  We consider a tower of generalized rook monoid algebras over the field
$\mathbb{C}$ of complex numbers and observe that the Bratteli diagram
associated to this tower is a simple graph. We construct simple modules and
describe Jucys-Murphy elements for generalized rook monoid algebras.
  Over an algebraically closed field $\Bbbk$ of positive characteristic $p$,
utilizing Jucys-Murphy elements of rook monoid algebras, for $0\leq i\leq p-1$,
we define the corresponding $i$-restriction and $i$-induction functors along
with two extra functors. On the direct sum $\mathcal{G}_{\mathbb{C}}$ of the
Grothendieck groups of module categories over rook monoid algebras over
$\Bbbk$, these functors induce an action of the tensor product of the universal
enveloping algebra $U(\widehat{\mathfrak{sl}}_p(\mathbb{C}))$ and the monoid
algebra $\mathbb{C}[\mathcal{B}]$ of the bicyclic monoid $\mathcal{B}$.
Furthermore, we prove that $\mathcal{G}_{\mathbb{C}}$ is isomorphic to the
tensor product of the Fock space representation of
$U(\widehat{\mathfrak{sl}}_{p}(\mathbb{C}))$ and the unique
infinite-dimensional simple module over $\mathbb{C}[\mathcal{B}]$, and also
exhibit that $\mathcal{G}_{\mathbb{C}}$ is a bialgebra. Under some natural
restrictions on the characteristic of $\Bbbk$, we outline the corresponding
result for generalized rook monoids.
