This is the seventh post in the Cartesian frames sequence.

Here, we introduce three new binary operations on Cartesian frames, and discuss their properties.

1. Tensor

Our first multiplicative operation is the tensor product, $\otimes$ .

One way we can visualize our additive operations from before, $\oplus$ and $&$ , is to imagine two robots (say, a mining robot $Agent (C)$ and a drilling robot $Agent (D)$ ) that have an override mode allowing an AI supervisor to take over that robot's decisions.

$C \oplus D$ represents the supervisor deciding which robot to take control of, then selecting that robot's action. (The other robot continues to run autonomously.)
$C & D$ represents something in the supervisor's environment (e.g., its human operator) deciding which robot the supervisor will take control of. Then the supervisor selects that robot's action (while the other robot runs autonomously).

$C \otimes D$ represents an AI supervisor that controls both robots simultaneously. This lets $Agent (C \otimes D)$ direct $Agent (C)$ and $Agent (D)$ to work together as a team.

Definition: Let $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ be Cartesian frames over $W$ . The tensor product of $C$ and $D$ , written $C \otimes D$ , is given by $C \otimes D = (A \times B, hom (C, D^{*}), ⋄)$ , where $hom (C, D^{*})$ is the set of morphisms $(g, h) : C \to D^{*}$ (i.e., the set of all pairs $(g : A \to F, h : B \to E)$ such that $b ⋆ g (a) = a \cdot h (b)$ for all $a \in A$ , $b \in B$ ), and $⋄$ is given by $(a, b) ⋄ (g, h) = b ⋆ g (a) = a \cdot h (b)$ .

Let us meditate for a moment on why this definition represents two agents working together on a team. The following will be very informal.

Let Alice be an agent with Cartesian frame $C = (A, E, \cdot)$ , and let Bob be an agent with Cartesian frame $D = (B, F, ⋆)$ . The team consisting of Alice and Bob should have agent $A \times B$ , since the team's choices consist of deciding what Alice does and also deciding what Bob does.

The environment is a bit more complicated. Starting from Alice, to construct the team, we want to internalize Bob's choices: instead of just being choices in $A$ 's environment, Bob's choices will now be additional options for the team $A \times B$ .

To do this, we want to first see Bob as being embedded in Alice's environment. This embedding is given by a function $h : B \to E$ , which extends each $b \in B$ to a full environment $e \in E$ ....