Lattice Paths and Submonoids of $$\mathbb Z^2$$ Z 2

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set $$X\subseteq \mathbb Z^2$$ , there are two naturally associated monoids: $$\mathscr {F}_X$$ , the monoid of all X-walks/paths; and $$\mathscr {A}_X$$ , the monoid of all endpoints of X-walks starting from the origin O. For each  $${A\in \mathscr {A}_X}$$ , write $$\pi _X(A)$$ for the number of X-walks from O to A. Calculating the numbers  $$\pi _X(A)$$ is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schroder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers $$\pi _X(A)$$ , geometrical properties of the step set X, algebraic properties of the monoid  $$\mathscr {A}_X$$ , and combinatorial properties of a certain bi-labelled digraph naturally associated to X. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results.