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It is demonstrated that there is antiferromagnetic spin-spin interaction between nearest electrons in the two-dimensional Wigner crystal. This is also valid for the Wigner liquid—the state with destroyed long-range order but preserved short-range order. We calculate the value of the antiferromagnetic interaction ~both analytically and numerically! and discuss a possible magnetic state of the Wigner crystal. This state can be strongly influenced by the spin-Peierls mechanism.