Continuing this blog post: The set of all pointfree funcoids on upper semilattices with least elements is exactly a certain algebraic structure defined by propositional formulas.

Really just add the identities defining a pointfree funcoid to the identities of an upper semilattice with least element.

I will list the exact list of identities defining a pointfree funcoid on a poset with least element:

- $latex x \leq x$
- $latex x \leq y \land y \leq z \Rightarrow x \leq z$
- $latex x \leq y \land y \leq x \Rightarrow x=y$
- $latex \bot \leq y$
- $latex \lnot(\bot [f] y)$
- $latex \lnot(x [f] \bot)$
- $latex x \cup y \leq z \Leftrightarrow x \leq z \land y \leq z$
- $latex i\cup j[f]k \Leftrightarrow i[f]k\lor j[f]k$
- $latex k[f]i\cup j \Leftrightarrow k[f]i\lor k[f]j$

(Here $latex [f]$ is a single relational symbol.)

This can be even generalized for upper semilattices with no requirement to have the least element, if we allow the least element constant symbol to be a partial function.

Note that for the general case of a pointfree funcoid between two upper semilattices, we have a partial algebraic structure because the operations are different for each of the two upper semilattices.

I am curious what are applications of this curious fact.

There may be a (non-fatal) error in the above: It seems that the above bijectively correspond to such pointfree funcoids only in special cases like if our semilattice is a boolean algebra. The construction is interesting nevertheless!

The above can be generalized for arbitrary posets: Instead of the symbol $latex \bot$ use the unary predicate symbol $latex \mathrm{Least}(x)$ that “determines” if $latex x$ is the least element and use the definition of free star in terms of arbitrary posets, as in the definition of staroids. In fact this way we would define 2-staroids, not pointfree funcoids, but that’s also interesting. I am going to tell this in details in a later blog post.