# Posets with two partial (self-)distributive operations

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 $$a, b, c \in X$$:

1. $$a \circ b$$ and $$a \ast b$$ are defined iff $$a \sqsubset b$$ or $$a \sqsupset b$$.
2. $$a \sqsubset b \implies b \neq a \circ b \neq a \ast b \neq b \neq a = b \circ a = b \ast a$$.
3. $$a \rhd (b \rhd' c) = (a \rhd b) \rhd' (a \rhd c)$$, where $${\rhd}, {\rhd'} \in \{{\circ}, {\ast}\}$$.
4. $$a \sqsubset b \implies c \rhd a \sqsubset c \rhd b$$, 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.

• Some motivation, and some example would be very welcome. – YCor Jul 7 at 8:35
• Added (a sketch of) an example and some background, although I doubt the latter is very useful in this case. I hope to find more examples of such and similar structures and their properties in order to be able to implement them in software. – Anton Salikhmetov Jul 7 at 10:05
• I don't understand what you mean by parenthesis-free products such as $c \rhd a \sqsubset c \rhd b$ or $4 \sqsupset 1 \sqsubset 2 \sqsubset 3$. – YCor Jul 7 at 10:07
• My intention was that $\sqsubset$ would denote the partial order, not an operation. – Anton Salikhmetov Jul 7 at 10:13