Funcoid is a “Structure” in the Sense of Math Logic

A few seconds ago I realized that certain cases of pointfree funcoids can be described as a structure in the sense of mathematical logic, that is as a finite set of operations and relational symbols.

Precisely, if a pointfree funcoid $f$ is defined on a lattice (or semilattice) with a least element $\bot$ , then because a lattice (semilattice) is a relational structure defined by a set of propositional formula, then we have the following operations and relations:

relation $[f]$
meet and join operations on our lattice
the least element of the lattice (a constant symbol)

and identities:

the standard identities of lattice considered as an algebraic structure (and also the obvious identities for the least element)
$\lnot(\bot [f] y)$
$\lnot(x [f] \bot)$
$i\cup j[f]k \Leftrightarrow i[f]k\lor j[f]k$
$k[f]i\cup j \Leftrightarrow k[f]i\lor k[f]j$