Equational theories of idempotent semifields

Abstract This paper provides answers to several open problems about equational theories of idempotent semifields. In particular, it is proved that (i) no equational theory of a non‐trivial class of idempotent semifields has a finite basis; (ii) there are continuum‐many equational theories of classes of idempotent semifields; and (iii) the equational theory of the class of idempotent semifields is co‐NP‐complete. This last result is also used to determine the complexity of deciding the existence of a right order on a free group or free monoid satisfying finitely many given inequalities.