  Given any cancellative monoid $\mathcal{M}$, we study the Hankel system
determined by its multiplication table. We prove that the Hankel system admits
self-absorption property provided that the monoid $\mathcal{M}$ has the local
algebraic structure: \[ \big(ax = by, cx=dy, az=bw \,\, \text{in
$\mathcal{M}$}\big)\Longrightarrow \big(cz=dw \,\, \text{in
$\mathcal{M}$}\big). \] Our result holds for all group-embeddable monoids and
goes beyond. In particular, it works for all cancellative Abelian monoids and
most common non-Abelian cancellative monoids such as $$
\mathrm{SL}_d(\mathbb{N}): = \big\{[a_{ij}]_{1\le i,j\le d}\in
\mathrm{SL}_d(\mathbb{Z})\big| a_{ij} \in \mathbb{N}\big\}. $$ The Hankel
system determined by the multiplication table of a monoid is further
generalized to that determined by level sets of any abstract two-variable map.
We introduce an algebraic notion of lunar maps and establish a stronger
hereditary self-absorption property for the corresponding generalized Hankel
systems. As a consequence, we prove the self-absorption property for arbitrary
spatial compression of the regular representation system
$\{\lambda_G(g)\}_{g\in G}$ of any discrete group $G$, as well as the Hankel
system $\{\Gamma_\ell^\Phi\}$ determined by the level sets of any rational map
of the form $\Phi(x,y)=a x^m + b y^n$ with $a,b,m,n\in \mathbb{Z}^*$: \[
\Gamma_\ell^{\Phi}(x, y)= \mathbf{1}(a x^m + b y^n= \ell), \quad x, y\in
\mathbb{N}^*, \, \ell\in \Phi (\mathbb{N}^*\times \mathbb{N}^*). \] The
self-absorption property is applied to the study of completely bounded Fourier
multipliers between Hardy spaces. Further applications are: i) exact complete
bounded norm of the Carleman embedding in any dimension; ii) mixed
Fourier-Schur multiplier inequalities with critical exponent $4/3$; iii)
failure of hyper-complete-contractivity for the Poisson semigroup.
