  Ultradiscretization with negative values is a long-standing problem and
several attempts have been made to solve it. Among others, we focus on the
symmetrized max-plus algebra, with which we ultradiscretize the discrete
sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon
equation has already been proposed by previous studies, but the equation and
the solutions obtained here are considered to directly correspond to the
discrete counterpart. We also propose a noncommutative discrete analogue of the
sine-Gordon equation, reveal its relations to other integrable systems
including the noncommutative discrete KP equation, and construct multisoliton
solutions by a repeated application of Darboux transformations. Moreover, we
derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and
its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra.
As a result, we have a complete set of commutative and noncommutative versions
of continuous, discrete, and ultradiscrete sine-Gordon equations.
