Cube-like structures generated by filters (Q2496141)

It is well known that the class lattices of faces of \(n\)-cubes and the class of lattices of closed intervals of finite Boolean algebras are equal. This class was first studied axiomatically by N.\, Metropolis and G.-C.\, Rota (1978), using a bounded semilattice with the binary reflection operator \(\Delta \). A cubic algebra is a join-semilattice with a binary operation \(\Delta \) satisfying the axioms \qquad \(x\leq y\) implies \(\Delta (y,x)\vee x = y\) \qquad \(x\leq y\leq z\) implies \(\Delta (z,\Delta (y,x)) = \Delta (\Delta (z,y),\Delta (z,x))\) \qquad \(x\leq y\) implies \(\Delta (y,\Delta (y,x)) = x\) \qquad \(x\leq y\leq z\) implies \(\Delta (z,x)\leq \Delta (z,y)\) If \(xy = \Delta (1,\Delta (x\vee y, y))\vee y\) then \((xy)y = x\vee y\) and \(x(yz) = y(xz)\). An MR-algebra is a cubic algebra satisfying the axiom \(a,b<x\) implies \(\Delta (x,a)\vee b<x\) iff \(a\wedge b\) does not exist. In the paper the authors extended this study to filter algebras, a larger class of MR-algebras that includes all commutable MR-algebras. Using a naturally defined subclass of the class of maximal filters, the authors investigate the structure of the automorphism group of a cube. This class of filters admits a topology similar to Stone topology. They show that not every MR-algebra needs to be a filter algebra.