Equational theories of idempotent semifields

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.