $\mathbb{Z}$ are considered. It is shown that both these semigroups do not admit an irreducible system of generators. In their subsemigroups of cofinite functions all irreducible systems of generators are described. The last semigroups are presented in terms of generators and relations.
Keywords: Order-preserving transformation, monotonic function, generating system, generators and relations, word problem. 
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     author = {Vadym Doroshenko},
     title = {Generators and relations for the semigroups of increasing functions on $\mathbb N$ and $\mathbb Z$},
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Vadym Doroshenko. Generators and relations for the semigroups of increasing functions on $\mathbb N$ and $\mathbb Z$. Algebra and discrete mathematics, no. 4 (2005), pp. 1-15. http://geodesic.mathdoc.fr/item/ADM_2005_4_a0/