1 ▁Tra ht man ▁published ▁a ▁proof ▁of ▁upper ▁bound ▁, ▁but ▁then ▁he ▁found ▁an ▁error ▁in ▁it . ▁The ▁conject ure ▁holds ▁in ▁many ▁partial ▁cases , ▁see ▁for ▁instance , ▁K ari ▁and ▁Tra ht man . ▁ ▁Other ▁work ▁The ▁finite ▁basis ▁problem ▁for ▁sem igr oups ▁of ▁order ▁less ▁than ▁six ▁in ▁the ▁theory ▁of ▁sem igr oups ▁was ▁pos ed ▁by ▁Alfred ▁Tar ski ▁in ▁ 1 9 6 6 , ▁and ▁repeated ▁by ▁An atol y ▁M alt se v ▁and ▁L . ▁N . ▁She vr in . ▁In ▁ 1 9 8 3 , ▁Tra ht man ▁solved ▁this ▁problem ▁by ▁prov ing ▁that ▁all ▁sem igr oups ▁of ▁order ▁less ▁than ▁six ▁are ▁finit ely ▁based . ▁ ▁In ▁the ▁theory ▁of ▁vari eties ▁of ▁sem igr oups ▁and ▁universal ▁al gebras ▁the ▁problem ▁of ▁existence ▁of ▁covering ▁elements ▁in ▁the ▁lattice ▁of ▁vari eties ▁ ▁was ▁pos ed ▁by ▁Evans ▁in ▁ 1 9 7 1 . ▁The ▁positive ▁solution ▁of ▁the ▁problem ▁was ▁found ▁by ▁Tra ht man . ▁He ▁also ▁found ▁a ▁six - element ▁sem igr oup ▁that ▁generates ▁a ▁variety ▁with ▁a ▁continu um ▁of ▁sub vari eties , ▁and ▁vari eties ▁of ▁sem igr oups ▁having ▁no ▁ir reducible ▁base ▁of ▁ident ities . ▁ ▁The ▁theory ▁of ▁locally ▁test able ▁autom ata ▁can ▁be ▁based ▁on ▁the ▁theory ▁of ▁vari eties ▁of ▁locally ▁test able ▁sem igr oups . ▁Tra ht man ▁found ▁the ▁precise ▁estimation