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 $latex f$ is defined on a lattice (or semilattice) with a least element $latex \bot$, 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 $latex [f]$
  • 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)
  • $latex \lnot(\bot [f] y)$
  • $latex \lnot(x [f] \bot)$
  • $latex i\cup j[f]k \Leftrightarrow i[f]k\lor j[f]k$
  • $latex k[f]i\cup j \Leftrightarrow k[f]i\lor k[f]j$

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

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