Classical, quantum and statistical mechanics may all be seen as varieties of matrix mechanics for different rigs. In quantum mechanics we use linear algebra over the ring of complex numbers; in classical mechanics everything is formally the same, but we instead use the rig , where the addition is and the multiplication is .
Statistical mechanics (or better, ‘thermal statics’) is matrix mechanics over a rig that depends on the temperature? . As , the rig reduces to and thermal statics reduces to classical statics, just as quantum dynamics reduces to classical dynamics as Planck’s constant approaches zero.
I suppose matrix mechanics over a distributive lattice could be thought as a kind of minimax (or maximin) calculation - David.