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On Mode Reducts of Semimodules

The problem of embedding modes as subreducts into semimodules over commutative semirings raises the question of characterizing those semimodules that may contain modes as subreducts. We initiate a study of such semimodules by providing certain classes of semimodules with this property. First we characterize mode reducts of commutative monoids, showing that they are equivalent to certain ternary algebras coming from the regularizations of varieties of integral affine spaces, and to some n-ary semilattices. Then we generalize these results, characterizing mode reducts of semimodules over commutative unital rings and semirings obtained from them by adding a new zero element.