A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.
Two semigroups are said to be distinct if they are neither isomorphic nor anti-isomorphic. Although there exist 1373 distinct monoids of order six, only two are known to be non-finitely based. In the present dissertation, the finite basis property of the other 1371 distinct monoids of order six is verified. Since it is long established that all semigroups of order five or less are finitely based, the two known non-finitely based monoids of order six are the only examples of minimal order.
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