Finding MV algebras with Sage

asked 2020-10-12 12:59:47 -0500

Bern gravatar image

updated 2020-10-12 13:52:48 -0500

slelievre gravatar image

I would like to find with Sage in a quick way all finite lattices having a partial operation $x\cdot y$, defined for $x\geq y$, satisfying the following properties:

(a) $\forall x\geq y\geq z:\ x\cdot z \leq x\cdot y$ and $(x\cdot y)\cdot (x\cdot z) = y\cdot z$

(b) $\forall x\geq y, z:\ x\cdot (y \wedge z) = x\cdot y \wedge x\cdot z$

(c) $(x \vee y)\cdot y = x\cdot (x \wedge y) $

(d) $\forall x\geq y:\ y\leq x\cdot y$ and $(x\cdot y)\cdot y = x$

This is equivalent to having the structure of an MV algebra (see Prop. 44 on page 34 of https://pnp.mathematik.uni-stuttgart.de/iaz/iaz1/Rump/32-35.pdf).

Such lattices are always distributive.

I would be grateful for any help.

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