Central European Journal of Mathematics
Semigroup, Monoid, Finitely based, Varieties, Hereditarily Finitely Based
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.
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Lee, Edmond W. H., "Finite Basis Problem for 2-Testable Monoids" (2011). Mathematics Faculty Articles. 2.