J. Aust. Math. Soc.  81 (2006), 63-96
Polyhedral convex cones and the equational theory of the bicyclic semigroup

F. Pastijn
  Department of Mathematics, Statistics and Computer Science
  Marquette University
  Milwaukee WI 53201-1881
  USA
  francisp@mscs.mu.edu


Abstract
To any given balanced semigroup identity $v \approx w$ a number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup
\[ BC= \langle a,b \mid a^2b=aba=a,\ ab^2=bab=b \rangle \]
or in the semigroup
\[E=\langle a,b \mid a^2b=a,\ ab^2=b \rangle. \]
The semigroups $BC$ and $E$ deserve our attention because a semigroup variety contains a simple semigroup which is not completely simple (respectively, which is idempotent free) if and only if this variety contains $BC$ (respectively, $E$). Therefore, for a given identity $v \approx w$ it is decidable whether or not the variety determined by $v \approx w$ contains a simple semigroup which is not completely simple (respectively, which is idempotent free).
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