Abbreviation: DLO

A \emph{distributive lattice with operators} is a structure $\mathbf{A}=\langle A,\vee,\wedge,f_i\ (i\in I)\rangle$ such that

$\langle A,\vee,\wedge\rangle$ is a distributive lattice

$f_i$ is \emph{join-preserving} in each argument: $f_i(\ldots,x\vee y,\ldots)=f_i(\ldots,x,\ldots)\vee f_i(\ldots,y,\ldots)$

Let $\mathbf{A}$ and $\mathbf{B}$ be distributive lattices with operators of the same signature. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a distributive lattice homomorphism and preserves all the operators:

$h(f_i(x_0,\ldots,x_{n-1}))=f_i(h(x_0),\ldots,h(x_{n-1}))$

Example 1:

Bounded distributive lattices with operators

Distributive lattice-ordered semigroups

Distributive lattices