$S$-acts over a well-ordered monoid with modular congruence lattice

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$-act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$-acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M. S. Kazak, which describes $S$-acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of $S$-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.