It is easy to think up interesting, natural models of the algebraic theory presented as follows, such that in these models, $x^\dagger$ is always the multiplicative inverse of $x$.

Question.What are some interesting, natural models in which $x^\dagger$ is not necessarily the multiplicative inverse?

**Sorts.** $U$

**Functions.** $$\wedge : U \times U \rightarrow U, \qquad \vee : U \times U \rightarrow U$$

$$1 : U, \qquad \times : U \times U \rightarrow U, \qquad (x \mapsto x^{\dagger}) : U \rightarrow U$$

**Axioms.** (I write $\times$ concatenatively).

- $(U,\wedge,\vee)$ is a distributive lattice
- $(U,1,\times)$ is a commutative monoid
- Multiplication distributes over both $\wedge$ and $\vee$
- $1^\dagger = 1$
- $(xy)^\dagger = x^\dagger y^\dagger$
- $x^{\dagger\dagger} = x$
- $(x \wedge y)^\dagger = x^\dagger \vee y^\dagger$
- $(x \vee y)^\dagger = x^\dagger \wedge y^\dagger$

**Examples of such things in which $x^\dagger$ is always the multiplicative inverse.**

$\mathbb{Q}_{>0}$ or $\mathbb{R}_{>0}$ with $\wedge$ and $\vee$ interpreted as $\mathrm{min}$ and $\mathrm{max}$ respectively.

$\mathbb{Z}$ with $(1,\times,x \mapsto x^\dagger)$ interpreted as $(0,+,x \mapsto -x)$, and $\wedge$ and $\vee$ interpreted as $\mathrm{min}$ and $\mathrm{max}$ respectively.

Cartesian products of these