A subresiduated lattice ordered commutative monoid (srl-monoid for short) is a pair \(({\textbf {A}},Q)\) where \({\textbf {A}}=(A,\wedge ,\vee ,\cdot ,e)\) is a lattice ordered monoid and Q is a subalgebra of A such that for each \(a,b\in A\) the set \(\{q \in Q: a \cdot q \le b\}\) has maximum, which will be denoted by \(a\rightarrow b\) . The srl-monoids can be regarded as algebras \((A,\wedge ,\vee ,\cdot ,\rightarrow ,e)\) of type (2, 2, 2, 2, 0). This class of algebras is a variety which properly contains the varieties of commutative residuated lattices and subresiduated lattices respectively. In this paper, we study some aspects of the lattice of congruences of any srl-monoid, which is order isomorphic to the lattice of its strongly convex subalgebras. As application of this study we characterize simple and subdirectly irreducible algebras, we prove that every srl-monoid has the congruence extension property and we describe compatible functions. Then we focus our attention on a family of compatible functions, which will be called monotone modal operators, and considering as monotone modal operator the identity map, we introduce and study the variety of strong srl-monoids and some of its subvarieties.
