Let $(X, {\sqsubset}, {\circ}, {\ast})$ be a set $X$ with a strict partial order $\sqsubset$ and two partial binary operations $\circ$ and $\ast$ such that for any $a, b, c \in X$:
- $a \circ b$ and $a \ast b$ are defined iff $a \sqsubset b$ or $a \sqsupset b$.
- $a \sqsubset b \implies b \neq a \circ b \neq a \ast b \neq b \neq a = b \circ a = b \ast a$.
- $a \rhd (b \rhd' c) = (a \rhd b) \rhd' (a \rhd c)$, where ${\rhd}, {\rhd'} \in \{{\circ}, {\ast}\}$.
- $a \sqsubset b \implies c \rhd a \sqsubset c \rhd b$, where ${\rhd} \in \{{\circ}, {\ast}\}$.
Question. Are structures similar to $(X, {\sqsubset}, {\circ}, {\ast})$ studied or mentioned in the literature?
For example, take $X = \mathbb Z$ with a strict partial order $\sqsubset$, and two partial binary operations $\circ$ and $\ast$ defined as follows: \begin{gather*} a \sqsubset b \implies a \circ b = -(a \ast b); \\ 2 \circ 3 = 4 \sqsupset 1 \sqsubset 2 \sqsubset 3; \\ 1 \circ 2 = 5 \sqsubset 1 \circ 3 = 6; \\ 1 \circ 4 = 5 \circ 6 = 7. \end{gather*}
Both the structure and example are constructed empirically as an attempt to solve the problem of matching fans without use of so-called oracle in the context of optimal reduction of $\lambda$-expressions.
