I've discovered a parametric structure that can be used to create models of certain logical systems. The idea is simple, so I'm probably not the first one to invent it. Please help me identify the official name of this structure if it has one. Forgive me If I'm not precise with terminology.
The structure is characterized by the following properties:
- A base set with total order, $min$, $max$, $0$ element and negation $-$. For the sake of simplicity I will assume the integer numbers $\mathbb{Z}$.
- Three operations, conjunction $\wedge _{s}$, disjunction $\vee _{t}$ where $s, t \in \mathbb{Z}$, and negation $\sim$. They are defined as follows:
$$ a \wedge _{s} b = \left\{\begin{matrix} a & \text{if $a \leq s$} \\ min(a,b) & \text{otherwise} \end{matrix}\right. $$
$$ a \vee _{t} b = \left\{\begin{matrix} a & \text{if $a \geq t$} \\ max(a,b) & \text{otherwise} \end{matrix}\right. $$
$$ \sim a = -a $$
If we constrain ourselves to the set $\left \{ -l,l \right \}$ where $l \in \mathbb{N}$ and choose $u < -l$ and $v > l$ such that $u,v \in \mathbb{Z}$, we get a Boolean algebra, which is a model of propositional logic:
$$ \wedge = \wedge _{u} \quad \vee = \vee _{v} \quad \neg = \sim \quad \mathbf{1} = l \quad \mathbf{0} = -l $$
If we instead choose $-l \leq u < l$ and $-l < v \leq l$, we get a model of shortcut logic, similar to the && and || operators found in some programming languages. Other applications are interesting too, for example I implemented backtracking using it in my experimental programming language.
Before I go and write an article on this structure to clarify the background of my programming language, I'd like to know if it has been studied, so I don't have to reinvent the wheel.