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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.

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  • $\begingroup$ Some motivation, and some example would be very welcome. $\endgroup$ – YCor Jul 7 at 8:35
  • $\begingroup$ 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. $\endgroup$ – Anton Salikhmetov Jul 7 at 10:05
  • $\begingroup$ I don't understand what you mean by parenthesis-free products such as $c \rhd a \sqsubset c \rhd b$ or $4 \sqsupset 1 \sqsubset 2 \sqsubset 3$. $\endgroup$ – YCor Jul 7 at 10:07
  • $\begingroup$ My intention was that $\sqsubset$ would denote the partial order, not an operation. $\endgroup$ – Anton Salikhmetov Jul 7 at 10:13

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