# 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)
• $\lnot(\bot [f] y)$
• $\lnot(x [f] \bot)$
• $i\cup j[f]k \Leftrightarrow i[f]k\lor j[f]k$
• $k[f]i\cup j \Leftrightarrow k[f]i\lor k[f]j$

What are applications of the curious fact that every funcoid is a structure defined by propositional formulas? I don’t yet know.