The Study Of The Monoid P₂¹ Of Order Five

The existing results show that only by solving the hereditarily finitely based problem and the counting problem of subvarieties of the five order monoid P21=(a,b|a2b=a2,b2=ba=b>? {1},it can be determined the smallest degree of the smallest semigroup which generates the semigroup variety with continuum many subvarieties,with countably infinitely many subvarieties or without the property of hereditarily finitely based.Therefore,the hereditarily finitely based problem and the counting problem of subvarieties about P21 have been proposed many times by M.Jackson and E.W.H.Lee in many literatures.In this paper,we examine the hereditarily finitely based problem of P21.By analyzing identities satisfied by all its subvarieties,it is shown that P21 is hereditarily finitely based,and then the variety generated by P21 has countably infinitely many subvarieties,which completely answers the open question of M.Jackson and E.W.H.Lee.Hence,the smallest order of monoid without the hereditarily finitely based property is six;the smallest order of monoid which generates monoid varieties with countably infinitely many subvarieties is five;while the smallest order of monoid which generates monoid varieties with continuum many subvarieties is six.