In the context of a research project in applied computer science I've come across an algebraic structure that I would informally describe as a "module over a boolean algebra," although it is not actually a module. This structure consists of a boolean algebra $B$ together with a set $V$ and two operations $\odot$ and $\oplus$ with the following signatures:
$$ \odot : B \times V \to V \\ \oplus : V \times V \to V $$
Additionally, I believe the structure should satisfy the following laws:
- $(V, \oplus)$ is a commutative monoid. We denote its identity by $\textbf{0} \in V$.
- For any $v \in V$ we have $0 \odot v = \textbf{0}$
- For any $v \in V$ we have $1 \odot v = v$
- For any $a, b \in B$ and $v \in V$ we have $a \odot (b \odot v) = (a \land b) \odot v$
- For any $a, b \in B$ and $v \in V$ we have $(a \lor b) \odot v = (a \odot v) \oplus (b \odot v)$
- For any $a \in B$ and $u, v \in V$ we have $a \odot (u \oplus v) = (a \odot u) \oplus (a \odot v)$
In my project I am working with particular concrete examples of this structure, and the laws above are largely reverse-engineered from those particular examples. I am therefore interested in any similar structures you know of; the laws do not need to be an exact match.
Does anyone know of a standard name for an algebraic structure like this?
Edit: The example as it arises in my project is essentially the following. Fix some $n \in \mathbb{N}$ and let $B$ be the boolean algebra of functions $\{0, \ldots, n\} \to \{0, 1\}$. I then have some set $S$ such that $V$ is given by the set of functions $S \to B$. The operations $\odot$ and $\oplus$ are then defined by pointwise $\land$ and pointwise $\lor$, respectively:
$$(a \odot f)(s) = (a \land f(s)) \\ (f \oplus g)(s) = (f(s) \lor g(s)) $$
I understand that this example as formulated above is essentially just the boolean algebra of functions $\{0, \ldots, n\} \times S \to \{0, 1\}$. However, in my actual application I do not directly manipulate explicit representations of these functions, but instead work with compact implicit representations. I became interested in this structure as a way to keep track of the properties that these implicit representations are supposed to satisfy.
Edit: Using the notation of a semimodule, with $\oplus$ represented by $+$ and $\odot$ represented by juxtaposition, the axioms above are basically just the axioms of a semimodule over $B$:
- $(V, +)$ is a commutative monoid. We denote its identity by $0_V \in V$.
- For any $v \in V$ we have $0 v = 0_V$
- For any $v \in V$ we have $1 v = v$
- For any $a, b \in B$ and $v \in V$ we have $a (b v) = (a \land b) v$
- For any $a, b \in B$ and $v \in V$ we have $(a \lor b) v = a v + b v$
- For any $a \in B$ and $u, v \in V$ we have $a (u + v) = a u + a v$
I am keeping the $\land$ and $\lor$ notation because $+$ on a boolean algebra can be ambiguous between "or" and "xor."
A module over a Boolean algebra is a special case of a semimodule over a semiring. For this reason, I strongly encourage you to drop your symbols $\odot$, $\oplus$, $\vee$ and $\wedge$, and rewrite your axioms as given in a much simpler way in the linked page.
EDIT (answer to the comments). There is a large literature on idempotent semirings, also called dioids (see for instance Gunawardena, An introduction to Idempotency ), with plenty of applications in various domains. However, apart from the two-element Boolean algebra, these idempotent semirings are rarely Boolean algebras and I am not aware of a specific term for a semimodule over a Boolean algebra.
(2) Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets, so I tried to look at semimodules over $\mathcal{P}(S)$ for some set $S$. However, since the negation operation is not used in the definition of a semimodule, exploiting the fact that the set of "scalars" is a Boolean algebra is not clear to me. That being said, there are plenty of ways to extend the structure of a field of sets, in particular relational structures, and this might be a source of inspiration.